Properties of a Triangle

In $∆ABC$, $AB=AC$ and $BC$ is produced to $D$. From $D$, $DE$ is drawn perpendicular to $BA$ produced and $DF$ is drawn perpendicular to $AC$ produced. Prove that $BD$ bisects $\angle EDF$

In $∆ABC$, $AB=AC$ and $BC$ is produced to $D$. From $D$, $DE$ is drawn perpendicular to $BA$ produced and $DF$ is drawn perpendicular to $AC$ produced. Prove that $BD$ bisects $\angle EDF$

Does the median bisect the angles?

The figure below consists of a square and an equilateral triangle connected together with a common side.

What is the measure of $x$?

In the diagram below, points $Q \text{and} S$ lie on a circle with centre $O$. $SR$ is a tangent to the circle at $S$. $\angle QRS= 40° \text{and} \angle SOQ= 80°$. Prove that $∆QSR$ is isosceles.

$ABC$ is an isosceles triangle with $\angle A={40}^{\circ }$ and $AB=AC.$

$ABDE$ is a square drawn on the side $AB$ of $∆ABC.$

Find the measure of angles marked as $a,b,c,d.$

In $△ABC$, $AD\perp BC$ at $D$ and $AE$ is the bisector of $\angle A$. If $\angle B=62°$ and $\angle C=36°$, then what is the measure of $\angle DAE$?

If one of the three angles of the equilateral triangle measures $60°$, then find the other two angles.

All the sides of an equilateral triangle are

All the sides of an equilateral triangle are equal.

All the three angles of the equilateral triangle have the same measure.

Prove that equilateral triangle is equiangular.

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 $\mathrm{O}.$ Is $\mathrm{O}\mathrm{B}=\mathrm{O}\mathrm{C}$ (Yes/No).

An exterior angle of a triangle is $110°$ and one of the interior opposite angle is $30°$. Then the other interior angle is $K\times 40°$. Find$K$.

$ABCD$ is a square and $ABE$ is an equilateral triangle outside the square then

In the figure, $\mathrm{PA}$ is a tangent to the circle, $\mathrm{PBC}$ is secant and $\mathrm{AD}$ bisects angle $\mathrm{BAC}.$ Show that triangle $\mathrm{PAD}$ is an isosceles triangle. Also, show that $\angle \mathrm{CAD}=\frac{1}{2} [\angle \mathrm{PBA}-\angle \mathrm{PAB}].$

Tangent at $\mathrm{P}$ to the circumcircle of triangle $\mathrm{PQR}$ is drawn. If this tangent is parallel to side $\mathrm{QR}$, show that $△\mathrm{PQR}$ is isosceles.
 

$\mathrm{AB}$ is the diameter and $\mathrm{AC}$ is a chord of a circle with centre $\mathrm{O}$ such that angle $\mathrm{BAC}=30°.$ The tangent to the circle at $\mathrm{C}$ intersects $\mathrm{AB}$ produced in $\mathrm{D}.$ Show that $\mathrm{BC}=\mathrm{BD}.$
 

In the given figure, if $x=k°$ then, find the value of $k$.

In $\Delta ABC$ if $AB=BC$, then