In the given figure, $OA \cdot OB = OC \cdot OD$. Show that $\angle A = \angle C$ and $\angle B = \angle D$.
Generated by claude-sonnet-4-6 · 2026-06-15 10:22 · grounding rag
Model Answer
Given: OA · OB = OC · OD
$$\Rightarrow \frac{OA}{OC} = \frac{OD}{OB} \quad \cdots(1)$$
Also, ∠AOD = ∠COB (Vertically opposite angles) $\cdots$(2)
From (1) and (2), △AOD ~ △COB (SAS similarity criterion)
∴ ∠A = ∠C and ∠D = ∠B (Corresponding angles of similar triangles) [Proved]
Source: Chapter 6, Section 6.4, Example 6
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Explanation
- The key step is rearranging OA·OB = OC·OD into a ratio form: OA/OC = OD/OB (sides including the vertically opposite angle are proportional).
- Vertically opposite angles at O are always equal — state this explicitly.
- These two facts together satisfy SAS similarity, which then gives the equal corresponding angles.
- Examiners award marks for: (1) correct ratio, (2) vertically opposite angles, (3) naming SAS criterion, (4) concluding with corresponding angles.