EXERCISE

In an isosceles triangle $\mathrm{A}\mathrm{B}\mathrm{C},$ with $\mathrm{A}\mathrm{B}=\mathrm{A}\mathrm{C},$ the bisectors of $\mathrm{\angle }\mathrm{B}$ and $\mathrm{\angle }\mathrm{C}$ intersect each other at $\mathrm{O}.$ Join $\mathrm{A}$ to $\mathrm{O}.$ Show that: $\mathrm{A}\mathrm{O}$ bisects $\mathrm{\angle }\mathrm{A}$.
 

$\mathrm{A}\mathrm{B}\mathrm{C}$ is an isosceles triangle in which altitudes $\mathrm{B}\mathrm{E}$ and $\mathrm{C}\mathrm{F}$ are drawn to equal sides $\mathrm{A}\mathrm{C}$ and $\mathrm{A}\mathrm{B}$ respectively as shown in figure. Show that these altitudes are equal.

$\mathrm{A}\mathrm{B}\mathrm{C}$ is a triangle in which altitudes $\mathrm{B}\mathrm{E}$ and $\mathrm{C}\mathrm{F}$ to sides $\mathrm{A}\mathrm{C}$ and $\mathrm{A}\mathrm{B}$ are equal as shown in figure. Show that $\mathrm{\Delta ABE}\cong \mathrm{\Delta ACF}$.

 

$\mathrm{A}\mathrm{B}\mathrm{C}$ is a triangle in which altitudes $\mathrm{B}\mathrm{E}$ and $\mathrm{C}\mathrm{F}$ to sides $\mathrm{A}\mathrm{C}$ and $\mathrm{A}\mathrm{B}$ are equal as shown in figure. Show that $\mathrm{A}\mathrm{B}=\mathrm{A}\mathrm{C},$ i.e. $\mathrm{A}\mathrm{B}\mathrm{C}$ is an isosceles triangle.

$\mathrm{A}\mathrm{B}\mathrm{C}$ and $\mathrm{D}\mathrm{B}\mathrm{C}$ are two isosceles triangles on the same base $\mathrm{B}\mathrm{C}$ as shown in the figure. Show that $\mathrm{\angle }\mathrm{A}\mathrm{B}\mathrm{D}=\mathrm{\angle }\mathrm{A}\mathrm{C}\mathrm{D}$.

$\mathrm{\Delta }\mathrm{A}\mathrm{B}\mathrm{C}$ is an isosceles triangle in which $\mathrm{A}\mathrm{B}=\mathrm{A}\mathrm{C}.$ Side $\mathrm{B}\mathrm{A}$ is produced to $\mathrm{D}$ such that $\mathrm{A}\mathrm{D}=\mathrm{A}\mathrm{B}$ as shown in figure. Show that $\mathrm{\angle }\mathrm{B}\mathrm{C}\mathrm{D}$ is a right angle.

$\mathrm{A}\mathrm{B}\mathrm{C}$ is a right angled triangle in which $\mathrm{\angle }\mathrm{A}=90^{\mathrm{o}}$ and $\mathrm{A}\mathrm{B}=\mathrm{A}\mathrm{C}.$ Find $\mathrm{\angle }\mathrm{B}$ and $\mathrm{\angle }\mathrm{C}.$

Show that the angles of an equilateral triangle are $60^{\mathrm{o}}$ each.