Criteria for Congruence of Triangles

In triangle $∆\mathrm{ABC}, \angle \mathrm{B}=\angle \mathrm{C}$ and $\mathrm{D}$ is the midpoint of $\mathrm{BC}$, show that $∆\mathrm{ABD}\cong ∆\mathrm{ACD}$.

$ABC$ is an equilateral triangle, $AD\text{ and} BE$ are perpendiculars to $BC\text{ and }AC$ respectively. Prove that,

$\left(i\right)$  $AD= BE$ $\left(ii\right) BD=CE$

In the given figure, $RQ$ and $SP$ are equal perpendiculars to $PQ$. The length of $RO$ is always equal to the length of

Explain ASA congruency

Three angles of one triangle are respectively equal to the three angles of the other. Further any one side of the two triangles being equal is sufficient to prove the two triangles congruent.

"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?

Rita says, 'For two triangles to be congruent, any three parameters of the six ($3$ sides and $3$ angles) should be equal.'

Give examples in favour of and against her statement.

In the given figure, $AB= EF, BC= DE, AB\perp BD\text{ and} FE\perp CE$. Prove that $∆ABD\cong ∆FEC$

In the given figure, $\mathrm{BD}$ is the angle bisector of $\angle \mathrm{B}$ and $\mathrm{BD}\perp \mathrm{AC}$. Prove that $\mathrm{ABC}$ is an isosceles triangle.

In figure, $BC$ and $\overline{AD}$ intersect at $E$. 

Prove that $△CED\cong △BEA$

In the figure below, O is the mid-point of AB and CD, prove that $AC=BD$.

In the given figure, two circles with centres $O\text{ and} O\text{'}$ intersects at $X \text{and} Y$. $PQ$ is a chord of one circle parallel to $XY$ (as shown in the figures). Prove that $O\text{'}O \text{or} OO\text{'}$ produced bisects $PQ$.

In the given figure, $∆\mathrm{LMN}$ is an isosceles triangle with $\mathrm{ML}=\mathrm{NL}$. Also $\mathrm{OL}$ is the angular bisector of $\angle \mathrm{MLN}$.

Use the ASA criterion to prove that $∆\mathrm{MLO}\cong ∆\mathrm{NLO}$. Also show that $\mathrm{O}$ is the mid-point of $\mathrm{MN}$.

$\mathrm{QR}$ is a tangent at $\mathrm{Q}$ to a circle whose center is $\mathrm{P}$, $\mathrm{PR} \parallel \mathrm{AQ}$, where $\mathrm{AQ}$ is a chord through $\mathrm{A}$, the end point of the diameter $\mathrm{AB}$. Prove that $\mathrm{BR}$ is tangent at $\mathrm{B}$.

In the figure, is $△ACD\cong △BCD$? If yes, write the condition of congruence.

If line segment $\mathrm{AB} \mathrm{and} \mathrm{CD}$ bisect each other at $O$ then using side angle side congruence rule prove that $△AOC$$\cong △BOD$.

$△ABC and △DBC$ are two isosceles triangles. Show that $AP$ is the perpendicular bisector of side $BC$.

In $△ABC, \angle A=90°$ and $AB=AC$. Bisector of $\angle A$ meets $BC$ at $D$. Prove $BC=2AD$.

In Fig. X and Y are two points on equal sides $AB$ and $AC$ of a $∆ABC$ such that $AX = AY$. Prove that $XC = YB$.
  

$AD$ and $BC$ are equal perpendiculars to a line segment $AB$. Then $CD$ _____ $AB$. [bisects/trisects]