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# ABCD is a rectangle in which diagonal BD bisects ∠B. Show that ABCD is a square.

**Solution:**

Given, ABCD is a rectangle

The diagonal BD bisects ∠B.

We have to show that ABCD is a square.

Join the diagonal AC

Considering triangles BAD and BCD,

Since BD is the bisector of ∠B

∠ABD = ∠CBD

∠A = ∠C = 90°

Common side = BD

The ASA congruence rule states that if any two consecutive angles of a triangle along with a non-included side are equal to the corresponding consecutive angles and the non-included side of another triangle, the two triangles are said to be congruent.

By ASA criteria, the triangles BAD and BCD are congruent.

The Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem states that when two triangles are congruent, then their corresponding sides and angles are also congruent or equal in measurements.

By CPCTC,

AB = BC

AD = CD

We know that the opposite sides of a rectangle are equal.

AB = CD

BC = AD

So, AB = BC = CD = AD

Therefore, ABCD is a square.

**✦ Try This: **Two parallel lines l and m are intersected by a transversal t. Show that the quadrilateral formed by the bisectors of interior angles is a rectangle.

**☛ Also Check:** NCERT Solutions for Class 9 Maths Chapter 8

**NCERT Exemplar Class 9 Maths Exercise 8.4 Problem 15**

## ABCD is a rectangle in which diagonal BD bisects ∠B. Show that ABCD is a square.

**Summary:**

ABCD is a rectangle in which diagonal BD bisects ∠B. It is shown that ABCD is a square

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