Corresponding Parts of Congruent Triangles (CPCT)

$ABCD$ is a paralleogram. $M$ is the mid-point of $BC$. Show that, $DC=CP$

$PQR$ is an equilateral triangle. $QM$ and $RN$ are medians. Prove that, $QM=RN$

Prove that if altitudes from two vertices of a triangle to the opposite sides are equal, then the triangle is isosceles.

$ABCD$ is a quadrilateral in which $AP$ and $CQ$ are perpendicular to diagonal $BD$ and $AP=CQ$. Prove that, $BD$ bisects $AC$

In $∆ABC$, $AB=AC$ and $AP=AQ$. Prove that, $CP=BQ$.

In the given figure, $AD=BC$,$AC=BD$. Prove that, $\angle ADB=\angle ACB$

Prove that, $∆ABD\cong ∆CBD$. Find the values of $x$ and $y$, if $\angle ABD=35°$, $\angle CBD=\left(3x+5\right)°$, $\angle ADB=\left(y-3\right)°$, $\angle CDB=25°$

In $∆AOB$ and $∆COD$, $\angle B=\angle C$ and $O$ is the mid-point of $BC$. Find the values of $x$ and $y$ if $AB=3x$ units, $CD=y+2$ units, $AO=x+2$ units, $DO=y$ units.

$PS$ bisects $\angle QPR$ and $PS\perp QR$. If $PQ=2x$ units, $PR=\left(3y+8\right)$ units, undefined units and $SR=2y$ units. Find the value of $x$ and $y$

In the given triangles, $AC=DF, BD=CE$ and $\angle ACB=\angle FDE$. Prove that, $\angle A=\angle F$

In the quadrilateral $ABCD$, $AD=CD$ and $\angle A=90°=\angle C$. Prove that, $AB=BC$

In $∆\mathrm{ABC}$, $\mathrm{AB}=\mathrm{AC}$, and $\mathrm{CP}$ and $\mathrm{BQ}$ are altitudes. Prove that, $\mathrm{CP}=\mathrm{BQ}$

In $∆\mathrm{PQR}$, $\mathrm{PQ}=\mathrm{PR}$, $\mathrm{QN}=\mathrm{RM}$. Prove that, $\angle \mathrm{QPM}=\angle \mathrm{RPN}$