In the given figure ∠ ADE = ∠ ACB and AD/DB = AE/EC. Prove that ΔABC is an isosceles triangle.

∠ ABC = ∠ ACB AB = AC, so Δ ABC is isosceles.

MathsCBSE Class 10 · Basic · Board exam 2025, set 430/4/1 · 2 marks · Very short answer

CBSE Class 10 Mathematics Basic 2025: question 25

Set in the CBSE Class 10 board exam of 2025, paper 430/4/1, section B, question 25, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q25 · 2 marks · Very short answerTriangles

In the given figure $∠ A D E = ∠ A C B$ and $\frac{A D}{D B} = \frac{A E}{E C}$ . Prove that $Δ$ ABC is an isosceles triangle.

Figure in the paper: Triangle ABC with A at the top. D is on side AB and E on side AC; DE is drawn parallel to BC. Angles at D and C are marked equal. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 430/4/1.

Hints

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

Hint 1

What does the equal-ratio condition tell you about DE and BC?

Hint 2

Once DE ∥ BC, look for corresponding angles at D and B.

Hint 3

Equal base angles give equal opposite sides.

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.

Since $\frac{A D}{D B} = \frac{A E}{E C}$ , we get $D E ∥ B C$ [converse of Basic Proportionality Theorem].
½ mark
Therefore $∠ A D E = ∠ A B C$ [corresponding angles, DE ∥ BC, AB is the transversal].
½ mark
It is given that $∠ A D E = ∠ A C B$ , so $∠ A B C = ∠ A C B$ .
½ mark
Sides opposite equal angles are equal, so $A C = A B$ . Hence $Δ A B C$ is isosceles.
½ mark

Answer: $∠ A B C = ∠ A C B \Rightarrow A B = A C$ , so $Δ A B C$ is isosceles.

Where the marks are

$D E ∥ B C$ by converse of BPT (½ mark)
$∠ A D E = ∠ A B C$ (corresponding angles) (½ mark)
$∠ A B C = ∠ A C B$ (½ mark)
$A B = A C$ , so the triangle is isosceles (½ mark)

Revise the chapter

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

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 430/4/1 (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 2025: question 25

Hints

Step-by-step solution

Revise the chapter

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