NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.1 – Part 3

Co-interior Angles

When two lines are cut by a third line (transversal) co-interior angles are between the pair of lines on the same side of the transversal. If the lines are parallel the co-interior angles are supplementary (add up to 180 degrees).

Q-5 Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

Solution:

• Given, ABCD is a quadrilateral in which diagonals AC and BD bisect each other at right angle at O.

• To prove, Quadrilateral ABCD is a square.

Proof,

In ,

• (Diagonals bisect each other)

• (Vertically opposite)

• (Diagonals bisect each other)

Therefore,

• by Side-Angle-Side congruence condition.

• Thus, by Corresponding Parts of Congruent Triangle…….equation(1)

Also,

• (Alternate interior angles)

Now, In ,

• (Diagonals bisect each other)

• (Vertically opposite)

• (Common)

Therefore, by Side-Angle-Side congruence condition

Thus, by Corresponding Parts of Congruent Triangle………equation(2)

Also,

• and

• (From equation (1))

• by Corresponding Parts of Congruent Triangle

• (co-interior angles)

• ( )

• ……..equation(3)

• That is, one of the interior angle is right angle.

Thus, from (1), (2) and (3) given quadrilateral ABCD is a square.

Q-6 Diagonal AC of a parallelogram ABCD bisects (see Fig.). Show that

1. it bisects also,

2. ABCD is a rhombus.

Solution:

Given, Parallelogram ABCD and its diagonal AC, it’s bisects

In ,

• (parallel sides are equal in a parallelogram)

• (parallel sides are equal in a parallelogram)

• (Common side)

Therefore, by SSS congruence condition.

Thus,

• by Corresponding Parts of Congruent Triangle

• And (AC id diagonal)

• (AC id diagonal)

Therefore, AC bisects also.

Since, (Proved)

• (Opposite sides of equal angles of a triangle are equal)

• Also, (parallel sides are equal in a parallelogram)

Thus, ABCD is a rhombus.

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