NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.1 – Part 3 (For CBSE, ICSE, IAS, NET, NRA 2022)

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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
  • 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

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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