- Congruent objects are exact copies of one another.
- If two line segments have the same (i.e., equal) length, they are congruent. Also, if two line segments are congruent, they have the same length.
- If two angles have the same measure, they are congruent. Also, if two angles are congruent, their measures are same.
1. SSS congruence criteria: Triangles are congruent if three sides of the one are equal to the three corresponding sides of the other.
2. SAS congruence criteria: Triangles are congruent if two sides and the angle included between them in one of the triangle are equal to the corresponding sides and the angle included between them of the other triangle.
3. ASA congruence criteria: Two triangles are congruent if two angles and the side included between them in one of the triangles are equal to the corresponding angles and the side included between them of the other triangle.
Emphasis will also be laid upon the topic- Congruence Among Right-Angled Triangles.
RHS congruence criteria: If under a correspondence, the hypotenuse and one side of a right-angled triangle are respectively equal to the hypotenuse and one side of another right-angled triangle, then the triangles are congruent.
Later the chapter Congruence of Triangles concludes with a summary.
Page No 137:
Question 1:
Complete the following statements:
(a) Two line segments are congruent if __________.
(b) Among two congruent angles, one has a measure of 70°; the measure of the other angle is __________.
(c) When we write ∠A = ∠ B, we actually mean __________.
Answer:
(a) They have the same length
(b) 70°
(c) m ∠A = m ∠B
Page No 137:
Question 2:
Give any two real-life examples for congruent shapes.
Answer:
(i) Sheets of same letter pad
(ii) Biscuits in the same packet
Page No 137:
Question 3:
If ΔABC ≅ ΔFED under the correspondence ABC ↔ FED, write all the Corresponding congruent parts of the triangles.
Answer:
If these triangles are congruent, then the corresponding angles and sides will be equal to each other.
∠A ↔ ∠F
∠B ↔ ∠E
∠C ↔ ∠D
Page No 137:
Question 4:
If ΔDEF ≅ ΔBCA, write the part(s) of ΔBCA that correspond to
(i) ∠E (ii) (iii) ∠F (iv)
Answer:
(i) ∠C
(ii)
(iii) ∠A
(iv)
Video Solution for congruence of triangles (Page: 137 , Q.No.: 4)
NCERT Solution for Class 7 math - congruence of triangles 137 , Question 4
Page No 149:
Question 1:
Which congruence criterion do you use in the following?
(a) Given: AC = DF
AB = DE
BC = EF
So, ΔABC ≅ ΔDEF
(b) Given: ZX = RP
RQ = ZY
∠PRQ = ∠XZY
So, ΔPQR ≅ ΔXYZ
(c) Given: ∠MLN = ∠FGH
∠NML = ∠GFH
ML = FG
So, ΔLMN ≅ ΔGFH
(d) Given: EB = DB
AE = BC
∠A = ∠C = 90°
So, ΔABE ≅ ΔCDB
Answer:
(a) SSS, as the sides of ΔABC are equal to the sides of ΔDEF.
(b) SAS, as two sides and the angle included between these sides of ΔPQR are equal to two sides and the angle included between these sides of ΔXYZ.
(c) ASA, as two angles and the side included between these angles of ΔLMN are equal to two angles and the side included between these angles of ΔGFH.
(d) RHS, as in the given two right-angled triangles, one side and the hypotenuse are respectively equal.
Page No 149:
Question 2:
You want to show that ΔART ≅ ΔPEN,
(a) If you have to use SSS criterion, then you need to show
(i) AR = (ii) RT = (iii) AT =
(b) If it is given that ∠T = ∠N and you are to use SAS criterion, you need to have
(i) RT = and (ii) PN =
(c) If it is given that AT = PN and you are to use ASA criterion, you need to have
(i) ? (ii) ?
Answer:
(a)
(i) AR = PE
(ii) RT = EN
(iii) AT = PN
(b)
(i) RT = EN
(ii) PN = AT
(c)
(i) ∠ATR = ∠PNE
(ii) ∠RAT = ∠EPN
Video Solution for congruence of triangles (Page: 149 , Q.No.: 2)
NCERT Solution for Class 7 math - congruence of triangles 149 , Question 2
Page No 150:
Question 3:
You have to show that ΔAMP ≅ AMQ.
In the following proof, supply the missing reasons.
- |
Steps |
- |
Reasons |
(i) |
PM = QM |
(i) |
… |
(ii) |
∠PMA = ∠QMA |
(ii) |
… |
(iii) |
AM = AM |
(iii) |
… |
(iv) |
ΔAMP ≅ ΔAMQ |
(iv) |
… |
Answer:
(i) Given
(ii) Given
(iii) Common
(iv) SAS, as the two sides and the angle included between these sides of ΔAMP are equal to two sides and the angle included between these sides of ΔAMQ.
Page No 150:
Question 4:
In ΔABC, ∠A = 30°, ∠B = 40° and ∠C = 110°
In ΔPQR, ∠P = 30°, ∠Q = 40° and ∠R = 110°
A student says that ΔABC ≅ ΔPQR by AAA congruence criterion. Is he justified? Why or why not?
Answer:
No. This property represents that these triangles have their respective angles of equal measure. However, this gives no information about their sides. The sides of these triangles have a ratio somewhat different than 1:1. Therefore, AAA property does not prove the two triangles congruent.
Page No 150:
Question 5:
In the figure, the two triangles are congruent.
The corresponding parts are marked. We can write ΔRAT ≅ ?
Answer:
It can be observed that,
∠RAT = ∠WON
∠ART = ∠OWN
AR = OW
Therefore, ΔRAT ΔWON, by ASA criterion.
Video Solution for congruence of triangles (Page: 150 , Q.No.: 5)
NCERT Solution for Class 7 math - congruence of triangles 150 , Question 5
Page No 150:
Question 6:
Complete the congruence statement:
ΔBCA ≅?
ΔQRS ≅?
Answer:
Given that, BC = BT
TA = CA
BA is common.
Therefore, ΔBCA ΔBTA
Similarly, PQ = RS
TQ = QS
PT
= RQ
Therefore, ΔQRS ΔTPQ
Page No 150:
Question 7:
In a squared sheet, draw two triangles of equal areas such that
(i) The triangles are congruent.
(ii) The triangles are not congruent.
What can you say about their perimeters?
Answer:
(i)
Here, ΔABC and ΔPQR have the same area and are congruent to each other also. Also, the perimeter of both the triangles will be the same.
(ii)
Here, the two triangles have the same height and base. Thus, their areas are equal. However, these triangles are not congruent to each other. Also, the perimeter of both the triangles will not be the same.
Video Solution for congruence of triangles (Page: 150 , Q.No.: 7)
NCERT Solution for Class 7 math - congruence of triangles 150 , Question 7
Page No 150:
Question 8:
Draw a rough sketch of two triangles such that they have five pairs of congruent parts but still the triangles are not congruent.
Answer:
Consider two triangles $\u25b3$ABC and $\u25b3$XYZ.
In $\u25b3$ABC and $\u25b3$XYZ,
$\angle \mathrm{A}=\angle \mathrm{X}=40\xb0\phantom{\rule{0ex}{0ex}}\angle \mathrm{B}=\angle \mathrm{Y}=80\xb0\phantom{\rule{0ex}{0ex}}\angle \mathrm{C}=\angle \mathrm{Z}=60\xb0\phantom{\rule{0ex}{0ex}}\mathrm{AB}=\mathrm{YZ}\phantom{\rule{0ex}{0ex}}\mathrm{AC}=\mathrm{XY}$
The given triangles have five pairs of congruent parts. But these two triangles are not congruent by any criterion of congruence.
Page No 150:
Question 9:
If ΔABC and ΔPQR are to be congruent, name one additional pair of corresponding parts. What criterion did you use?
Answer:
BC = QR
ΔABC ΔPQR (ASA criterion)
Page No 151:
Question 10:
Explain, why
ΔABC ≅ ΔFED
Answer:
Given that, ∠ABC = ∠FED (1)
∠BAC = ∠EFD (2)
The two angles of ΔABC are equal to the two respective angles of ΔFED. Also, the sum of all interior angles of a triangle is 180º. Therefore, third angle of both triangles will also be equal in measure.
∠BCA = ∠EDF (3)
Also, given that, BC = ED (4)
By using equation (1), (3), and (4), we obtain
ΔABC ≅ ΔFED (ASA criterion)
Video Solution for congruence of triangles (Page: 151 , Q.No.: 10)
NCERT Solution for Class 7 math - congruence of triangles 151 , Question 10
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