In the given figure, OA · OB = OC · OD. Show that ∠ A = ∠ C and ∠ B = ∠ D.

△ AOD ~ △ COB (SAS), so ∠ A = ∠ C and ∠ B = ∠ D

MathsCBSE Class 10 · Standard · Board exam 2025, set 30/2/1 · 2 marks · Very short answer

CBSE Class 10 Mathematics Standard 2025: question 24(b)

Set in the CBSE Class 10 board exam of 2025, paper 30/2/1, section B, question 24(b), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q24(b) · 2 marks · Very short answerTriangles

In the given figure, $O A \cdot O B = O C \cdot O D$ . Show that $∠ A = ∠ C$ and $∠ B = ∠ D$ .

Figure in the paper: Two line segments AB and CD intersect at point O. A is at the upper-left, D at the lower-left, C at the upper-right and B at the lower-right. Triangles OAD and OCB are formed by joining A to D and C to B. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/2/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Turn the product equation into a ratio of sides.

Hint 2

Which two triangles have sides OA, OD and OC, OB? What angle do they share at O?

Hint 3

Use the SAS similarity criterion, then match the corresponding angles.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Given $O A \cdot O B = O C \cdot O D$ .
Dividing both sides by $O C \cdot O B$ : $\frac{O A}{O C} = \frac{O D}{O B}$ .
½ mark
In $△ A O D$ and $△ C O B$ : $∠ A O D = ∠ C O B$ [vertically opposite angles].
½ mark
Two sides are proportional and the included angles are equal, so $△ A O D \sim △ C O B$ [SAS similarity criterion].
½ mark
Corresponding angles of similar triangles are equal (A↔C, D↔B). Therefore $∠ A = ∠ C$ and $∠ D = ∠ B$ . Hence proved.
½ mark

Answer: $△ A O D \sim △ C O B$ (SAS), so $∠ A = ∠ C$ and $∠ B = ∠ D$

Where the marks are

Rewrites the given product as the ratio $\frac{O A}{O C} = \frac{O D}{O B}$ (1/2 mark)
States $∠ A O D = ∠ C O B$ as vertically opposite angles (1/2 mark)
Concludes $△ A O D \sim △ C O B$ by SAS (1/2 mark)
Deduces $∠ A = ∠ C$ and $∠ B = ∠ D$ (1/2 mark)

Revise the chapter

Chapter: Triangles (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Triangles, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 30/2/1 (Standard): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Standard 2025: question 24(b)

Hints

Step-by-step solution

Revise the chapter

More questions from Triangles