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

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:

Quadrilateral poly1 Quadrilateral poly1: Polygon A, B, C, D Segment a Segment a: Segment [A, B] of Quadrilateral poly1 Segment b Segment b: Segment [B, C] of Quadrilateral poly1 Segment c Segment c: Segment [C, D] of Quadrilateral poly1 Segment d Segment d: Segment [D, A] of Quadrilateral poly1 Segment f Segment f: Segment [A, C] Segment g Segment g: Segment [B, D] Point A A = (-1.88, 3.64) Point A A = (-1.88, 3.64) Point A A = (-1.88, 3.64) Point B B = (1.88, 3.64) Point B B = (1.88, 3.64) Point B B = (1.88, 3.64) Point C C = (1.88, 0.48) Point C C = (1.88, 0.48) Point C C = (1.88, 0.48) Point D D = (-1.88, 0.48) Point D D = (-1.88, 0.48) Point D D = (-1.88, 0.48) Point O Point O: Intersection point of f, g Point O Point O: Intersection point of f, g Point O Point O: Intersection point of f, g

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 ΔAOBandΔCOD ,

  • AO=CO (Diagonals bisect each other)

  • AOB=COD (Vertically opposite)

  • OB=OD (Diagonals bisect each other)

Therefore,

  • ΔAOBΔCOD by Side-Angle-Side congruence condition.

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

    Also,

  • OAB=OCD (Alternate interior angles)

  • ABCD

Now, In ΔAODandΔCOD ,

  • AO=CO (Diagonals bisect each other)

  • AOD=COD (Vertically opposite)

  • OD=OD (Common)

Therefore, ΔAODΔCOD by Side-Angle-Side congruence condition

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

Also,

  • AD=BC and AD=CD

  • AD=BC=CD=AB (From equation (1))

  • ADC=BCD by Corresponding Parts of Congruent Triangle

  • ADC+BCD=180° (co-interior angles)

  • 2ADC=180° ( ADC=BCD )

  • ADC=90° ……..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 A (see Fig.). Show that

  1. it bisects C also,

  2. ABCD is a rhombus.

Quadrilateral poly1 Quadrilateral poly1: Polygon D, C, B, A Segment a Segment a: Segment [D, C] of Quadrilateral poly1 Segment b Segment b: Segment [C, B] of Quadrilateral poly1 Segment c Segment c: Segment [B, A] of Quadrilateral poly1 Segment d Segment d: Segment [A, D] of Quadrilateral poly1 Segment f Segment f: Segment [C, A] Point D D = (-1.58, 3.74) Point D D = (-1.58, 3.74) Point D D = (-1.58, 3.74) Point C C = (2.36, 3.74) Point C C = (2.36, 3.74) Point C C = (2.36, 3.74) Point B B = (1.26, 1.42) Point B B = (1.26, 1.42) Point B B = (1.26, 1.42) Point A A = (-2.84, 1.46) Point A A = (-2.84, 1.46) Point A A = (-2.84, 1.46)

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 A

In ΔADCandΔCBA ,

  • AD=CB (parallel sides are equal in a parallelogram)

  • DC=BA (parallel sides are equal in a parallelogram)

  • AC=CA (Common side)

Therefore, ΔADCΔCBA by SSS congruence condition.

Thus,

  • ACD=CAB by Corresponding Parts of Congruent Triangle

  • And CAB=CAD (AC id diagonal)

  • ACD=BCA (AC id diagonal)

Therefore, AC bisects C also.

Since, ACD=CAD (Proved)

  • AD=CD (Opposite sides of equal angles of a triangle are equal)

  • Also, AB=BC=CD=DA (parallel sides are equal in a parallelogram)

Thus, ABCD is a rhombus.

Explore Solutions for Mathematics

Sign In