Page 50 - Mathematics Class - X

3. Place the triangle ABC on PQR such that vertex A falls on vertex P and side AB falls alongside PQ and
side AC falls alongside PR as shown in Fig. (b).

||
4. In Fig. (b), ∠B = ∠Q. Since corresponding angles are equal, BC QR.

PB PC AB AC
5. By BPT, = or =
BQ CR BQ CR
Fig. (b)
BQ CR
or =
AB AC

BQ + AB CR + AC
or = [Adding 1 on both sides]
AB AC

AQ AR PQ PR AB AC
or = or = or = ...(i)
AB AC AB AC PQ PR

Alternate Method
6. Place the ∆ABC on ∆PQR such that vertex B falls on vertex Q, and side BA falls alongside QP and side BC
falls alongside QR as shown in Fig. (c).

7. In Fig. (c), ∠C = ∠R. Since corresponding angles are equal, AC || PR.

AP CR BP BR
8. By BPT, = ; or = [Adding 1 on both sides]
AB BC AB BC
Fig. (c)
PQ QR AB BC
or = ; or = . ..(ii)
AB BC PQ QR

From (i) and (ii), AB = AC = BC
PQ PR QR

We find that when the corresponding angles of two triangles are equal, then their corresponding sides are
proportional. Hence, the two triangles are similar. This is AAA criterion for similarity of triangles.

Case 2
9. Cut out two triangles ABC and PQR with their corresponding sides proportional using a coloured paper.
P (A)
i.e. AB = BC = AC
PQ QR PR
B C
Q R
Fig. (d)

10. Place the ∆ABC on ∆PQR such that vertex A falls on vertex P and side AB falls alongside PQ. Observe that
side AC falls alongside PR as shown in Fig. (d).

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