In Fig. 6.31, OA⋅OB=OC⋅OD. Show that ∠A=∠C and ∠B=∠D. [Figure: segments AB and CD intersect at point O; A and D on one side, C and B on the other, forming triangles AOD and COB.]
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Step-by-Step Solution
Step 1: Rearrange the given relation
We are given the product relationship OA⋅OB=OC⋅OD. To use similarity criteria involving ratios of sides, we rearrange this equation by dividing both sides by OC⋅OB to form a proportion relating the sides.
Step 2: Identify vertically opposite angles
From the figure shown, the line segments AB and CD intersect at point O. Angles ∠AOD and ∠COB are vertically opposite angles formed by the intersection of these two lines. Vertically opposite angles are always equal.
Step 3: Apply SAS similarity criterion
We have established that OCOA=OBOD (ratio of two sides are equal) and ∠AOD=∠COB (the included angles are equal). According to the SAS (Side-Angle-Side) similarity criterion, if two sides and the included angle of one triangle are proportional to the corresponding two sides and included angle of another triangle, then the two triangles are similar. Therefore, △AOD is similar to △COB.
Step 4: State corresponding angles of similar triangles
Since △AOD∼△COB, their corresponding angles are equal. Angle A in △AOD corresponds to angle C in △COB, and angle D in △AOD corresponds to angle B in △COB. Thus, we conclude that ∠A=∠C and ∠B=∠D.