"Two triangles with a pair of equal angles are congruent."

Why is it necessary to have the side between the two angles be of the same length for both the triangles?

In the figure, in $△ABC, AB=AC=10 \text{cm} \text{and} BC=12 \text{cm}.$ $P \text{and} Q$ are the midpoints of $AB \text{and} AC$ respectively. $PM \text{and} RN$ are perpendiculars on $SQ$. If $BS : SR : RC=1 : 2 : 1$, then the length of $MN$ is:

In the adjoining figure, $\mathrm{X}$ and $\mathrm{Y}$ are respectively two points on equal sides $\mathrm{AB}$ and $\mathrm{AC}$ of $∆\mathrm{ABC}$ such that  $\mathrm{AX}=\mathrm{AY}$. Prove that $\mathrm{CX}=\mathrm{BY}$.

In the adjoining figure, $\mathrm{BM}\perp \mathrm{AC}$ and $\mathrm{DN}\perp \mathrm{AC}$. If $\mathrm{BM}=\mathrm{DN}$, prove that $\mathrm{AC}$ bisects $\mathrm{BD}$.

In the adjacent figure, $ABCD$ is a rectangle. If $BM$ and $DN$ are perpendiculars from $B$ and $D$ on $AC$, prove that $△BMC\cong △DNA$. Is it true that $BM=DN$?

In figure, $\angle P\cong \angle R$, $\mathrm{seg} PQ\cong \mathrm{seg} RQ$. Prove that, $△PQT\cong △RQS$

In $∆\mathrm{ABC}, \mathrm{AB}=\mathrm{AC}$ and the bisectors of $\angle \mathrm{B}$ and $\angle \mathrm{C}$ meet at a point $\mathrm{O}$. prove that $\mathrm{BO}=\mathrm{CO}$ and the ray $\mathrm{AO}$ is the bisector of $\angle \mathrm{A}$.

In the given figure, $ABCD$ is a quadrilateral in which $AD=BC$ and $\angle ADC=\angle BCD$. Show that the points $A, B, C, D$ lie on a circle.

Observe the information shown in pairs of triangles given below. State the test by which the two triangles are congruent. Write the remaining congruent parts of the triangles.

From the information shown in the figure.,

 In $△PTQ=△STR$

 $\angle PTQ=\angle STR$ ..... vertically opposite angles

 seg $TQ\cong$ seg $TR$

 $△PTQ\cong △STR$....._____ test

$\angle TPQ\cong$ _____ .....corresponding angles of congruent triangles.

_____ $\cong \angle TRS$ .....corresponding angles of congruent triangles.

seg $PQ\cong$ _____ ..... corresponding sides of congruent triangles.

Diagonals of parallelogram $ABCD$ intersect at $O$ as shown in Fig. $XY$contains $O$, and $X,$$Y$are points on opposite sides of the parallelogram. Give reasons for the following:

$∆BOY\cong ∆DOX$.

Now, state if $XY$ is bisected at $O$.

In figure, segment $PT$ is the bisector of $\angle QPR$. A line through $R$ intersects ray $QP$ at point $S$ and $RS\parallel PT$. Prove that $PS=PR$.

In the given figure, $\mathrm{ABCD}$ is a quadrilateral in which $\mathrm{AB}\parallel \mathrm{DC}$ and $\mathrm{P}$ is the midpoint of $\mathrm{BC}$. On producing, $\mathrm{AP}$ and $\mathrm{DC}$ meet at $Q$. Prove that $\mathrm{AB}=\mathrm{CQ}$.

$∆ABC$ is isosceles in which $AB=AC$. Segment $BD$ and segment $CE$ are medians. Show that $BD=CE$.

In the given figure, $\mathrm{OA}=\mathrm{OB}$ and $\mathrm{OP}=\mathrm{OQ}$. Prove that

$\mathrm{PX}=\mathrm{QX}$

In the given figure, $\mathrm{ABC}$ is an equilateral triangle, $\mathrm{PQ}\parallel \mathrm{AC}$ and $\mathrm{AC}$ is produced to $\mathrm{R}$ such that $\mathrm{CR}=\mathrm{BP}$. Prove that $\mathrm{QR}$ bisects $\mathrm{PC}$.

In the given figure, if $x=y$ and $\mathrm{AB}=\mathrm{CB}$, then prove that $\mathrm{AE}=\mathrm{CD}$.