Q. $ABCD$ is a quadrilateral in which $AD=BC$ and $\angle DAB=\angle CBA$ . Prove that
(i) $△ABD\cong △BAC$
(ii) $BD=AC$
(iii) $\angle ABD=\angle BAC$

Q. Prove that the angles of an equilateral triangle are ${60}^{\circ }$ each.

Q.

ABC is an isosceles triangle in which altitudes $BE$ and $CF$ are drawn to equal sides $AC$ and $AB$ respectively. Show that $\Delta ABE\cong \Delta ACF$

Q. In given figure $AD$ and $BC$ are equal perpendiculars to a line segment $AB$. Show that $CD$ bisects $AB.$

Q. $l$ and $m$ are two parallel lines intersected by another pair of parallel lines $p$ and $q$. Show that $△ABC\cong △CDA$

Q. In given figures two sides $AB$ and $BC$ and median $AM$ of one triangle $ABC$ are respectively equal to sides $PQ$ and $QR$ and median $PN$ of $△PQR$. Show that:
(i) $△ABM\cong △PQN$
(ii) $△ABC\cong △PQR$

Q.

$ABC$ is a triangle in which altitudes $BE$ and $CF$ to sides $AC$ and $AB$ are equal (see Fig.). Show that
(i) $△ABE\cong △ACF$
(ii) $AB=AC$, i.e., $ABC$ is an isosceles triangle.

Q. $AD$ is an altitude of an isosceles triangle $ABC$ in which $AB=AC$. Show that
(i) $AD$ bisects $BC$ (ii) $AD$ bisects $\angle A$

Q. $ABC$ is a right angled triangle in which $\angle A={90}^{\circ }$ and $AB=AC$. Find $\angle B$ and $\angle C$.

Q. $△ABC$ and $△DBC$ are two isosceles triangles on the same base $BC$ and vertices $A$ and $D$ are on the same side of $BC$. If $AD$ is extended to intersect $BC$ at $P$, show that
(i) $△ABD\cong △ACD$
(ii) $△ABP\cong △ACP$
(iii) $AP$ bisects $\angle A$ as well as $△D$.
(iv) $AP$ is the perpendicular bisector of $BC$.

Q. $AB$ is a line segment and $P$ is its mid-point. $D$ and $E$ are points on the same side of $AB$ such that $\angle BAD=\angle ABE$ and $\angle EPA=\angle DPB$. Show that
(i) $△DAP\cong △EBP$
(ii) $AD=DE$

Q. $BE$ and $CF$ are two equal altitudes of a triangle $ABC$. Using RHS congruence rule, prove that the triangle $ABC$ is isosceles.

Q. In right angled triangle $ABC$, right angled at $C,M$ is the mid-point of hypotenuse $AB$. $C$ is joined to $M$ and produced to a point $D$ such that $DM=CM$. Point $D$ is joined to point $B$. Show that:
(i) $△AMC\cong △BMD$
(ii) $\angle DBC$ is a right angle.
(iii) $△DBC\cong △ACB$
(iv) $CM=\frac{1}{2}AB$

Q. $△ABC$ is an isosceles triangle in which $AB=AC$. Sides $BA$ is produced to $D$ such that $AD=AB$. Show that $\angle BCD$ is a right angle.

Q. In given figure $ABC$ and $DBC$ are two isosceles triangles on the same base $BC$. Show that $\angle ABD=\angle ACD$.

Q. In Fig, $AC=AE,AB=AD$ and $\angle BAD=\angle EAC$. Show that $BC=DE$

Q. Line $l$ is the bisector of an angle $\angle A$ and $B$ is any point on $l$. $BP$ and $BQ$ are perpendiculars from $B$ to the arms of $\angle A$. Show that:
(i) $△ABP\cong △AQB$
(ii) $BP=BQ$ or $B$ is equidistant from the arms of $\angle A$.

Q. In an isosceles triangle $ABC$, with $AB=AC$, the bisectors of $\angle B$ and $\angle C$ intersect each other at $O$. Join $A$ to $O$. Show that :
(i) $OB=OC$ (ii) $AO$ bisects $\angle A$

Q.

In quadrilateral $ACBD,AC=AD$ and $AB$ bisects $\angle A$. Show that $△ABC\cong △ABD$. What can you say about $BC$ and $BD$?

Q. In $△ABC,AD$ is the perpendicular bisector of $BC$. Show that $△ABC$ is an isosceles triangle in which $AB=AC$.