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Mathematics · Triangles · NCERT Examples

Example 3

A complete, independently verified solution for NCERT Example 3.

Mathematics · Chapter 6 · NCERT Examples

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NCERT Example

Example 3

converse of BPT · isosceles triangle · parallel lines

In Figure 6.16, PS/SQ = PT/TR and ∠PST = ∠PRQ. Prove that PQR is an isosceles triangle.

Triangle PQR with S on PQ, T on PR, segment ST, and the marked angles PST and PRQ.
Figure 6-16 from the supplied NCERT chapter.

Solution

Board-exam working:

Given:

PSSQ=PTTR\frac{PS}{SQ}=\frac{PT}{TR}

Since S and T divide PQ and PR in the same ratio, ST ∥ QR by the converse of BPT.

Therefore, ∠PST = ∠PQR because they are corresponding angles.

But ∠PST = ∠PRQ is given. Hence ∠PQR = ∠PRQ.

Sides opposite equal angles of a triangle are equal, so PR = PQ.

Final answer

Therefore, △PQR is isosceles with PQ = PR.