Q. In the given figure, if $x=y$ and $AB=CB$ then prove that $AE=CD$.

Q. "If two angles and a side of one triangle are equal to two angles and a side of another triangle, then the two triangles must be congruent". Is the statement true? Why?

Q. In the given figure, $O$ is a point in the interior of $1M$ a square $ABCD$ such that $OAB$ is an equilateral triangle. Show that $OCD$ is an isosceles triangle.

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. $ABCD$ is a quadrilateral such that diagonal $AC$ bisects the angles $\angle A$ and $\angle C$ prove that $AB=AD$ and $CB=CD$.

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. "If two sides and an angle of one triangle are equal to two sides and an angle of another triangle, then the two triangles must be congruent". Is the statement true? Why?

Q. The bisectors of $\angle B$ and $\angle C$ of an isosceles $\Delta ABC$ with$AB=AC$ intersect each other at point $O.$ Shows that the exterior angle adjacent to $\Delta ABC$ is equal to $\Delta BOC$

Q. In a $△ABC,D$ is the mid point of side $AC$ such that $BD=\frac{1}{2}AC$. Show that $\angle ABC$ is a right angle.

Q. In the given figure, $AB\parallel CD$ and $O$ is the midpoint of $AD$.
Show that (i) $△AOB\cong △DOC$
(ii) $O$ is the midpoint of $BC$.

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

Q. $AD$ and $BC$ are equal perpendicular to a line segment $AB$. Show that $CD$ bisects $AB$.

Q. The image of an object placed at a point A before a plane mirror $LM$ is seen at the point $B$ by an observer at $D$ as shown in Fig. Prove that the image is as far behind the mirror as the object is in front of the mirror.

Q. $△ABC$ is a right triangle right angled at $A$ such that $AB=AC$ and bisector of $\angle C$ intersects the side $AB$ at $D$. prove that $AC+AD=BC$.

Q. $P$ is a point on the bisector of $\angle ABC$. The line through $P$ parallel to $BA$ meets $BC$ at $Q$. Prove that $\Delta BPQ$ is an isosceles triangle.

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

Q. In the given figure, $OA=OB$ and $OP=OQ$. Prove that
(i) $PX=QX$,
(ii) $AX=BX$.

Q. In the given figure, $ABCD$ is a quadrilateral in which $AB\parallel DC$ and $P$ is the midpoint of $BC$. On producing, $AP$ and $DC$ meet at $Q$. Prove that
(i) $AB=CQ$,
(ii) $DQ=DC+AB$.