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 $\angle EDF$

The circumcentre of triangle $ABC$is $O$. If angle $OBA=35°$ and $\angle OCA=25°$, then the value of angle $OBC$ is

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 in which $AB=AC$ and $LM\hspace{0.17em}$ is parallel to $BC$. If $\angle A=60°$, find $\angle LMC$.

If the altitude from one vertex of a triangle bisects the opposite side. Prove that the triangle is isosceles.

In a $△PQR$, $X \text{and} Y$ are the points on $PQ \text{and} QR$ respectively. If $PQ = QR$ and $QX = RY$, show that $PX= RY$

In the given figure, if $AB=AC$ and $DB=DC$, then find the ratio of $\angle ABD \text{to} \angle ACD$.

In the given figure, $AB=AC$ and $\angle 1=\angle 2$, then prove that $\angle PBC=\angle PCB$.

In $△PQR$, if $\angle R=\angle P$, $QR=4 \mathrm{cm}$ and $PR=5\text{ cm}$, then find the length of $PQ$.

In the figure given below, two isosceles right-angled triangles, $DEF$ and $HGI$ are on the same base $\overline{DH}$. 

 $\overline{DH}$ is parallel to $\overline{FI}$.

If $DE=GH=9 \mathrm{cm}$ and $DH=20 \mathrm{cm}$, then the area of the quadrilateral $FEGI$ is

If $\angle ACB$ is a right angle and $AC=CD$ and $CDEF$ is a parallelogram. If $\angle FEC=10°$, the $\angle BDE$ equals

In the given figure, $ABCD$ is a square and $EF$ is parallel to diagonal $BD$ and $EM=FM$.

Prove that $DF=BE$

In $∆ABC$, the sides $AB, AC$ are equal and the base $BC$ is produced to any point $D$. From $D, DE$ is drawn perpendicular to $BA$ produced and $DF$ perpendicular to $AC$ produced. Prove that $BD$ bisects $\angle EDF$.

In $∆ABC, AB=AC$ and $\angle B=\frac{2}{5}$th of $\angle A$. Find the measure of $\angle A$.

In the given figure, $\angle BAC=85°, CA=CB$ and $BD=CD$. Find the measures of $\angle x, \angle y$ and $\angle z$.

In right angled $∆ABC$ right angled at $B$, such that $\angle BCA=2\angle CAB$, show that hypotenuse $AC=2BC$.

$ABC$ 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 $a,b,c,d.$