Question

State AA criterion of similarity of two triangles and use it to prove the following.

In the given figures of $△ A BC$ and $△ PQR$ , $A D$ and $PS$ are angle bisectors of $∠ B A C$ and $∠ RPQ$ respectively.

If $△ A BC \sim △ PQR$ , prove that $△ A C D \sim △ PRS$ .

Approach

State AA criterion: if two angles of one triangle equal two angles of another, they are similar. Given $△ A BC \sim △ PQR$ , so $∠ B A C = ∠ RPQ$ . Since $A D$ and $PS$ bisect these angles, $∠ C A D = ∠ RPS$ . Also, $∠ A C D = ∠ PRS$ (corresponding angles). By AA, $△ A C D \sim △ PRS$ .

Step-by-step working

1
State AA criterion
1 mark
2
$△ A BC \sim △ PQR \Rightarrow ∠ B A C = ∠ RPQ$
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3
Bisectors: $∠ C A D = ∠ RPS$
1 mark
4
$∠ A C D = ∠ PRS$ (corresponding)
1 mark
5
By AA, $△ A C D \sim △ PRS$
1 mark

Concepts used

aa similarityangle bisectortrianglesproof

Approach

Step-by-step working

Concepts used

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