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Mathematics · Triangles · NCERT Examples

Example 6

A complete, independently verified solution for NCERT Example 6.

Mathematics · Chapter 6 · NCERT Examples

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NCERT Example

Example 6

SAS similarity · vertically opposite angles

In Figure 6.31, OA × OB = OC × OD. Show that ∠A = ∠C and ∠B = ∠D.

Segments AB and CD intersect through O, forming triangles AOD and COB.
Figure 6-31 from the supplied NCERT chapter.

Solution

Board-exam working:

Given: OA × OB = OC × OD.

Dividing both sides by OC × OB:

OAOC=ODOB...(1)\frac{OA}{OC}=\frac{OD}{OB}\qquad ...(1)

Also, ∠AOD = ∠COB because they are vertically opposite angles. ...(2)

From (1) and (2), the two sides including the equal angle are proportional.

Therefore, △AOD ∼ △COB by SAS similarity, with A ↔ C and D ↔ B.

Corresponding angles of similar triangles are equal, so ∠A = ∠C and ∠D = ∠B.

Final answer

∠A = ∠C and ∠B = ∠D.