Congruence of Triangles

$UY$ and $XV$ are equal and perpendiculars to a line segment $XY$. Show that $UV$ bisects $XY$.

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]

If $AD$ is an altitude of an isosceles triangle $ABC$ in which $AB = AC$. Then:

$ABC$ is an isosceles triangle in which altitudes $BE$ and $CF$ are drawn to equal sides $AC$ and $AB$ respectively. Then:

Given that $△ABC\cong △PQR$, so explain what it means and if side $BC=7$ units, then find the length of side $QR$.

Given that $△ABC\cong △PQR$, so explain what it means and if side $AC=5$ units, then find the length of side $PR$.

If $E$ and $F$ are the midpoints of equal sides $AB$ and $AC$ of a triangle $ABC$. Then:

What are the Rules of Congruency?

Line $l$ is the bisector of an angle $\mathrm{\angle }A$ and $B$ is any point on $l$. $BP$ and $BQ$ are perpendiculars from $B$ to the arms of $\mathrm{\angle }A$. Is $△APB\cong △AQB$ (Yes/No) ?

In quadrilateral $ACBD, AC=AD$ and $AB$ bisects $\angle A$ (see given figure). If $△ABC\cong △ABD.$ Is $BC=$$BD$ (Yes/No) ?

$\mathrm{A}\mathrm{B}\mathrm{C}$ is an isosceles triangle in which altitudes $\mathrm{B}\mathrm{E}$ and $\mathrm{C}\mathrm{F}$ are drawn to equal sides $\mathrm{A}\mathrm{C}$ and $\mathrm{A}\mathrm{B}$ respectively as shown in figure. The altitudes of the triangle are _____. (equal/unequal)

Which of the following is not a criterion for congruence of triangles?

These shapes are congruent.

Which of the following figures are congruent to the square $ABCD$.

In figure $OA=OB$ and $OD=OC$. Show that $AD\parallel BC$.

In figure $OA=OB$ and $OD=OC$. Show that $△AOD\cong △BOC$.

Define the congruency of two triangles.

How can two squares have the same angles but not be congruent?

Two squares are congruent if both of them have the same length of the _____.