In Fig. 6.9, if ∠1 = ∠2 and △NSQ ⩬ △MTR, then prove that △PTS ~ △PRQ
Solution:
Given, the triangles NSQ and MTR are similar.
Also, ∠1 = ∠2
We have to prove that the triangles PTS and PRQ are similar.
Corresponding parts of congruent triangles(CPCT) state that if two or more triangles are congruent, then all of their corresponding angles and sides are congruent as well.
Given, △NSQ ⩬ △MTR
By CPCT, ∠NQS = ∠MRT
Also, ∠PRQ = ∠PQR ------------- (1)
We know that the sum of all the three interior angles of a triangle is always equal to 180°
In △PST,
∠P + ∠1 + ∠2 = 180°
Given, ∠1 = ∠2
So, ∠P + ∠1 + ∠1 = 180°
∠P + 2∠1 = 180° -------------------- (2)
In △PQR,
∠P + ∠PQR + ∠PRQ = 180°
From (1), ∠PRQ = ∠PQR
∠P + ∠PQR + ∠PQR = 180°
∠P + 2∠PQR = 180° -------------------- (3)
Equating (2) and (3) as the RHS are same,
∠P + 2∠1 = ∠P + 2∠PQR
Cancelling out common terms,
∠PQR = ∠1
So, ∠PST = ∠PQR --------- (4)
In △PTS and △PRQ,
∠P = ∠P = common angle.
From (4) ∠PST = ∠PQR
AAA criterion states that if two angles of a triangle are respectively equal to two angles of another triangle, then by the angle sum property of a triangle their third angle will also be equal.
By AAA criterion, the triangles PTS and PRQ are similar.
Therefore, △PTS ~ △PRQ
✦ Try This: In the given figure, ∠1 = ∠2 and AC/BD = CB/CE . Prove that ∆ ACB ~ ∆ DCE.
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 6
NCERT Exemplar Class 10 Maths Exercise 6.3 Problem 3
In Fig. 6.9, if ∠1 = ∠2 and △NSQ ⩬ △MTR, then prove that △PTS ~ △PRQ
Summary:
In Fig. 6.9, if ∠1 = ∠2 and △NSQ ⩬ △MTR, it is proved that △PTS ~ △PRQ
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