Question

State the SAS criteria of similarity of two triangles. In the given figure, it is given that OA · OC = OB · OD. Use the SAS criteria to prove that AD||CB.

Approach

SAS criterion: If in two triangles, one pair of corresponding sides are proportional and the included angles are equal, triangles are similar. Given $O A \cdot OC = OB \cdot O D$ , we have $\frac{O A}{O D} = \frac{OB}{OC}$ . With $∠ A O D = ∠ BOC$ (vertically opposite), $△ A O D \sim △ BOC$ by SAS. Corresponding angles give $∠ O A D = ∠ OCB$ , which are alternate interior angles, so AD||CB.

Step-by-step working

1
Correct figure drawn
1 mark
2
$O A \cdot OC = OB \cdot O D \Rightarrow \frac{O A}{O D} = \frac{OB}{OC}$
1 mark
3
$∠ A O D = ∠ BOC$ (vertically opposite)
0.5 marks
4
$△ A O D \sim △ BOC$ (SAS)
1 mark
5
$∠ O A D = ∠ OCB$ (corresponding angles)
0.5 marks
6
Alternate interior angles equal $\Rightarrow$ AD||CB
1 mark

Concepts used

similaritysas criterionparallel lines

Approach

Step-by-step working

Concepts used

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