NCERT Class 9 Solutions: Triangles (Chapter 7) Exercise 7.4 – Part 3
Interior Angle and Exterior Angle
Interior Angle: interior angle (or internal angle) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex.
Exterior Angle: exterior angle (or external angle) is an angle formed by one side of a simple polygon and a line extended an adjacent side
Q5 In the figure and PS bisects . Prove that
Solution:
Given, and PS bisects

To prove,

Proof,
……equation (1) ( as angle opposite to larger side is larger.) ……equation (2) (PS bisects )
…..equation (3) (exterior angle of a triangle equals to the sum of opposite interior angles) ….equation(4) (exterior angle of a triangle equals to the sum of opposite interior angles)

Adding (1) and (2) [from (1), (2), (3) and (4)]
Q6 Show that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest.
Solution:

Let l is a line segment and B is a point lying .We drew a line AB perpendicular to l. Let C be any other point on l.

To prove,

Proof,In ,

Now, ( must be acute angle. (Angle sum property of a triangle)
or
(Side opposite to the larger angle is larger.)