R. D. Sharma Solutions for Chapter: Congruent Triangles, Exercise 3: EXERCISE

In an isosceles triangle, if the vertex angle is twice the sum of the base angles, calculate the angles of the triangle.

Prove that each angle of an equilateral triangle is $60°$.

Angles$A, B, C$ of a triangle$ABC$ are equal to each other. Prove that $∆ABC$ is equilateral.

$ABC$ is a right-angled triangle in which $\angle A=90°$ and $AB=AC.$ Find $\angle B and \angle C.$

$PQR$ is a triangle in which $PQ=PR$ and $S$ is any point on the side $PQ$. Through $S$, a line is drawn parallel to $QR$ and intersecting $PR$ at $T$. Prove that $PS=PT$.

In a $∆ABC,$ it is given that $AB=AC$ and the bisectors of $\angle B and \angle C$ intersect at $O$. If $M$ is a point on $BO$ produced, prove that $\angle MOC=\angle ABC.$

$P$ is a point on the bisector of an angle $\angle \mathrm{ABC}$. If the line through $P$ parallel to $AB$meets $BC$ at $Q$, prove that  $∆BPQ$ is isosceles.

$ABC$ is a triangle in which $\angle B=2 \angle C$ . $D$ is a point on $BC$such that $AD$bisects $\angle BAC$ and $AB=CD .$Prove that $\angle BAC=72°$