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#### Question 1:

Two opposite angles of a parallelogram are (3x − 2)° and (50 − x)°. Find the measure of each angle of the parallelogram.

It is given that the two opposite angles of a parallelogram are and .

We know that the opposite angles of a parallelogram are equal.

Therefore,

…… (i)

Thus, the given angles become

Also, .

Therefore the sum of consecutive interior angles must be supplementary.

That is;

Since opposite angles of a parallelogram are equal.

Therefore,

And

Hence the four angles of the parallelogram are , , and .

#### Question 2:

If an angle of a parallelogram is two-third of its adjacent angle, find the angles of the parallelogram.

Let one of the angle of the parallelogram as

We know that the sum of adjacent angles of the parallelogram is supplementary.

Therefore,

Since, opposite angles of a parallelogram are equal.

Therefore, the four angles in sequence are ,,and.

#### Question 3:

Find the measure of all the angles of a parallelogram, if one angle is 24° less than twice the smallest angle.

Let the smallest angle of the parallelogram be

Therefore, according to the given statement other angle becomes .

Also, the opposite angles of a parallelogram are equal.

Therefore, the four angles become ,,and.

According to the angle sum property of a quadrilateral:

Thus, the other angle becomes

Hence, the four angles of the parallelogram are , , and .

#### Question 4:

The perimeter of a parallelogram is 22 cm. If the longer side measures 6.5 cm what is the measure of the shorter side?

Let the shorter side of the parallelogram be cm.

The longer side is given ascm.

Perimeter of the parallelogram is given as 22 cm

Therefore,

Hence, the measure of the shorter side is cm.

#### Question 5:

In a parallelogram ABCD, ∠D  = 135°, determine the measures of ∠A and ∠B.

It is given that ABCD is a parallelogram with

We know that the opposite angles of the parallelogram are equal.

Therefore,

Also, and are adjacent angles, which must be supplementary.

Therefore,

Hence , and .

#### Question 6:

ABCD is a parallelogram in which ∠A = 70°. Compute ∠B, ∠C and ∠D.

It is given that ABCD is a parallelogram with

We know that the opposite angles of the parallelogram are equal.

Therefore,

Also, and are adjacent angles, which must be supplementary.

Therefore,

Also, and are opposite angles of a parallelogram.

Therefore,

Hence, the angles of a parallelogram are , , and .

#### Question 7:

In the given figure, ABCD is a parallelogram in which ∠A = 60°. If the bisectors of ∠A and ∠B meet at P, prove that AD = DP, PC = BC and DC = 2AD.

The figure is given as follows:

It is given that ABCD is a parallelogram.

Thus,

Opposite angles of a parallelogram are equal.

Therefore,

Also, we have AP as the bisector of

Therefore,

…… (i)

Similarly,

…… (ii)

We have ,

From (i)

Thus, sides opposite to equal angles are equal.

Similarly,

From (ii)

Thus, sides opposite to equal angles are equal.

Also,

#### Question 8:

In the given figure, ABCD is a parallelogram in which ∠DAB = 75° and ∠DBC = 60°. Compute ∠CDB and ∠ADB.

The figure is given as follows:

It is given that ABCD is a parallelogram.

Thus

And are alternate interior opposite angles.

Therefore,

…… (i)

We know that the opposite angles of a parallelogram are equal. Therefore,

Also, we have

Therefore,

…… (ii)

In

By angle sum property of a triangle.

From (i) and (ii),we get:

Hence, the required value for is

And is .

#### Question 9:

In the given figure, ABCD is a parallelogram and E is the mid-point of side BC. If DE and AB when produced meet at F, prove that AF = 2AB.

Figure is given as follows:

It is given that ABCD is a parallelogram.

DE and AB when produced meet at F.

We need to prove that

It is given that

Thus, the alternate interior opposite angles must be equal.

In and , we have

(Proved above)

(Given)

(Vertically opposite angles)

Therefore,

(By ASA Congruency )

By corresponding parts of congruent triangles property, we get

DC = BF …… (i)

It is given that ABCD is a parallelogram. Thus, the opposite sides should be equal. Therefore,

…… (ii)

But,

From (i), we get:

From (ii), we get:

Hence proved.

#### Question 10:

Which of the following statements are true (T) and which are false (F)?

(i) In a parallelogram, the diagonals are equal.

(ii) In a parallelogram, the diagonals bisect each other.

(iii) In a parallelogram, the diagonals intersect each other at right angles.

(iv) In any quadrilateral, if a pair of opposite sides is equal, it is a parallelogram.

(v) If all the angles of a quadrilateral are equal, it is a parallelogram.

(vi) If three sides of a quadrilateral are equal, it is a parallelogram.

(vii) If three angles of a quadrilateral are equal, it is a parallelogram.

(viii) If all the sides of a quadrilateral are equal it is a parallelogram.

(i) Statement: In a parallelogram, the diagonals are equal.

False

(ii) Statement: In a parallelogram, the diagonals bisect each other.

True

(iii) Statement: In a parallelogram, the diagonals intersect each other at right angles.

False

(iv) Statement: In any quadrilateral, if a pair of opposite sides is equal, it is a parallelogram.

False

(v) Statement: If all the angles of a quadrilateral are equal, then it is a parallelogram.

True

(vi) Statement: If three sides of a quadrilateral are equal, then it is not necessarily a parallelogram.

False

(vii) Statement: If three angles of a quadrilateral are equal, then it is no necessarily a parallelogram.

False

(viii) Statement: If all sides of a quadrilateral are equal, then it is a parallelogram.

True

#### Question 1:

In a parallelogram ABCD, determine the sum of angles ∠C and ∠D.

The parallelogram can be drawn as:

We have ,thus and are consecutive interior angles.

These must be supplementary.

Therefore,

#### Question 2:

In a parallelogram ABCD, if ∠B = 135°, determine the measure of its other angles.

Since ABCD is a parallelogram with .

Opposite angles of a parallelogram are equal.

Therefore,

Also, let

Similarly,

We know that the sum of the angles of a quadrilateral is .

Hence the measure of other angles are , and .

#### Question 3:

ABCD is a square. AC and BD intersect at O. State the measure of ∠AOB.

The figure can be drawn as follows:

In and,

(Sides of a square are equal)

(Diagonals of a parallelogram bisect each other)

(Common)

So, by SSS Congruence rule, we have

Also,

(Corresponding parts of congruent triangles are equal)

But, (Linear pairs)

We have,

Hence, the required measure of is .

#### Question 4:

ABCD is a rectangle with ∠ABD = 40°. Determine ∠DBC.

The rectangle is given as follows with

We have to find .

An angle of a rectangle is equal to .

Therefore,

Hence, the measure for is .

#### Question 5:

The sides AB and CD of a parallelogram ABCD are bisected at E and F. Prove that EBFD is a parallelogram.

Figure is given as follows:

It is given that ABCD is a parallelogram.

E is the mid point of AB

Thus,

,

...... (i)

Similarly,

……(ii)

From (i) and (ii)

Also,

Thus,

Therefore, EBFD is a parallelogram.

#### Question 6:

P and Q are the points of trisection of the diagonal BD of a parallelogram ABCD. Prove that CQ is parallel to AP. Prove also that AC bisects PQ.

Figure can be drawn as follows:

We have P and Q as the points of trisection of the diagonal BD of parallelogram ABCD.

We need to prove that AC bisects PQ. That is, .

Since diagonals of a parallelogram bisect each other.

Therefore, we get:

and

P and Q as the points of trisection of the diagonal BD.

Therefore,

and

Now, and

Thus,

AC bisects PQ.

Hence proved.

#### Question 7:

ABCD is a square E, F, G and H are points on AB, BC, CD and DA respectively. such that AE = BF = CG = DH. Prove that EFGH is a square.

Square ABCD is given:

E, F, G and H are the points on AB, BC, CD and DA respectively, such that :

We need to prove that EFGH is a square.

Say,

As sides of a square are equal. Then, we can also say that:

In and ,we have:

(Given)

(Each equal to 90°)

(Each equal to y )

By SAS Congruence criteria, we have:

Therefore, EH = EF

Similarly, EF= FG, FG= HG and HG= HE

Thus, HE=EF=FG=HG

Also,

and

But,

and

Therefore,

i.e.,

Similarly,

Thus, EFGH is a square.

Hence proved.

#### Question 8:

ABCD is a rhombus, EABF is a straight line such that EA = AB = BF. Prove that ED and FC when produced meet at right angles.

Rhombus ABCD is given:

We have

We need to prove that

We know that the diagonals of a rhombus bisect each other at right angle.

Therefore,

,,

In A and O are the mid-points of BE and BD respectively.

By using mid-point theorem, we get:

Therefore,

In  A and O are the mid-points of BE and BD respectively.

By using mid-point theorem, we get:

Therefore,

and

Therefore, DOCG is a parallelogram.

Thus, opposite angles of a parallelogram should be equal.

Also, it is given that

Therefore,

Or,

Hence proved.

#### Question 9:

ABCD is a parallelogram, AD is produced to E so that DE = DC and EC produced meets AB produced in F. Prove that BF = BC

ABCD is a parallelogram, AD produced to E such that .

Also , AB produced to F.

We need to prove that

In , D and O are the mid-points of AE and AC respectively.

By using Mid-point Theorem, we get:

Since, BD is a straight line and O lies on AC.

And, C lies on EF

Therefore,

…… (i)

Also, is a parallelogram with .

Thus,

In and ,we have:

So, by ASA Congruence criterion, we have:

By corresponding parts of congruence triangles property, we get:

From (i) equation, we get:

Hence proved.

#### Question 1:

Three angles of a quadrilateral are respectively equal to 110°, 50° and 40°. Find its fourth angles.

Let the measure of the fourth angles be x°. We know that the sum of the angles of a quadrilateral is 360°.

Therefore,

Hence the measure of the fourth angle is .

#### Question 2:

In a quadrilateral ABCD, the angles A, B, C and D are in the ratio 1 : 2 : 4 : 5. Find the measure of each angles of the quadrilateral.

We have ,.

So, let ,

,

And

By angle sum property of a quadrilateral, we get:

Also,

And

Similarly,

Hence, the four angles are ,,and .

#### Question 3:

In a quadrilateal ABCD, CO and DO are the bisectors of ∠C and ∠D respectively. Prove that ∠COD =$\frac{1}{2}$ (∠A + ∠B).

The quadrilateral can be drawn as follows:

We have CO and DO as the bisectors of angles and respectively.

We need to prove that.

In ,We have,

…… (I)

By angle sum property of a quadrilateral, we have:

Putting in equation (I):

Hence proved.

#### Question 4:

The angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral.

We have, .

So, let ,

,

and

By angle sum property of a quadrilateral, we get:

Also,

And

Similarly,

Hence, the four angles are , , and  .

#### Question 1:

In a ΔABC, D, E and F are, respectively, the mid-points of BC, CA and AB. If the lengths of side AB, BC and CA are 7 cm, 8 cm and 9 cm, respectively, find the perimeter of ΔDEF.

is given with D,E and F as the mid-points of BC , CA and AB respectively as shown below:

Also, , and .

We need to find the perimeter of

In , E and F are the mid-points of CA and AB respectively.

Theorem states, the line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.

Therefore, we get:

Similarly, we get

And

Perimeter of

Hence, the perimeter of is .

#### Question 2:

In a triangle ∠ABC, ∠A = 50°, ∠B = 60° and C =∠70°. Find the measures of the angles of the triangle formed by joining the mid-points of the sides of this triangle.

It is given that D, E and F be the mid-points of BC , CA and AB respectively.

Then,

, and .

Now, and transversal CB and CA intersect them at D and E respectively.

Therefore,

[(Given)]

and

[(Given)]

Similarly,

Therefore,

[(Given)]

and

[(Given)]

Similarly,

Therefore,

[(Given)] and

[(Given)]

Now BC is a straight line.

Similarly,

and

Hence the measure of angles are , and.

#### Question 3:

In a triangle, P, Q and R are the mid-points of sides BC, CA and AB respectively. If AC = 21 cm, BC = 29 cm and AB = 30 cm, find the perimeter of the quadrilateral ARPQ.

It is given that P, Q and R are the mid-points of BC, CA and AB respectively.

Also, we have , and

We need to find the perimeter of quadrilateral ARPQ

In , P and R are the mid-points of CB and AB respectively.

Theorem states, the line segment joining the mid-points of any two sides of a traingle is parallel to the third side and equal to half of it.

Therefore, we get:

Similarly, we get

We have Q and R as the mid points of AC and AB respectively.

Therefore,

And

Perimeter of

Hence, the perimeter of quadrilateral ARPQ is.

#### Question 4:

In a ΔABC median AD is produced to X such that AD = DX. Prove that ABXC is a parallelogram.

is given with AD as the median extended to point X such that .

Join BX and CX.

We get a quadrilateral ABXC, we need to prove that it’s a parallelogram.

We know that AD is the median.

By definition of median we get:

Also, it is given that

Thus, the diagonals of the quadrilateral ABCX bisect each other.

Therefore, quadrilateral ABXC is a parallelogram.

Hence proved.

#### Question 5:

In a ABC, E and F are the mid-points of AC and AB respectively. The altitude AP to BC intersects FE at Q. Prove that AQ = QP.

is given with E and F as the mid points of sides AB and AC.

Also, intersecting EF at Q.

We need to prove that

In , E and F are the mid-points of AB and AC respectively.

Theorem states, the line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.

Therefore, we get:

Since, Q lies on EF.

Therefore,

This means,

Q is the mid-point of AP.

Thus, (Because, F is the mid point of AC and)

Hence proved.

#### Question 6:

In a ΔABC, BM and CN are perpendiculars from B and C respectively on any line passing through A. If L is the mid-point of BC, prove that ML = NL.

In , BM and CN are perpendiculars on any line passing through A.

Also.

We need to prove that

From point L let us draw

It is given that , and

Therefore,

Since, L is the mid points of BC,

Therefore intercepts made by these parallel lines on MN will also be equal

Thus,

Now in ,

And . Thus, perpendicular bisects the opposite sides.

Therefore, is isosceles.

Hence

Hence proved.

#### Question 7:

In the given figure, triangle ABC is right-angled at B. Given that AB = 9 cm, AC = 15 cm and D, E are the mid-points of the sides AB and AC respectively, calculate

(i) The length of BC

We have right angled at B.

It is given that and

D and E are the mid-points of sides AB and AC respectively.

(i) We need to calculate length of BC.

In right angled at B:

By Pythagoras theorem,

Hence the length of BC is .

(ii) We need to calculate area of .

In right angled at B, D and E are the mid-points of AB and AC respectively.

Theorem states, the line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.

Therefore, .

Thus, (Corresponding angles of parallel lines are equal)

And

area of

D is the mid-point of side AB .

Therefore, area of

Hence the area of is .

#### Question 8:

In the given figure, M, N and P are the mid-points of AB, AC and BC respectively. If MN = 3 cm, NP = 3.5 cm and MP = 2.5 cm, calculate BC, AB and AC.

We have as follows:

M, N and P are the mid-points of sides AB ,AC and BC respectively.

Also, , and

We need to calculate BC, AB and AC.

In , M and N are the mid-points of AB and AC respectively.

Theorem states, the line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.

Therefore,

Similarly,

And

Hence, the measure for BC, AB and AC is , and respectively.

#### Question 9:

ABC is a triangle and through A, B, C lines are drawn parallel to BC, CA and AB respectively intersecting at P, Q and R. Prove that the perimeter of ΔPQR is double the perimeter of ΔABC.

We have as follows:

Through A,B and C lines are drawn parallel to BC,CA and AB respectively intersecting at P,Q and R respectively.

We need to prove that perimeter of is double the perimeter of .

and

Therefore, is a parallelogram.

Thus,

Similarly,

is a parallelogram.

Thus,

Therefore,

Then, we can say that A is the mid-point of QR.

Similarly, we can say that B and C are the mid-point of PR and PQ respectively.

In ,

Theorem states, the line drawn through the mid-point of any one side of a triangle is parallel to the another side, intersects the third side at its mid-point.

Therefore,

Similarly,

Perimeter of is double the perimeter of

Hence proved.

#### Question 10:

In the given figure, BEAC. AD is any line from A to BC intersecting BE in H. P, Q and R are respectively the mid-points of AH, AB and BC. Prove that ∠PQR = 90°

is given with

AD is any line from A to BC intersecting BE in H.

P,Q and R respectively are the mid-points of AH,AB and BC.

We need to prove that

Let us extend QP to meet AC at M.

In , R and Q are the mid-points of BC and AB respectively.

Theorem states, the line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.

Therefore, we get:

…… (i)

Similarly, in,

…… (ii)

From (i) and (ii),we get:

and

We get, is a parallelogram.

Also,

Therefore, is a rectangle.

Thus,

Or,

Hence proved.

#### Question 11:

In the given figure, AB = AC and CP || BA and AP is the bisector of exterior ∠CAD of ΔABC. Prove that (i) ∠PAC = ∠BCA (ii) ABCP is a parallelogram.

We have the following given figure:

We have and and AP is the bisector of exterior angle of .

(i) We need to prove that

In ,

We have (Given)

Thus, (Angles opposite to equal sides are equal)

By angle sum property of a triangle, we get:

…… (i)

Now,

(AP is the bisector of exterior angle )

(Linear Pair)

…… (ii)

From equation (i) and (ii),we get:

(ii) We need to prove that is a parallelogram.

We have proved that

This means,

Also it is given that

We know that a quadrilateral with opposite sides parallel is a parallelogram.

Therefore, is a parallelogram.

#### Question 12:

ABCD is a kite having AB AD and BC = CD. Prove that the figure formed by joining the mid-points of the sides, in order, is a rectangle.

ABCD is a kite such that and

Quadrilateral PQRS is formed by joining the mid-points P,Q,R and S of sides AB,BC,CD and AD respectively.

We need to prove that Quadrilateral PQRS is a rectangle.

In , P and Q are the mid-points of AB and BC respectively.

Therefore,

and

Similarly, we have

and

Thus,

and

Therefore, PQRS is a parallelogram.

Now,

But, P and S are the mid-points of AB and AD

…… (I)

In : P and S are the mid-point of side AB and AD

By mid-point Theorem, we get:

Or,

In , P is the mid-point of side AB and

By Using the converse of mid-point theorem, we get:

M is the mid-point of AO

Thus,

…… (II)

In and , we have:

(Common)

[From (I)]

[From (II)]

By SSS Congruence theorem, we get:

By corresponding parts of congruent triangles property, we get:

But,

and

Therefore,

(, Corresponding angles should be equal)

Or,

We have proved that

Similarly, .

Then we can say that and

Therefore, is a parallelogram with

Or, we can say that is a rectangle.

We get:

Also, PQRS is a parallelogram.

Therefore, PQRS is a rectangle.

Hence proved.

#### Question 13:

Let ABC be an isosceles triangle in which AB = AC. If D, E, F be the mid-points of the sides BC, CA and AB respectively, show that the segment AD and EF bisect each other at right angles.

, an isosceles triangle is given with D,E and F as the mid-points of BC, CA and AB respectively as shown below:

We need to prove that the segment AD and EF bisect each other at right angle.

Let’s join DF and DE.

In , D and E are the mid-points of BC and AC respectively.

Theorem states, the line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.

Therefore, we get: Or

Similarly, we can get

Therefore, AEDF is a parallelogram

We know that opposite sides of a parallelogram are equal.

and

Also, from the theorem above we get

Thus,

Similarly,

It is given that , an isosceles triangle

Thus,

Therefore,

Also,

Then, AEDF is a rhombus.

We know that the diagonals of a rhombus bisect each other at right angle.

Therefore, M is the mid-point of EF and

Hence proved.

#### Question 14:

ABC is a triangle, D is a point on AB such that AD = $\frac{1}{4}$ AB and E is a point on AC such that AE = $\frac{1}{4}$ AC. Prove that DE = $\frac{1}{4}$ BC.

is given with D a point on AB such that .

Also, E is point on AC such that.

We need to prove that

Let P and Q be the mid points of AB and AC respectively.

It is given that

and

But, we have taken P and Q as the mid points of AB and AC respectively.

Therefore, D and E are the mid-points of AP and AQ respectively.

In , P and Q are the mid-points of AB and AC respectively.

Theorem states, the line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.

Therefore, we get and …… (i)

In , D and E are the mid-points of AP and AQ respectively.

Therefore, we get and …… (ii)

From (i) and (ii),we get:

Hence proved.

#### Question 15:

In the given figure, ABCD is a parallelogram in which P is the mid-point of DC and Q is a point on AC such that CQ = $\frac{1}{4}$ AC. If PQ produced meets BC at R, prove that R is a mid-point of BC.

Figure is given as follows:

ABCD is a parallelogram, where P is the mid-point of DC and Q is a point on AC such that

.

PQ produced meets BC at R.

We need to prove that R is a mid-point of BC.

Let us join BD to meet AC at O.

It is given that ABCD is a parallelogram.

Therefore, (Because diagonals of a parallelogram bisect each other)

Also,

Therefore,

In , P and Q are the mid-points of CD and OC respectively.

Theorem states, the line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.

Therefore, we get:

Also, in , Q is the mid-point of OC and

Therefore, R is a mid-point of BC.

Hence proved.

#### Question 16:

In the given figure, ABCD and PQRC are rectangles and Q is the mid-point of AC. Prove that

(i) DP = PC

(ii) PR = $\frac{1}{2}$ AC

Rectangles ABCD and PQRC are given as follows:

Q is the mid-point of AC.

In , Q is the mid-point of AC such that

Using the converse of mid-point theorem, we get:

P is the mid-point of DC

That is;

Similarly, R is the mid-point of BC.

Now, in , P and R are the mid-points of DC and BC respectively.

Then, by mid-point theorem, we get:

Now, diagonals of a rectangle are equal.

Therefore putting ,we get:

Hence Proved.

#### Question 17:

ABCD is a parallelogram, E and F are the mid-points of AB and CD respectively. GH is any line intersecting AD, EF and BC at G, P and H respectively. Prove that GPPH.

ABCD is a parallelogram with E and F as the mid-points of AB and CD respectively.

We need to prove that

Since E and F are the mid-points of AB and CD respectively.

Therefore,

,

And

,

Also, ABCD is a parallelogram. Therefore, the opposite sides should be equal.

Thus,

Also, (Because )

Therefore, BEFC is a parallelogram

Then, and …… (i)

Now,

Thus, (Because as ABCD is a parallelogram)

We get,

AEFD is a parallelogram

Then, we get:

…… (ii)

But, E is the mid-point of AB.

Therefore,

Using (i) and (ii), we get:

Hence proved.

#### Question 18:

RM and CN are perpendiculars to a line passing through the vertex A of a triangle ABC. IF L is the mid-point of BC, prove that LM = LN.

In BM and CN are perpendiculars on any line passing through A.

Also.

We need to prove that

From point L let us draw

It is given that , and

Therefore,

Since, L is the mid points of BC,

Therefore intercepts made by these parallel lines on MN will also be equal

Thus,

Now in ,

And . Thus, perpendicular bisects the opposite sides.

Therefore, is isosceles.

Hence

Hence proved.

#### Question 19:

Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

Figure can be drawn as:

Let ABCD be a quadrilateral such that P,Q ,R and S are the mid-points of side AB,BC,CD and DA respectively.

In , P and Q are the mid-points of AB and BC respectively.

Therefore,

and

Similarly, we have

and

Thus,

and

Therefore, PQRS is a parallelogram.

Since, diagonals of a parallelogram bisect each other.

Therefore, PR and QS bisect each other.

Hence proved.

#### Question 20:

Fill in the blanks to make the following statements correct:

(i) The triangle formed by joining the mid-points of the sides of an isosceles triangle is........

(ii) The triangle formed by joining the mid-points of the sides of a right triangle is ........

(iii) The figure formed by joining the mid-points of consecutive sides of a quadrilateral is ..........

(i) The triangle formed by joining the mid-points of the sides of an isosceles triangle is isosceles.

Explanation:

Figure can be drawn as: A

, an isosceles triangle is given.

F and E are the mid-points of AB and AC respectively.

Therefore,

…… (I)

Similarly,

…… (II)

And

…… (III)

Now, is an isosceles triangle.

From equation (II) and (III), we get:

Therefore, in two sides are equal.

Therefore, it is an isosceles triangle.

(ii) The triangle formed by joining the mid-points of the sides of a right triangle is right triangle.

Explanation:

Figure can be drawn as: A

right angle at B is given.

F and E are the mid-points of AB and AC respectively.

Therefore,

…… (I)

Similarly,

…… (II)

And

…… (III)

Now, DE || AB and transversal CB and CA intersect them at D and E respectively.

Therefore,

and

Similarly,

Therefore,

and

Similarly,

Therefore,

Now AC is a straight line.

Now, by angle sum property of ,we get:

Therefore,

But,

Then we have:

(iii) The figure formed by joining the mid-points of the consecutive sides of a quadrilateral is parallelogram.

Explanation:

Figure can be drawn as:

Let ABCD be a quadrilateral such that P, Q, R and S are the mid-points of side AB, BC, CD and DA respectively.

In , P and Q are the mid-points of AB and BC respectively.

Therefore,

and

Similarly, we have

and

Thus,

and

Therefore, PQRS is a parallelogram.

#### Question 1:

In a parallelogram ABCD, write the sum of angles A and B.

In Parallelogram ABCD, and are adjacent angles.

Thus, .

Then, we have and as consecutive interior angles which must be supplementary.

Hence, the sum of and is .

#### Question 2:

In a parallelogram ABCD, if ∠D = 115°, then write the measure of ∠A.

In Parallelogram ABCD , and are Adjacent angles.

We know that in a parallelogram, adjacent angles are supplementary.

#### Question 3:

PQRS is a square such that PR and SQ intersect at O. State the measure of ∠POQ.

PQRS is a square given as:

Since the diagonals of a square intersect at right angle.

Therefore, the measure of is .

#### Question 4:

In a quadrilateral ABCD, bisectors of angles A and B intersect at O such that ∠AOB = 75°, then write the value of ∠C + ∠D.

The quadrilateral can be drawn as follows:

We have AO and BO as the bisectors of angles and respectively.

In ,We have,

…… (1)

By angle sum property of a quadrilateral, we have:

Putting in equation (1):

……(2)

It is given that in equation (2), we get:

Hence, the sum of and is .

#### Question 5:

The diagonals of a rectangle ABCD meet at O, If ∠BOC = 44°, find ∠OAD.

The rectangle ABCD is given as:

We have,

(Linear pair)

Since, diagonals of a rectangle are equal and they bisect each other. Therefore, in , we have

(Angles opposite to equal sides are equal.)

Therefore,

Now,in , we have

Since, each angle of a rectangle is a right angle.

Therefore,

Thus,

Hence, the measure of is.

#### Question 6:

If PQRS is a square, then write the measure of ∠SRP.

The square PQRS is given as:

Since PQRS is a square.

Therefore,

and

Now, in , we have

That is, (Angles opposite to equal sides are equal)

By angle sum property of a triangle.

()

Hence, the measure of is.

#### Question 7:

If ABCD is a rectangle with ∠BAC = 32°, find the measure of ∠DBC.

Figure is given as :

Suppose the diagonals AC and BD intersect at O.

Since, diagonals of a rectangle are equal and they bisect each other.

Therefore, in , we have

Angles opposite to equal sides are equal.

Therefore,

But,

Now,

Hence, the measure of is .

#### Question 8:

If ABCD is a rhombus with ∠ABC = 56°, find the measure of ∠ACD.

The figure is given as follows:

ABCD is a rhombus.

Therefore,

ABCD is a parallelogram.

Thus,

[(Given)]

[]

Now in ,we have:

Hence the measure of is .

#### Question 9:

The perimeter of a parallelogram is 22 cm. If the longer side measures 6.5 cm, what is the measure of shorter side?

Let the shorter side of the parallelogram be cm.

The longer side is given ascm.

Perimeter of the parallelogram is given as 22 cm

Therefore,

Hence, the measure of the shorter side is cm.

#### Question 10:

If the angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13, then find the measure of the smallest angle.

We have, .

So, let ,

,

and

By angle sum property of a quadrilateral, we get:

Smallest angle is :

Hence, the smallest angle measures.

#### Question 11:

If the bisectors of two adjacent angles A and B of a quadrilateral ABCD intersect at a point O such that ∠C + ∠D = kAOB, then find the value of k.

The quadrilateral can be drawn as follows:

We have AO and BO as the bisectors of angles and respectively.

In ,We have,

…… (I)

By angle sum property of a quadrilateral, we have:

Putting in equation (I):

…… (II)

On comparing equation (II) with

We get .

Hence, the value for k is .

#### Question 12:

In a parallelogram ABCD, if ∠A = (3x − 20)°, ∠B = (y + 15)°, ∠C = (x + 40)°, then find the values of x and y.

In parallelogram ABCD, and are opposite angles.

We know that in a parallelogram, the opposite angles are equal.

Therefore,

We have and

Therefore,

Therefore,

Similarly,

Also,

Therefore,

By angle sum property of a quadrilateral, we have:

Hence the required values for x and y are and respectively.

#### Question 13:

If measures opposite angles of a parallelogram are (60 − x)° and (3x − 4)°, then find the measures of angles of the parallelogram.

Let ABCD be a parallelogram, with and .

We know that in a parallelogram, the opposite angles are equal.

Therefore,

Thus, the given angles become

Similarly,

Also, adjacent angles in a parallelogram form the consecutive interior angles of parallel lines, which must be supplementary.

Therefore,

Similarly,

Thus, the angles of a parallelogram are ,, and .

#### Question 14:

In a parallelogram ABCD, the bisector of ∠A also bisects BC at X. Find AB : AD.

Parallelogram ABCD is given as follows:

We have AX bisects bisecting BC at X.

That is,

We need to find

Since, AX is the bisector

That is,

…… (i)

Also, ABCD is a parallelogram

Therefore, and AB intersects them

…… (ii)

In by angle sum property of a triangle:

From (i) and (ii), we get:

…… (iii)

From (i) and (iii),we get:

Sides opposite to equal angles are equal. Therefore,

As X is the mid point of BC. Therefore,

Also, ABCD is a parallelogram, then,

Thus,

Hence the ratio of is .

#### Question 15:

In the given figure, PQRS is an isosceles trapezium. Find x and y.

Trapezium is given as follows:

We know that PQRS is a trapezium with

Therefore,

Hence, the required value for x is .

#### Question 16:

In the given figure, ABCD is a trapezium. Find the values of x and y.

The figure is given as follows:

We know that ABCD is a trapezium with

Therefore,

It is given that and .

Similarly,

Hence, the required values for x and y is and respectively.

#### Question 17:

In the given figure, PQRS is a rhombus in which the diagonal PR is produced to T. If ∠SRT = 152°, find x, y and z.

Rhombus PQRS is given.

Diagonal PR is produced to T.

Also, .

We know that in a rhombus, the diagonals bisect each other at right angle.

Therefore,

Now,

In , by angle sum property of a triangle, we get:

Or, (Because O lies on SQ)

We have, .Thus the alternate interior opposite angles must be equal.

Therefore,

In ,we have

Since opposite sides of a rhombus are equal.

Therefore,

Also,

Angles opposite to equal sides are equal.

Thus,

But

Thus,

Hence the required values for x,y and z are , and respectively.

#### Question 18:

In the given figure, ABCD is a rectangle in which diagonal AC is produced to E. If ∠ECD = 146°, find ∠AOB.

ABCD is a rectangle

With diagonal AC produced to point E.

We have

(Linear pair)

We know that the diagonals of a parallelogram bisect each other.

Thus

Also, angles opposite to equal sides are equal.

Therefore,

By angle sum property of a traingle

Also, and are vertically opposite angles.

Therefore,

Hence, the required measure for is .

#### Question 19:

In the given figure, ABCD and AEFG are two parallelograms. If ∠C = 58°, find ∠F.

ABCD and AEFG are two parallelograms as shown below:

Since ABCD is a parallelogram, with

We know that the opposite angles of a parallelogram are equal.

Therefore,

Similarly, AEFG is a parallelogram, with

We know that the opposite angles of a parallelogram are equal.

Therefore,

Hence, the required measure for is .

#### Question 20:

Complete each of the following statements by means of one of those given in brackets against each:
(i) If one pair of opposite sides are equal and parallel, then the figure is ........................
(parallelogram, rectangle, trapezium)

(ii) If in a quadrilateral only one pair of opposite sides are parallel, the quadrilateral is ................ (square, rectangle, trapezium)

(iii) A line drawn from the mid-point of one side of a triangle .............. another side intersects the third side at its mid-point. (perpendicular to parallel to, to meet)

(iv) If one angle of a parallelogram is a right angle, then it is necessarily a .................
(rectangle, square, rhombus)

(v) Consecutive angles of a parallelogram are ...................
(supplementary, complementary)

(vi) If both pairs of opposite sides of a quadrilateral are equal, then it is necessarily a ...............
(rectangle, parallelogram, rhombus)

(vii) If opposite angles of a quadrilateral are equal, then it is necessarily a ....................
(parallelogram, rhombus, rectangle)

(viii) If consecutive sides of a parallelogram are equal, then it is necessarily a ..................
(kite, rhombus, square)

(i) If one pair of opposite sides are equal and parallel, then the figure is parallelogram.

Reason:

In and,

(Given)

(Common)

(Because , Alternate interior angles are equal)

So, by SAS Congruence rule, we have

Also,

(Corresponding parts of congruent triangles are equal)

But, these are alternate interior angles, which are equal.

Thus, and

(ii) If in a quadrilateral only one pair of opposite sides are parallel, the quadrilateral is trapezium.

(iii) A line drawn from the mid-point of one side of a triangle parallel to another side intersects the third side at its mid-point.

Reason:

This is a theorem.

(iv) If one angle of a parallelogram is a right angle, then it is necessarily a rectangle.

Reason:

Let ABCD be the given parallelogram.

We have,

In a parallelogram, opposite angles are equal.

Therefore,

Similarly,

Also,

Thus, a parallelogram with all the angles being right angle and opposite sides being equal is a rectangle.

(v) Consecutive angles of a parallelogram are supplementary.

Reason:

Let ABCD be the given parallelogram.

Thus, .

Therefore,

Consecutive angles and are supplementary.

(vi) If both pairs of opposite sides of a quadrilateral are equal, then it is necessarily a parallelogram.

Reason:

ABCD is a quadrilateral in which and .

We need to show that ABCD is a parallelogram.

In and , we have

(Common)

(Given)

(Given)

So, by SSS criterion of congruence, we have

By corresponding parts of congruent triangles property.

…… (i)

And

Now lines AC intersects AB and DC at A and C,such that

(From (i))

That is, alternate interior angles are equal.

Therefore, .

Similarly, .

Therefore, ABCD is a parallelogram.

(vii) If opposite angles of a quadrilateral are equal, then it is necessarily a parallelogram.

Reason:

ABCD is a quadrilateral in which and .

We need to show that ABCD is a parallelogram.

Therefore,

…… (i)

Since sum of angles of a quadrilateral is

From equation (i), we get:

Similarly,

Now, line AB intersects AD and BC at A and B respectively

Such that

That is, sum of consecutive interior angles is supplementary.

Therefore, .

Similarly, we get .

Therefore, ABCD is a parallelogram.

(viii) If consecutive sides of a parallelogram are equal, then it is necessarily a rhombus.

We have ABCD, a parallelogram with .

Since ABCD is a parallelogram.

Thus,

And

But,

Therefore,all four sides of the parallelogram are equal, then it is a rhombus.

#### Question 1:

Mark the correct alternative in each of the following:
The opposite sides of a quadrilateral have

(a) no common point

(b) one common point

(c) two common points

(d) infinitely many common points

We can look at a quadrilateral as:

The opposite sides of the above quadrilateral AB and CD have no point in common.

Hence the correct choice is (a).

#### Question 2:

The consecutive sides of a quadrilateral have

(a) no common point

(b) one common point

(c) two common points

(d) infinitely many common points

We can look at a quadrilateral as:

The consecutive sides of the above quadrilateral AB and BC have one point in common.

Hence the correct choice is (b).

#### Question 3:

PQRS is a quadrilateral, PR and QS intersect each other at O. In which of the following cases, PQRS is a parallelogram?

(a) ∠P = 100°, ∠Q = 80°, ∠R = 95°

(b)  ∠P =85°, ∠Q = 85°, ∠R = 95°

(c) PQ = 7 cm, QR = 7 cm, RS = 8 cm, SP = 8 cm

(d) OP = 6.5 cm, OQ = 6.5 cm, OR = 5.2 cm, OS = 5.2 cm

Let us analyze each case one by one.

We have a quadrilateral named PQRS, with diagonals PR and QS intersecting at O.

(a) ,,

By angle sum property of a quadrilateral, we get:

Clearly,

And

Thus we have PQRS a quadrilateral with opposite angles are equal.

Therefore,

PQRS is a parallelogram.

(b) ,,

By angle sum property of a quadrilateral, we get:

Clearly,

And

Thus we have PQRS a quadrilateral with opposite angles are not equal.

Therefore,

PQRS is not a parallelogram.

(c) ,,,

Clearly,

And

Thus we have PQRS a quadrilateral with opposite sides are not equal

Therefore,

PQRS is not a parallelogram.

(c) ,,,

We know that the diagonals of a parallelogram bisect each other.

But, here we have

And

Therefore,

PQRS is not a parallelogram

Hence, the correct choice is (a).

#### Question 4:

Which  of the following quadrilateral is not a rhombus?

(a) All four sides are equal

(b) Diagonals bisect each other

(c) Diagonals bisect opposite angles

(d) One angle between the diagonals is 60°

Let us consider the rhombus ABCD as:

We have the following properties of a rhombus:

All four sides are equal.

Diagonals bisect each other at right angles.

Hence the correct choice is (d).

#### Question 5:

Diagonals necessarily bisect opposite angles in a

(a) rectangle

(b) parallelogram

(c) isosceles trapezium

(d) square

From the given choices, only in a square the diagonals bisect the opposite angles.

Let us prove it.

Take the following square ABCD with diagonal AD.

In and:

(Opposite sides of a square are equal.)

(Common)

(Opposite sides of a square are equal.)

Thus,

(By SSS Congruence Rule)

By Corresponding parts of congruent triangles property we have:

Therefore, in a square the diagonals bisect the opposite angles.

Hence the correct choice is (d).

#### Question 6:

The two diagonals are equal in a

(a) parallelogram

(b) rhombus

(c) rectangle

(d) trapezium

Two diagonals are equal only in a rectangle.

This can be proved as follows:

The rectangle is given as ABCD, with the two diagonals as AD and BC.

In and:

(Opposite sides of a rectangle are equal.)

(Common)

(Each angle in a rectangle is a right angle)

Thus,

(By SAS Congruence Rule)

By Corresponding parts of congruent triangles property we have:

Therefore, in a rectangle the two diagonals are equal.

Hence the correct choice is (c).

#### Question 7:

We get a rhombus by joining the mid-points of the sides of a

(a) parallelogram

(b) rhombus

(c) rectangle

(d) triangle

We get a rhombus by joining the mid-points of the sides of a rectangle.

It is given a rectangle ABCD in which P,Q,R and S are the mid-points AB,BC,CD and DA respectively.

PQ,QR,RS and SP are joined.

In , P and Q are the mid-points AB and BC respectively.

Therefore,

and ……(i)

Similarly, In , R and S are the mid-points CD and AD respectively.

Therefore,

and ……(ii)

From (i) and (ii), we get

and

Therefore, is a parallelogram. …… (iii)

Now is a rectangle.

Therefore,

…… (iv)

In and , we have:

(P is the mid point of AB)

(Each is a right angle)

(From equation (iv))

So, by SAS congruence criteria, we get:

By Corresponding parts of congruent triangles property we have:

…… (v)

From (iii) and (v) we obtain that is a parallelogram such that and

Thus, the two adjacent sides are equal.

Thus, is a rhombus.

Hence the correct choice is (c).

#### Question 8:

The bisectors of any two adjacent  angles of a parallelogram intersect at

(a) 30°

(b) 45°

(c) 60°

(d) 90°

Let the figure be as follows:

ABCD is a parallelogram.

We have to find

AN and AD is the bisectors of and .

Therefore,

…… (i)

And

…… (ii)

We know that .

Therefore, the sum of consecutive interior angles must be supplementary.

From (i) and (ii), we get

…… (iii)

By angle sum property of a triangle:

Hence the correct choice is (d).

#### Question 9:

The bisectors of the angle of a parallelogram enclose a

(a) parallelogram

(b) rhombus

(c) rectangle

(d) square

We have ABCD, a parallelogram given below:

Therefore, we have

Now, and transversal AB intersects them at A and B respectively. Therefore,

Sum of consecutive interior angle is supplementary. That is;

We have AR and BR as bisectors of and respectively.

…… (i)

Now, in , by angle sum property of a triangle, we get:

From equation (i), we get:

Similarly, we can prove that.

Now, and transversal ADintersects them at A and D respectively. Therefore,

Sum of consecutive interior angle is supplementary. That is;

We have AR and DP as bisectors of and respectively.

…… (ii)

Now, in , by angle sum property of a triangle, we get:

From equation (i), we get:

We know that and are vertically opposite angles, thus,

Similarly, we can prove that.

Therefore, PQRS is a rectangle.

Hence, the correct choice is (c).

#### Question 10:

The figure formed by joining the mid-points of the adjacent sides of a quadrilateral is a

(a) parallelogram

(b) rectangle

(c) square

(d) rhombus

Figure can be drawn as:

Let ABCD be a quadrilateral such that P,Q,R and S are the mid-points of side AB,BC,CD and DA respectively.

In,P and Q are the mid-points of AB and BC respectively.

Therefore,

and

Similarly, we have

and

Thus,

and

Therefore, PQRSis a parallelogram.

The figure formed by joining the mid-points of the adjacent sides of a quadrilateral is a

parallelogram.

Hence the correct choice is (a).

#### Question 11:

The figure formed by joining the mid-points of the adjacent sides of a rectangle is a

(a) square

(b) rhombus

(c) trapezium

(d) none of these

We get a rhombus by joining the mid-points of the sides of a rectangle.

It is given that rectangle ABCD in which P,Q,R and S are the mid-points AB,BC,CD and DA respectively.

PQ,QR,RS and SP are joined.

In , P and Q are the mid-points AB and BC respectively.

Therefore,

and ……(i)

Similarly, In , R and S are the mid-points CD and AD respectively.

Therefore,

and ……(ii)

From (i) and (ii), we get

and

Therefore, is a parallelogram. …… (iii)

Now is a rectangle.

Therefore,

…… (iv)

In and , we have:

(P is the mid point of AB)

(Each is a right angle)

(From equation (iv))

So, by SAS congruence criteria, we get:

By Corresponding parts of congruent triangles property we have:

…… (v)

From (iii) and (v) we obtain that is a parallelogram such that .

Thus, the two adjacent sides are equal.

Thus, is a rhombus .

Hence the correct choice is (b).

#### Question 12:

The figure formed by joining the mid-points of the adjacent sides of a rhombus is a

(a) square

(b) rectangle

(c) trapezium

(d) none of these

Figure is given as :

A rhombus ABCD is given in which P, Q, R and S are the mid-points of sides AB, BC, CD and DA respectively.

In , P and Q are the mid-points AB and BC respectively.

Therefore,

and ……(i)

Similarly, In , R and S are the mid-points CD and AD respectively.

Therefore,

and ……(ii)

From (i) and (ii), we get

and

Therefore, is a parallelogram. …… (iii)

Now, we shall find one of the angles of a parallelogram.

Since ABCD is a rhombus

Therefore,

(Sides of rhombus are equal)

(P and Q are the mid-points AB and BC respectively)

In , we have

(Angle opposite to equal sides are equal)

Therefore, ABCD is a rhombus

…… (iii)

Also,

…… (iv)

Now, in and , we have

[From (iii)]

[From (iv)]

And ( is a parallelogram)

So by SSS criteria of congruence, we have

By Corresponding parts of congruent triangles property we have:

…… (v)

Now,

And

Therefore,

From (ii), we get

From (v), we get

Therefore, …… (vi)

Now, transversal PQ cuts parallel lines SP and RQ at P and Q respectively.

[Using (vi)]

Thus, is a parallelogram such that .

Therefore, is a rectangle.

Hence the correct choice is (b).

#### Question 13:

The figure formed by joining the mid-points of the adjacent sides of a square is a

(a) rhombus

(b) square

(c) rectangle

(d) parallelogram

We get a square by joining the mid-points of the sides of a square.

It is given a square ABCD in which P, Q, R and S are the mid-points AB, BC, CD and DA respectively.

PQ, QR, RS and SP are joined.

Join AC and BD.

In , P and Q are the mid-points AB and BC respectively.

Therefore,

and ……(i)

Similarly, In ΔADC, R and S are the mid-points CD and AD respectively.

Therefore,

and ……(ii)

From (i) and (ii), we get

and

Therefore, is a parallelogram. …… (iii)

Now is a square.

Therefore,

…… (iv)

Similarly,

…… (v)

In and , we have:

(From equation (iv))

(Each is a right angle)

(From equation (v))

So, by SAS congruence criteria, we get:

By Corresponding parts of congruent triangles property we have:

…… (vi)

From (iii) and (vi) we obtain that is a parallelogram such that .

But, is a parallelogram.

So,

and …… (vii)

Now, (From equation (i))

Therefore,

…… (viii)

Since P and S are the mid-points AB and AD respectively

Therefore,

…… (ix)

Thus, in quadrilateral PMON, we have:

and (From equation (viii) and (ix))

Therefore, quadrilateral PMON is a parallelogram.

Also,

(Because )

(Because diagonals of a square are perpendicular)

Therefore, is a quadrilateral such that ,

and .Also, .

Hence, is a square.

Hence the correct choice is (b).

#### Question 14:

The figure formed by joining the mid-points of the adjacent sides of a parallelogram is a

(a) rectangle

(b) parallelogram

(c) rhombus

(d) square

It is given a parallelogram ABCD in which P,Q,R and S are the mid-points AB, BC, CD and DA respectively.

PQ, QR, RS and SP are joined.

In , P and Q are the mid-points AB and BC respectively.

Therefore,

and ……(i)

Similarly, In , R and S are the mid-points CD and AD respectively.

Therefore,

and ……(ii)

From (i) and (ii), we get

and

Therefore, is a parallelogram.

Hence the correct choice is (b).

#### Question 15:

If one angle of a parallelogram is 24° less than twice the smallest angle, then the measure of the largest angle of the parallelogram is

(a) 176°

(b) 68°

(c) 112°

(d) 102°

Let the smallest angle of the parallelogram be

Therefore, according to the given statement other angle becomes.

Also, the opposite angles of a parallelogram are equal.

Therefore, the four angles become ,,and.

According to the angle sum property of a quadrilateral:

Thus, the other angle becomes

Thus, the largest angle of the parallelogram are is .

Hence the correct choice is (c).

#### Question 16:

In a parallelogram ABCD, if ∠DAB = 75° and ∠DBC = 60°, then ∠BDC =

(a) 75°

(b) 60°

(c) 45°

(d) 55°

Parallelogram can be drawn as :

We know that the opposite angles of a parallelogram are equal.

Therefore,

By angle sum property of a triangle:

Thus, measures.

Hence the correct choice is (c).

#### Question 17:

ABCD is a parallelogram and E and F are the centroids of triangles ABD and BCD respectively, then EF =

(a) AE

(b) BE

(c) CE

(d) DE

Parallelogram ABCD is given with E and F are the centroids of and.

We have to find EF.

We know that the diagonals of a parallelogram of bisect each other.

Thus, AC and BD bisect each other at point O.

Also, median is the line segment joining the vertex to the mid-point of the opposite side of the triangle. Therefore, the centroids E and F lie on AC.

Now, the centroid divides each median into two segments whose lengths are in the ratio 2:1, with the longest one nearest the vertex.

Then, in ΔABD, we get:

Or,

and …… (I)

Similarly, in ,we get:

and …… (II)

Also,

From (I) and (II), we get:

And …… (III)

Also, from (II) and (III), we get :

…… (IV)

Now, from (I),

From (IV), we get:

From(III):

Hence the correct choice is (a).

#### Question 18:

ABCD is a parallelogram, M is the mid-point of BD and BM bisects ∠B. Then ∠AMB =

(a) 45°

(b) 60°

(c) 90°

(d) 75°

Figure is given as follows:

ABCD is a parallelogram.

It is given that :

BM bisects

Therefore,

But,

(Alternate interior opposite angles as)

Therefore,

In,

Sides opposite to equal angles are equal.

Thus,

Also,

(Opposite sides of a parallelogram are equal)

Thus,

ABCD is a parallelogram with

Therefore,

ABCD is a rhombus.

And we know that diagonals of the rhombus bisect each other at right angle.

Thus,

Hence the correct choice is (c).

#### Question 19:

ABCD is a parallelogram and E is the mid-point of BC. DE and AB when produced meet at F. Then, AF =

(a) $\frac{3}{2}AB$

(b) 2 AB

(c) 3 AB

(d) $\frac{5}{4}AB$

Parallelogram ABCD is given with E as the mid-point of BC.

DE and AB when produced meet at F

We need to find AF.

Since ABCD is a parallelogram, then

Therefore,

Then, the alternate interior angles should be equal.

Thus, …… (I)

In and :

(From(I))

(E is the mid-point of BC)

(Vertically opposite angles)

(by ASA Congruence property)

We know that the corresponding angles of congruent triangles should be equal.

Therefore,

But,

(Opposite sides of a parallelogram are equal)

Therefore,

…… (II)

Now,

From (II),we get:

Hence the correct choice is (b).

#### Question 20:

If an angle of a parallelogram is two-third of its adjacent angle, the smallest angle of the parallelogram is

(a) 108°

(b) 54°

(c) 72°

(d) 81°

Let one of the angle of the parallelogram as

We know that the sum of adjacent angles of the parallelogram is supplementary.

Therefore,

Therefore, the smallest angle of the parallelogram as

Hence, the correct choice is (c).

#### Question 21:

If the degree measures of the angles of quadrilateral are 4x, 7x, 9x and 10x, what is the sum of the measures of the smallest angle and largest angle?

(a) 140°

(b) 150°

(c) 168°

(d) 180°

We have,

,

,

and

By angle sum property of a quadrilateral, we get:

Smallest angle:

Also, Largest angle:

Thus, the sum of the smallest and the largest becomes:

Hence, the correct choice is (c).

#### Question 22:

In a quadrilateral ABCD, ∠A + ∠C is 2 times ∠B + ∠D. If ∠A = 140° and ∠D = 60°, then ∠B =

(a) 60°

(b) 80°

(c) 120°

(d) None of these

By angle sum property of a quadrilateral we get:

But,we have

Then,

The two equations so formed cannot give us the value for with a given value of .

Hence the correct choice is (d).

#### Question 23:

If the diagonals of a rhombus are 18 cm and 24 cm respectively, then its side is equal to

(a) 16 cm

(b) 15 cm

(c) 20 cm

(d) 17 cm

Let ABCD be rhombus with diagonals AC and BD 18cm and 24cm respectively.

We know that diagonals of the rhombus bisect each other at right angles.

Therefore,

Similarly,

Also, is a right angled triangle.

By Pythagoras theorem, we get:

Hence the correct choice is (b).

#### Question 24:

The diagonals  AC and BD of a rectangle ABCD intersect each other at P. If ∠ABD = 50°, then ∠DPC =

(a) 70°

(b) 90°

(c) 80°

(d) 100°

Figure can be drawn as follows:

The diagonals AC and BD of rectangle ABCD intersect at P.

Also, it is given that

We need to find

It is given that

Therefore, (Because P lies on BD)

Also, diagonals of rectangle are equal and they bisect each other.

Therefore,

Thus, (Angles opposite to equal sides are equal)

[(Given)]

By angle sum property of a triangle:

But and are vertically opposite angles.

Therefore, we get:

Hence the correct choice is (c).

#### Question 25:

ABCD is a parallelogram in which diagonal AC bisects ∠BAD. If ∠BAC = 35°, then ∠ABC =

(a) 70°

(b) 110°

(c) 90°

(d) 120°

ABCD is a parallelogram in which AC bisects.

It is given that

Therefore,

Since,

Therefore,

Hence, the correct choice is (b).

#### Question 26:

In a rhombus ABCD, if ∠ACB = 40°, then ∠ADB =

(a) 70°

(b) 45°

(c) 50°

(d) 60°

Rhombus ABCD is given as follows:

It is given that.

Therefore, (Because O lies on AC)

We know that the diagonals of a rhombus intersect at right angle.

Therefore,

By angle sum property of a triangle, we get:

Since, O lies on BD

Also ,

Therefore,

Hence, the correct choice is (c).

#### Question 27:

In ΔABC, ∠A = 30°, ∠B = 40° and ∠C = 110°. The angles of the triangle formed by joining the mid-points of the sides of this triangle are

(a) 70°, 70°, 40°

(b) 60°, 40°, 80°

(c) 30°, 40°, 110°

(d) 60°, 70°, 50°

It is given that D, E and F be the mid-points of BC , CA and AB respectively.

Then,

, and .

Now, and transversal CB and CA intersect them at D and E respectively.

Therefore,

[(Given)]

and

[(Given)]

Similarly,

Therefore,

[(Given)]

and

[(Given)]

Similarly,

Therefore,

[ (Given)]

and

[ (Given)]

Now BC is a straight line.

Similarly,

and

Hence the correct choice is (c).

#### Question 28:

The diagonals of a parallelogram ABCD intersect at O. If ∠BOC = 90° and ∠BDC = 50°, then ∠OAB =

(a) 40°

(b) 50°

(c) 10°

(d) 90°

ABCD is a parallelogram with diagonals AC and BD intersect at O.

It is given that and.

We need to find

Now,

(Linear pair)

Since, O lies on BD.

Therefore,

By angle sum property of a triangle, we get:

Since, O lies on AC.

Therefore,

Also,

Therefore,

Hence the correct choice is (a).

#### Question 29:

ABCD is a trapezium in which AB || DC. M and N are the mid-points of AD and the respectively. If AB = 12 cm, MN = 14 cm, then CD =

(a) 10 cm

(b) 12 cm

(c) 14 cm

(d) 16 cm

The given trapezium ABCD can be drawn as follows:

Here, .

M and N are the mid-points of AD and BC respectively.

We have,.

We need to find CD.

Join MN to intersect AC at G.

We have and a line MN formed by joining the mid-points of sides AD and BC.

Thus, we can say that

In M is the mid-point of AD and

Therefore, G is the mid point of AC

By using the converse of mid-point theorem, we get:

…… (i)

In , N is the mid point of BC and

By using the converse of mid-point theorem, we get:

…… (ii)

…… (iii)

On substituting and in (iii),we get:

Hence the correct choice is (d).

#### Question 30:

Diagonals of a quadrilateral ABCD bisect each other. If ∠A= 45°, then ∠B =

(i) 115°

(ii) 120°

(iii) 125°

(iv) 135°

We know that the diagonals of a parallelogram bisect each other.

Thus, the given quadrilateral ABCD is a parallelogram.

and are consecutive interior angles, which must be supplementary.

Therefore, we have

Hence the correct choice is (d).

#### Question 31:

P is the mid-point of side BC of a parallelogram ABCD such that ∠BAP = ∠DAP. If AD = 10 cm, then CD =

(a) 5 cm

(b) 6 cm

(c) 8 cm

(d) 10 cm

Parallelogram ABCD is given such that

We haveand

We need to find the measure of side CD.

Since and AP as transversal

But it is given that

Therefore, we get:

Also, sides opposite to equal angles are equal.

Then,

…… (I)

Also,

Substituting in (I), We get:

It is given that , this means opposite side ,as ABCD is a parallelogram. Therefore,

Or,

Hence the correct choice is (a).

#### Question 32:

In ΔABC, E is the mid-point of median AD such that BE produced meets AC at F. IF AC = 10.5 cm, then AF =

(a) 3 cm

(b) 3.5 cm

(c) 2.5 cm

(d) 5 cm

is given with E as the mid-point of median AD.

Also, BE produced meets AC at F.

We have , then we need to find AF.

Through D draw .

In , E is the mid-point of AD and .

Using the converse of mid-point theorem, we get:

…… (i)

In , D is the mid-point of BC and .

…… (ii)

From (i) and (ii),we have:

, …… (iii)

Now,

From (iii) equation, we get:

…… (iv)

We have .Putting this in equation (iv), we get:

Hence the correct choice is (b).

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