ABC is a triangle in which altitudes $BE$ and $CF$ to sides $AC$ and $AB$ are equal (see Fig. 7.32). Show that (i) $\triangle ABE \cong \triangle ACF$

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

In an isosceles triangle ABC, with $AB = AC$, the bisectors of $\angle B$ and $\angle C$ intersect each other at $O$. Join $A$ to $O$. Show that : (i) $OB = OC$

ABC is a triangle in which altitudes $BE$ and $CF$ to sides $AC$ and $AB$ are equal (see Fig. 7.32). Show that (ii) $AB = AC$, i.e., ABC is an isosceles triangle.

In $\triangle ABC$, $AD$ is the perpendicular bisector of $BC$ (see Fig. 7.30). Show that $\triangle ABC$ is an isosceles triangle in which $AB = AC$.

$\triangle ABC$ is an isosceles triangle in which $AB = AC$. Side $BA$ is produced to $D$ such that $AD = AB$ (see Fig. 7.34). Show that $\angle BCD$ is a right angle.

In an isosceles triangle ABC, with $AB = AC$, the bisectors of $\angle B$ and $\angle C$ intersect each other at $O$. Join $A$ to $O$. Show that : (ii) $AO$ bisects $\angle A$

ABC is a right angled triangle in which $\angle A = 90^{\circ}$ and $AB = AC$. Find $\angle B$ and $\angle C$.

ABC and DBC are two isosceles triangles on the same base $BC$ (see Fig. 7.33). Show that $\angle ABD = \angle ACD$.

ABC is an isosceles triangle in which altitudes $BE$ and $CF$ are drawn to equal sides $AC$ and $AB$ respectively (see Fig. 7.31). Show that these altitudes are equal.