# Triangles

## Class 9 NCERT Maths

### NCERT

1   In quadrilateral $ACBD, AC = AD$ and $AB$ bisects $\angle A$ (See the given figure). Show that $\Delta ABC \cong \Delta ABD$. What can you say about $BC$ and $BD$?

##### Solution :

In $\Delta ABC$ and $\Delta CDA,$$\\ \angle BAC = \angle DCA (Alternate interior angles, as p \parallel q)\\ AC = CA (Common)\\ \angle BCA = \angle DAC (Alternate interior angles, as l \parallel m)\\ \therefore \Delta ABC \cong \Delta CDA (By ASA congruence rule) 5 Line l is the bisector of an angle \angle A and \angle B is any point on l. BP and BQ are perpendiculars from B to the arms of \angle A (see the given figure). Show that:\\ (i) \Delta APB \cong \Delta AQB$$\\$ (ii)$BP = BQ$ or $B$ is equidistant from the arms of $\angle A$.

##### Solution :

In $\Delta APB$ and $\Delta AQB,$$\\ \angle APB = \angle AQB (Each 90^o)$$\\$ $\angle PAB = \angle QAB$ ($l$ is the angle bisector of $\angle A$)$\\$ $AB = AB$ (Common)$\\$ $\therefore \Delta APB \cong \Delta AQB$ (By $AAS$ congruence rule)$\\$ $\therefore BP = BQ$ (By $CPCT)$$\\$ Or, it can be said that B is equidistant from the arms of $\angle A$.