Question

State AA criterion of similarity of two triangles and use it to prove the following.
In the given figures of $△ A BC$ and $△ PQR$ , AD 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 and prove the AA similarity criterion: if two angles of one triangle are respectively equal to two angles of another triangle, then the triangles are similar. In $△ A BC \sim △ PQR$ , $∠ A = ∠ P$ , $∠ B = ∠ Q$ , $∠ C = ∠ R$ . In $△ A C D$ and $△ PRS$ , $∠ C A D = \frac{1}{2} ∠ B A C = \frac{1}{2} ∠ RPQ = ∠ RPS$ , $∠ C = ∠ R$ , so $△ A C D \sim △ PRS$ by AA.

Step-by-step working

1
Statement: If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar
1 mark
2
Given $△ A BC \sim △ PQR$ , so $∠ A = ∠ P$ , $∠ B = ∠ Q$ , $∠ C = ∠ R$
1 mark
3
In $△ A C D$ and $△ PRS$ : $∠ C A D = \frac{1}{2} ∠ B A C = \frac{1}{2} ∠ RPQ = ∠ RPS$
1 mark
4
$∠ C = ∠ R$ (from similarity)
0.5 marks
5
$△ A C D \sim △ PRS$ by AA similarity criterion
1.5 marks

Concepts used

trianglesaa similaritysimilar trianglesangle bisectorprove

Approach

Step-by-step working

Concepts used

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