Manipur Board Solutions for Chapter: Lines and Angles, Exercise 4: EXERCISE 6.4

If the bisectors of the base angles of a triangle enclose an angle of $135°$, prove that the triangle is a right triangle.

In any regular polygon of $n$ sides, prove that each of its angles measures $\frac{\left(n-2\right)}{n}\times 180°$.

The sides $AB\hspace{0.17em}\text{and} AC$ of the $∆ABC$ are produced to $P \text{and} Q$ respectively. The bisectors of $\angle PBC \text{and} \angle QCB$ intersects at $O$. Prove that $\angle BOC=90°-\frac{1}{2}\angle BAC$.

An exterior angle of a triangle is $120°$ and one of the interior opposite angles is $20°$. Find the other two angles of the triangle.

$ABC$ is a triangle. The bisector of the exterior angle at $B$ and the bisector of $\angle C$ intersect each other at $D$. Prove $\angle BDC=\frac{1}{2}\angle A$.

In a $∆ABC, AD$ bisects $\angle A \text{and} \angle C>\angle B$. Prove that $\angle ADB>\angle ADC$.

The side $BC$ of a $∆ABC$ is produces to $D$ to form the exterior angle  $\angle ACD$. The bisector of $\angle BAC$ at $BC \text{at} E$. Prove that $\angle ABC+\angle ACD=2\angle AEC$.

$ABC$ is an isosceles triangle in which $AB=AC$ and $AE$ bisects the exterior angle $CAD \text{at} A$. Prove that $AE\parallel BC$.