A diagonal of a parallelogram bisects one of its angle. Prove that it will bisect its opposite angle also.

Solution:

Consider a parallelogram ABCD

Join the diagonal AC

AC bisects angle A

We have to prove that AC will bisect its opposite angle also.

Since AC bisects angle A

∠CAB = ∠CAD ------------------ (1)

We know that the opposite sides of a parallelogram are parallel and congruent.

So, AB||CD and AC is a transversal.

We know that the alternate interior angles are equal.

∠CAB = ∠ACD ----------------- (2)

Similarly, AD||BC and AC is transversal.

∠DAC = ∠ACB ----------------- (3)

From (1), (2) and (3),

∠BCA = ∠DCA

This implies AC bisects the opposite angle C.

Therefore, it is proven that AC bisects its opposite angle also.

✦ Try This: ABCD is a parallelogram in which diagonal AC bisects ∠BAD. If ∠BAC =35°, then ∠ABC is equal to

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

NCERT Exemplar Class 9 Maths Exercise 8.4 Sample Problem 4

A diagonal of a parallelogram bisects one of its angle. Prove that it will bisect its opposite angle also.

Summary:

The (interior) bisector of an angle, also called the internal angle bisector, is the line or line segment that divides the angle into two equal parts. A diagonal of a parallelogram bisects one of its angles. It is proven that it will bisect its opposite angle also

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