Exercise 8.4

1. Which congruence criterion do you use in the following?

(i)

(i) Given: AC = DF AB = DE BC = EF So, ΔABC ≅ ΔDEF

Solution:

Given: AC =

AB =

BC =

Therefore, ΔABC ≅ ΔDEF by criterion

(ii)

(ii) Given: ZX = RP RQ = ZY ∠PRQ ≅ ∠XZY So, ΔPQR ≅ ΔXYZ

Solution:

Given: ZX =

RQ =

∠PRQ = ∠

Therefore, ΔPQR ≅ ΔXYZ by criterion

(iii)

(iii) Given: ∠MLN ≅ ∠FGH ∠NML ≅ ∠GFH ML = FG So, ΔLMN ≅ ΔGFH

Solution:

Given: ∠MLN = ∠

∠NML = ∠

ML =

Therefore, ΔLMN ≅ ΔGFH by criterion

(iv)

(iv) Given: EB = DB AE = BC ∠A = ∠C = 90° So, ΔABE ≅ ΔCDB

Solution:

Given: EB =

AE =

∠A = ∠ = 90°

Therefore, ΔABE ≅ ΔCDB by criterion

2. You want to show that ΔART ≅ ΔPEN.

(i)

(i) If you have to use SSS criterion, then you need to show: (a) AR = ? (b) RT = ? (c) AT = ?

Solution:

For SSS criterion, we need to show: (a) AR =

(b) RT =

(c) AT =

Therefore, by SSS criterion, all three pairs of sides must be equal

(ii)

(ii) If it is given that ∠T = ∠N and you are to use SAS criterion, you need to have: (a) RT = ? (b) PN = ?

Solution:

Given that ∠T = ∠N, for SAS criterion we need: (a) RT =

(b) AT =

Therefore, by SAS criterion, we need two sides and the included angle

(iii)

(iii) If it is given that AT = PN and you are to use ASA criterion, you need to have: (a) ? = ? (b) ? = ?

Solution:

Given that AT = PN, for ASA criterion we need: (a) ∠ = ∠

(b) ∠ = ∠

Therefore, by ASA criterion, we need two angles and the included side

3. You have to show that ΔAMP ≅ ΔAMQ. In the following proof, supply the missing reasons.

Solution:

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?

Solution:

Let's analyze the given information:

In ΔABC: ∠A = °

∠B = °

∠C = °

In ΔPQR: {.reveal(when="blank-2")}∠P = ∠A = °

∠Q = ∠B = °

∠R = ∠C = °

The student is justified because AAA only proves triangles are similar, not congruent

5. In the figure, the two triangles are congruent. The corresponding parts are marked. We can write ΔRAT ≅ ?

Solution:

Let's analyze the given triangles:

Given that triangles are congruent and looking at corresponding parts: A corresponds to

R corresponds to

T corresponds to

Therefore, ΔRAT ≅ Δ

6. Complete the congruence statement.

Solution:

Let's match the corresponding parts: ΔABC ≅ Δ

A corresponds to

B corresponds to

C corresponds to

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?

Solution:

(i) If the triangles are congruent, their perimeters are .

(ii) If the triangles are not congruent but have equal areas, their are not necessarily equal.

Triangles with equal area can have different perimeters if their are different.

8. If ΔABC and ΔPQR are to be congruent, name one additional pair of corresponding parts. What criterion did you use?

Solution:

One additional pair of corresponding parts:

Criterion used: (Angle-Side-Angle) congruence

9. Explain, why ΔABC ≅ ΔFED.

Solution:

AB = (marked equal)

BC = (marked equal)

∠B = ∠ = 90°

Therefore, ΔABC ≅ ΔFED by criterion