NCERT Class 9 Solutions: Triangles (Chapter 7) Exercise 7.1 – Part 4
Cointerior angles
When two lines are cut by a third line (transversal) cointerior angles are between the pair of lines on the same side of the transversal. If the lines are parallel the cointerior angles are supplementary (add up to 180 degrees).
Q7 AB is a line segment and P is its midpoint. D and E are points on the same side of AB such that and (see Fig). Show that
Solution:
Given,P is midpoint of AB.

(Adding both sides)
In , (P is midpoint of AB) (Given)Therefore, (By AngleSideAngle congruence condition.

(By corresponding Part of Congruent Triangles).
Q8 In right triangle ABC, right angled at C, M is the midpoint 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 (see Fig). Show that:

ΔAMC ≅ ΔBMD

∠DBC is a right angle.

ΔDBC ≅ ΔACB

CM = 1/2 AB
Solution:
Given, right triangle ABC, right angled at C,M is the midpoint of hypotenuse AB.C is joined to M and produced to point D
M is the midpoint of AB and

In and (M is the midpoint) (Vertically opposite angles) (Given)
Therefore, (By SideAngleSide congruence condition).

(By Corresponding Parts of Congruent Triangles)Therefore, as alternate interior angles are equal.Now, (cointeriors angles)

In ΔDBC and ΔACB, (Common) (Right angles) (By Corresponding Parts of Congruent Triangles, already proved)
Therefore (By SideAngleSide congruence condition)

(M is midpoint)