Theorems and Related Questions in Isosceles Triangle

In Fig. 8.59, AB = BD, AD = CD and $\angle B = 90°.$ Find $\angle BAC.$(give answer in degree)
 

In Fig. 8.57, AB = BC, AD = CD, $\angle ABC = 110°$ $\angle ADC = 40°.$ Calculate $\angle BCD.$(give ans in degree)

In an isosceles triangle ABC, with AC=BC, the line CD bisects AB at D and $\angle$CAB=${52}^{°}$. Find $\angle$CBD.(Give answer in degree)

In an isosceles triangle $ABC$, with $AC=BC$, the line $CD$ bisects $AB$ at $D$ and $\angle CAB=$${52}^{°}$. Find $\angle DCB$(Give answer in degree).

In Fig., $AB=AC; BC=CD$, $\angle BAF={130}^{°}$ and $DE\parallel BC$. Calculate: $\angle DCE$(give answer in degree)

In Fig., AB=AC; BC=CD, $\angle$BAF=${130}^0$ and DE||BC. Calculate: $\angle CDE$(Give answer in degree)

The vertical angle of an isosceles triangle is ${70}^0$. Find each of its angles. (Give answer in degree)

D is a point on the side BC of a $△ABC$ such that AB=BD=AD=DC. Show that $\angle ADC=4\angle ACD.$

In $△PQR$, $\angle Q={90}^o$and $S$ is a point on $PR$ such that $\angle SQR=\angle SRQ$. Prove that $SR=SP$.

In a triangle $ABC$, the bisectors of Angle $B$ and angle $C$ meet at $O$. If $OB=OC$, then prove that triangle $ABC$ is an isosceles triangle.

In figure $AB=AC$ and Angle $BAD=$angle $CAE$. Prove that $AD=AE$.

In Figure$PQR$ is such that $PQ=PR$ and $S$ is a point on $QR$ produced. $ST$ is
perpendicular to $QP$ produced and $SM$ is perpendicular to $PR$ produced. Prove that QS bisects $\angle TSM$

Show that angles of an equilateral triangle are $60°$each.

In Fig. 8.58, $PQR$ is an isosceles triangle with $PQ=PR$. $QP$ is produced to $S$ such that $PQ=PS$. Prove that $\angle QRS$ is a right angle.

In Fig. $ABC$ is a triangle. The bisector of $\angle ABC$ meets $AC at D. A$ point $E$ lies on $BD$such that $AE=AD.$ Prove that $\angle BAE = \angle ACB.$

If the bisector of the exterior vertical angle of a triangle is parallel to the base, then show that the triangle is isosceles.

In given figure $AB=BC, CD\parallel AB$ and $CD$ is the bisector of $\angle ACE$. Prove the $△ABC$ is an equilateral triangle.

Through any point in the bisector of an angle, a straight line is drawn parallel to either arm of the angle. Prove that the triangle so formed is isosceles.

Two lines $AB$ and $CD$ intersect at $O$ such that $BC$ is equal and parallel to $AD$, Prove that the lines $AB$ and $CD$ bisects each other at $O$. 

$BE$ and $CF$ are, respectively, the perpendicular to the side $AC$ and $AB$ of a $△ABC$. If $BE=CF$, prove that $△ABC$ is isosceles.