O P Malhotra, S K Gupta and, Anubhuti Gangal Solutions for Chapter: Congruency of Triangles, Exercise 3: Exercise

If $\mathrm{M}\mathrm{P}=\mathrm{N}\mathrm{Q}$ prove that $∆\mathrm{s} \mathrm{O}\mathrm{M}\mathrm{P}$ and $\mathrm{O}\mathrm{N}\mathrm{Q}$ are congruent and hence prove that $\mathrm{M}\mathrm{N}$ and $\mathrm{P}\mathrm{Q}$ bisect each other at $\mathrm{O}$.

$\mathrm{\angle }\mathrm{A}\mathrm{B}\mathrm{C}$ is bisected by $\mathrm{B}\mathrm{X}$. $\mathrm{O}$ is any point on $\mathrm{B}\mathrm{X}$ and $\mathrm{P}\mathrm{O}\mathrm{Q}$ is perpendicular to $\mathrm{B}\mathrm{X}$. Prove that $\mathrm{P}\mathrm{O}=\mathrm{O}\mathrm{Q}$.

Prove that $\mathrm{O}\mathrm{X}$ bisect $\angle \mathrm{P}\mathrm{O}\mathrm{Q}$.

In the figure, $\mathrm{P}\mathrm{S}=\mathrm{Q}\mathrm{R}$ and $\mathrm{P}\mathrm{S}\parallel \mathrm{Q}\mathrm{R}$. Prove that $\mathrm{P}\mathrm{Q}=\mathrm{S}\mathrm{R}$.

In the given figure, triangle $\mathrm{A}\mathrm{B}\mathrm{C}$ and $\mathrm{D}\mathrm{C}\mathrm{B}$ are right-angled at $\mathrm{A}$ and $\mathrm{D}$ respectively and $\mathrm{A}\mathrm{C}=\mathrm{D}\mathrm{B}$. Prove that $\mathrm{\Delta }\mathrm{A}\mathrm{B}\mathrm{C}\cong \mathrm{\Delta }\mathrm{D}\mathrm{C}\mathrm{B}$.

Find the size of each angle of an equilateral triangle. This can be found from the following considerations:
Why is $\mathrm{\angle }\mathrm{B}=\mathrm{\angle }\mathrm{C}?$ Why is $\mathrm{\angle }\mathrm{C}=\mathrm{\angle }\mathrm{A}?$ Why $\mathrm{\angle }\mathrm{A}=\mathrm{\angle }\mathrm{B}$? $\therefore \mathrm{\angle }\mathrm{A}=\mathrm{\angle }\mathrm{B}=\mathrm{\angle }\mathrm{C}$.

Find the size of each angle of an equilateral triangle.
This can be found from the following considerations:
What $\mathrm{\angle }\mathrm{A}+\mathrm{\angle }\mathrm{B}+\mathrm{\angle }\mathrm{C}$ equal to?
$\therefore$ Each angle $=$_____?

If two angles of a triangle are equal, then prove that sides opposite those angles are also equal.