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Syllabus

1) ACBD is a rectangle

2) CD is parallel to the original parallel lines.

1. AQ = BP

2. PQ = CD

3. ABPQ is a parallelogram

^{o}, prove that all four sides are equal, as the resultant quadrilateral before proving this can also be taken to be a rectangle.Q. In figure given below, ABCD is a rectangle and diagonals intersect at O. AC is produced to E. If $\angle $ECD = 146$\xb0$, find the angles of triangle AOB.

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Q.5. (a) In figure (1) given below, ABCD is a parallelogram with perimeter 40. Find the values of x and y.

(b) In figure (2) given below, ABCD is a parallelogram. Find the values of x and y.

(c) In figure (3) given below, ABCD is a rhombus. Find x and y.

Angle AOD

Q.7. (a) In figure (1) given below, equilateral triangle EBC surmounts square ABCD. Find angle BED represented by x.

(b) In figure (2) given below, ABCD is a rectangle and diagonals intersect at O. AC is produced to E. If $\angle $ ECD = 146$\xb0$, find the angles of the $\u25b3$AOB.

(c) In figure (3) given below, ABCD is a rhombus and diagonals intersect at O. If $\angle $OAB : $\angle $OBA = 3 : 2, find the angles of the $\u25b3$AOD.

Q.9. In the following figure, OP, OQ and OR are drawn perpendiculars to the sides BC, CA and AB respectively of triangle ABC. Prove that :

$A{P}^{2}+B{P}^{2}+C{Q}^{2}=A{Q}^{2}+C{P}^{2}+B{R}^{2}$

14. (a) In figure ( 1) given below, A BCD is a parallelogram and X is mid-point of BC. The line AX produced meets DC produced at Q. The parallelogram ABPQ is completed. Prove that

(i) the triangles ABX and QCX are congruent-

(ii) DC=CQ=QP

18. In a parallelogram ABCD, the bisector of $\angle $A meets DC in E and AB= 2AD. Prove that

(i) BE bisects $\angle $ B

(ii) $\angle $ AEB = a right angle.

Example 6.In the adjoining figure, ABCD is a parallelogram. Find the values of x, y and z.Q10. In parallelogram ABCD, AP and AQ are perpendiculars from vertex of obtuse angle A as shown. If $\angle $

x: $\angle $y= 2 : 1 ; find the angles of the parallelogram.

16. ABCD is a square. ABO is an equilateral triangle inside the square. Find $\angle $DOC.

Q7. (a) In figure (1) given below, equilateral triangle EBC surmounts square ABCD. Find angle BED represented by x.

(b) In figure (2) given below, ABCD is a rectangle and diagonals intersect at O. AC is produced to E. If $\angle $ECD = 146°, find the angles of the $\u2206$AOB.

(c) In figure (3) given below, ABCD is a rhombus and diagonals intersect at O. If $\angle $OAB : $\angle $OBA = 3 : 2, find the angles of the $\u2206$AOD.

Prove that :RD and SC when produced meet at right angle.

Q.8.c. In figure (3) given below, ABCD is a kite and diagonals intersect at O. If $\angle DAB=112\xb0and\angle DCB=64\xb0$, find $\angle ODCand\angle OBA$.

Q.11. Use ruler and compasses to construct a parallelogram with diagonals 6 cm and 8 cm in length having given the acute angle between them is 60$\xb0$. Measure one of the longer sides.

Show that :

(i) $\angle POC=\left(22\frac{1}{2}\right)\xb0$

(ii) $\angle BDC=2\angle POC$

(iii)$\angle BOP=3\angle COP$