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

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Co-interior Angles

Give co-interior angles,also a+b=180

Give Co-Interior Angles

Give co-interior angles,also a+b=180

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:

ABCD Is a Quadrilaterl

ABCD is a quadrilaterl diagonal AC and BD bisect each other right angle at O.

  • 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.

Parallelogram ABCD With Diagonal Bisecting Angle

Parallelogram ABCD with diagonal AC which bisects ∠A

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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