State the SAS criteria of similarity of two triangles. In the given figure, it is given that OA · OC = OB · OD. Use the SAS criteria to prove that AD ∥ CB.

△ AOD ~ △ BOC (SAS), so ∠ DAB = ∠ CBA (alternate angles), hence AD ∥ CB.

MathsCBSE Class 10 · Basic · Board exam 2026, set 430/2/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2026: question 33

Set in the CBSE Class 10 board exam of 2026, paper 430/2/1, section D, question 33, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q33 · 5 marks · Long answerTriangles

State the SAS criteria of similarity of two triangles. In the given figure, it is given that $O A \cdot O C = O B \cdot O D$ . Use the SAS criteria to prove that $A D ∥ C B$ .

Figure in the paper: Two line segments AB and CD intersect at point O. A and D are on one side of O, and C and B on the other; A-O-B and C-O-D are straight segments. Points: A top-left, D bottom-left, C top-right, B bottom-right, O the intersection point. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 430/2/1.

Hints

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

Hint 1

Rewrite $O A \cdot O C = O B \cdot O D$ as a ratio of sides.

Hint 2

Which angle do triangles AOD and BOC share in size at O?

Hint 3

Similar triangles have equal corresponding angles; look for alternate angles on the transversal AB.

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.

SAS similarity criterion: if one angle of a triangle is equal to one angle of another triangle, and the sides including these angles are proportional, then the two triangles are similar.
1 mark
Given $O A \cdot O C = O B \cdot O D$ . Dividing both sides by $O B \cdot O C$ gives $\frac{O A}{O B} = \frac{O D}{O C}$ .
1 mark
In $△ A O D$ and $△ B O C$ : $\frac{O A}{O B} = \frac{O D}{O C}$ (shown above) and $∠ A O D = ∠ B O C$ [vertically opposite angles].
1 mark
$∴ △ A O D \sim △ B O C$ [SAS similarity], so the corresponding angles are equal: $∠ O A D = ∠ O B C$ , i.e. $∠ D A B = ∠ C B A$ .
1 mark
These are alternate interior angles made by the transversal AB with the lines AD and CB. Since they are equal, $A D ∥ C B$ . Hence proved.
1 mark

Answer: $△ A O D \sim △ B O C$ (SAS), so $∠ D A B = ∠ C B A$ (alternate angles), hence $A D ∥ C B$ .

Where the marks are

States the SAS similarity criterion correctly (1 mark)
Converts the given product into the ratio $O A / O B = O D / O C$ (1 mark)
Uses the vertically opposite angles at O (1 mark)
Concludes $△ A O D \sim △ B O C$ and the equal corresponding angles (1 mark)
Uses equal alternate angles to conclude $A D ∥ C B$ (1 mark)

Revise the chapter

Chapter: Triangles (notes, formulas, where marks are lost and every past question)
Topic: AAA and AA Similarity Criterion

Practise this chapterBoard-style questions from Triangles, one at a time, marked against the answer key and marking points.Back to the full paper2026 board paper, set 430/2/1 (Basic): 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 Basic 2026: question 33

Hints

Step-by-step solution

Revise the chapter

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