R Chandra Solutions for Chapter: Triangles, Exercise 2: TEST YOUR SKILLS

Prove that angles opposite to equal sides of an isosceles triangle are equal.

Hint: In $\mathrm{\Delta }\mathrm{A}\mathrm{B}\mathrm{C}$ , draw $\mathrm{A}\mathrm{D}$ an angle bisector of $\mathrm{\angle }\mathrm{A}$. Now, in $\mathrm{\Delta }\mathrm{A}\mathrm{B}\mathrm{D}\hspace{0.17em}$and $\hspace{0.17em}\mathrm{\Delta }\mathrm{A}\mathrm{C}\mathrm{D}$ .
$\mathrm{A}\mathrm{D}=\mathrm{A}\mathrm{D}$ [Common]
$\mathrm{A}\mathrm{B}=\mathrm{A}\mathrm{C}$ [Given]
$\mathrm{\angle }\mathrm{B}\mathrm{A}\mathrm{D}=\mathrm{\angle }\mathrm{C}\mathrm{A}\mathrm{D}$ [Construction]
$\hspace{0.17em}\Rightarrow \mathrm{\Delta }\mathrm{A}\mathrm{BD}\cong \mathrm{\Delta }\mathrm{A}\mathrm{C}\mathrm{D}$

Prove that the sides opposite to equal angles of a triangle are equal.

In the figure, the bisector $\mathrm{A}\mathrm{D}$ of $\mathrm{\angle }\mathrm{A}$ is perpendicular to the side $\mathrm{B}\mathrm{C}$ of the $\mathrm{\Delta }\mathrm{A}\mathrm{B}\mathrm{C}$. Show that $\mathrm{A}\mathrm{B}=\mathrm{A}\mathrm{C}$ and thus, $\mathrm{\Delta }\mathrm{A}\mathrm{B}\mathrm{C}$ is isosceles.

In the figure, $\mathrm{E}$ and $\mathrm{F}$ are respectively the midpoints of equal sides $\mathrm{A}\mathrm{B}$ and $\mathrm{A}\mathrm{C}$ of $∆\mathrm{A}\mathrm{B}\mathrm{C}$. Show that $\mathrm{B}\mathrm{F}=\mathrm{C}\mathrm{E}$.

In an isosceles triangle $\mathrm{A}\mathrm{B}\mathrm{C}$ with $\mathrm{A}\mathrm{B}=\mathrm{A}\mathrm{C},\mathrm{}\mathrm{D}\mathrm{}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{}\mathrm{E}$ are points on $\mathrm{B}\mathrm{C}$ such that $\mathrm{B}\mathrm{E}=\mathrm{C}\mathrm{D}$ (see figure). Show that $\mathrm{A}\mathrm{D}=\mathrm{A}\mathrm{E}.$

In the figure, $\mathrm{A}\mathrm{D}=\mathrm{A}\mathrm{E}$ and $\mathrm{D}\mathrm{}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{}\mathrm{E}$ are points on $\mathrm{B}\mathrm{C}$ such that $\mathrm{B}\mathrm{D}=\mathrm{E}\mathrm{C}$. Prove that $\mathrm{A}\mathrm{B}=\mathrm{A}\mathrm{C}$.