Congruence of Triangles
Question 1: Which congruence criterion do you use in the following?
(a) Given AC = DF
AB = DE
BC = EF
So, ΔABC ≈ ΔDEF
Answer: SSS criterion
(b) Given: ZX = RP
RQ = ZY
∠PRQ = ∠XZY
So, ΔPQR ≈ΔXYZ
Answer: SAS Criterion
(c) Given: ∠MLN = ∠FGH
∠NML = ∠GFH
ML = FG
So, ΔLMN ≈ ΔGFH
Answer: ASA Criterion
(d) Given: EB = DB
AE = BC
∠A = ∠C = 90°
So, ΔABE ≈ ΔCDB
Answer: SAS Criterion
Question 2: You want to show that ΔART ≈ΔPEN
(a) If you have to use SS criterion, then you need to show (i) AR = (ii) RT = (iii) AT =
Answer: (i) → PE, (ii) → EN (iii) → PN
(b) It is given that angle T = angle N and you are to use SAS criterion, you need to have (i) RT = and (ii) PN =
Answer: (i) → EN, (ii) → AT
(c) If it is given that AT = PN and you use ASA criterion, you need to have (i) ? (ii) ?
Answer: (i) ∠A = ∠P (ii) ∠T = ∠N
Question 3: You have to show that ΔAMP ≈ΔAMQ. In the following proof, supply the missing reasons.
|(a) PM = QM||Equal measure|
|(b) ∠PMA = ∠QMA||Equal measure|
|(c) AM = AM||Equal measure|
|(d) ΔAMP ≈ ΔAMQ||SAS criterion|
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: AAA is not a criterion for congruence. Hence, the student is wrong. This can be understood by taking the example of equilateral triangles. Each angle of an equilateral triangle is 60°; no matter what is the size of the triangle. So, all equilateral triangle need not be congruent.
Question 5: In the figure, two triangles are congruent. The corresponding parts are marked. We can write ΔRAT ≈…………..
Question 6: Complete the congruence statement: ΔBCA ≈ …………. ΔQRS ≈ ……………….
Answer: ΔBCA ≈ ΔBTA and ΔQRS ≈ ΔQPT
Question 7: In a squared sheet, draw two triangles of equal areas such that
(a) The triangles are congruent.
(b) The triangles are not congruent.
What can you say about their perimeters?
Answer: When triangles are congruent then their perimeters are equal. But when triangles are not congruent then their perimeters are different.
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: Five pairs of congruent parts can be three pairs of sides and two pairs of angles. In that case, SAS or ASA criterion would prove them to be congruent. Hence, such a figure is not possible.
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; ASA criterion
Question 10: Explain, why: ΔABC ≈ ΔFED
Answer: Angle B = angle E,
BC = ED
and angle F = angle A
Hence, as per ASA criterion, triangles are congruent.