State if the following is true or false.

In the given figure, $OA . OB = OC . OD,$ then $∠ A = ∠ C$ and $∠ B = ∠ D$ .

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True

False

answer is A.

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Detailed Solution

Given figure is,
Question Image

Also given that,
OA.OB=OC.OD……(1)
We know,
The SAS criterion for triangle similarity states that if two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar.
In triangle AOD and BOC,

O A O C = O D O B From equation(1) ∠ A O D = ∠ B O C Vertically Opposite angles

Therefore, the triangle AOD and BOC are similar by SAS criterion.
As the two triangles are similar then, the corresponding angles must be equal.
AOD and BOC are similar.
So, the corresponding angle A and angle B are equal and corresponding angle C and angle D are equal.