NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.1 – Part 5

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Classification of quadrilaterals, key properties, shape , ch …

Classification of Quadrilaterals by Key Property

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Q-9 In parallelogram ABCD, two points P and Q are taken on diagonal BD such that (see Fig.). Show that:

  1. APCQ is a parallelogram

Triangle of APD and CQB

Solution:

  1. In,

  2. (Given)

  3. (Alternate interior angles)

  4. (Opposite sides of a parallelogram)

Thus, by Side-Angle-Side congruence condition.

  1. By Corresponding parts of congruent triangles as .

  2. In ,

  3. (Given)

  4. (Alternate interior angles)

  5. AB = CD (Opposite sides of a parallelogram)

Thus, ΔAQB ≅ ΔCPD by Side-Angle-Side congruence condition.

  1. By Corresponding Parts of Congruent Triangles as.

  2. The diagonal of a parallelogram bisect each other.

  3. Given

  4. ....equation (1)

Also,………….equation (2) (diagonal of a parallelogram bisect each other)

From equation (1) and (2), APCQ is parallelogram

Q-10 ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on on diagonal BD (see Fig.).Show that

ABCD Parallelogram

Solution:

Given,

  • ABCD is parallelogram

  • AP and CQ are perpendiculars from A and C on diagonal BD

Solution (i)

In and ,

  • (Opposite side of parallelogram ABCD)

  • (Alternate interior angles)

Now,

  • (Equal to right angles as AP and CQ are perpendiculars)

  • (ABCD is a parallelogram)

  • Thus, by Angle-Angle-Side congruence condition.

Solution (ii)

by Corresponding Parts of Congruent Triangles as .