R S Aggarwal and Veena Aggarwal Solutions for Exercise 1: EXERCISE 9A

The line segments joining the midpoints $\mathrm{M}$ and $\mathrm{N}$ are parallel sides $\mathrm{AB}$ and $\mathrm{DC}$ respectively of a trapezium $\mathrm{ABCD}$ is perpendicular to both the sides $\mathrm{AB}$ and $\mathrm{DC}.$ Prove that $\mathrm{AD}=\mathrm{BC}.$

The bisectors $\angle B$ and $\angle C$ of an isosceles triangle with $AB=AC$ intersect each other at a point $O$. $BO$ is produced to meet $AC$ at a point $M$. Prove that $\angle MOC=\angle ABC.$

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

 $P$ is a point on the bisector of $\angle ABC$. If the line through $P$, parallel to $BA$ meets $BC$ at $Q$, prove that $△BPQ$ is an isosceles triangle.

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 the figure. Prove that the image is as far behind the mirror as the object is in front of the mirror.

In a $△ABC, D$ is the midpoint of side $AC$ such that $BD=½ AC$. Show that $\angle ABC$ is a right angle.

“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.”

“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.”