Question

Assertion: In ΔABC, M and N are midpoints of sides AB and AC. ∠BMN = 6x and ∠B = 2x. ∴ x = 22.5°.

Reason: Co-interior angles of parallel lines are supplementary.

Options

• Both A and R are true and R is the correct reason for A.

• Both A and R are true but R is the incorrect reason for A.

• A is true but R is false

• A is false but R is true.

Answer 1

Both A and R are true and R is the correct reason for A.

Explanation:

Assertion (A):

In ΔABC, M and N are midpoints of sides AB and AC.

Given:

• ∠BMN = 6x
• ∠B = 2x

We are to find:

• x = 22.5°.

Reason (R):

Co-interior angles of parallel lines are supplementary.

Understanding the Geometry:

• Since M and N are midpoints of AB and AC, the segment MN is parallel to BC by the Midpoint Theorem.
• In triangle ABC, consider ∠BMN and ∠B.
• ∠BMN and ∠B are co-interior angles on the same side of the transversal (segment BM) and between the lines MN || BC.

Using Co-Interior Angles:

• Co-interior angles formed between two parallel lines and a transversal add up to 180°.

So, ∠BMN + ∠B = 180°

⇒ 6x + 2x = 180°

⇒ 8x = 180°

⇒ x = 22.5°