Exercise 8.2

1. What additional information do you need to conclude that the two triangles given here under are congruent using SAS rule?

Solution:

Let's determine what additional information is needed for SAS congruence:

Given in ∆GHJ and ∆RTS:

∠H = ∠T (marked angles)

For SAS congruence, we need:

GH =

HJ =

Therefore, to prove ∆GHJ ≅ ∆RTS by SAS criterion, we need:

The equality of side GH with TR

The equality of side HJ with TS

These additional side pairs must be adjacent to the equal angles ∠H and ∠T

2. The map given below shows five different villages. Village M lies exactly halfway between the two pairs of villages A and B as well as P and Q. What is the distance between village A and village P (Hint: check if ΔPAM ≅ ΔQBM)

Solution:

Given that M is a of both AB and PQ:

AM =

PM =

Let's analyze triangles ΔPAM and ΔQBM:

• AM = MB (M is midpoint of AB)

• PM = MQ (M is midpoint of PQ)

• ∠PAM = ∠ (corresponding angles)

Looking at the measurements:

• AM = km (given)

• PM = km (given)

By congruence criterion, ΔPAM ≅ ΔQBM

Therefore, AP = = 3 km

3. Look at the pairs of triangles given below. Are they congruent? If congruent write the corresponding parts.

Solution:

Let's analyze each pair:

(i) Given triangles ABC and WDE: AB = = 6 cm

BC = = 4.5 cm

AC = = 7.5 cm

By criterion, ΔABC ≅ ΔWDE

(ii) Given triangles PQR and STU:

PQ = = 4 cm

QR = = 4 cm

PR = = 4 cm

By criterion, ΔPQR ≅ ΔSTU

(iii) Given triangles OSR and OWD:

OS = = 4.5 cm

SR = = 4.5 cm

OR = (common)

By criterion, ΔOSR ≅ ΔOWD

(iv) Given triangles ABC and PQR:

These triangles are as the sides are not equal

4. Which corresponding sides do we need to know to prove that the triangles are congruent using the SAS criterion?

Solution:

(i) For triangles ABC and PQR: Given: ∠ = ∠ = 40°

Need: and

Need: and

(ii) For triangles ABC and DCB:

Given: ∠ = ∠ = 35°

Need: and

Need: and (common side)

For SAS criterion, we need the two sides that include the given equal angle in each triangle.