Explain ASA congruency

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

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

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

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

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 $\mathrm{PC}$.

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

In $△ABC$ and $△DEF$, three equality relations between same parts are: $AB=ED$, $\angle B=\angle D$, $BC=DF$, then which of the following congruence conditions applies?

In the adjacent figure, $ABCD$ is a parallelogram and line segments $AE$ and $CF$ bisect the angles $A$ and $C$ respectively. Show that $AE\parallel CF$.

In $△PQR,PQ=10 \mathrm{cm},QR=12 \mathrm{cm},PR=8 \mathrm{cm}$. Find out the greatest and the smallest angle of the triangle.

In the adjoining figure, If $O$ is midpoint of $BM$, explain how one can find the breadth of the river without crossing it.