Q.1. The angles of quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral.

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Q.2. If the diagonals of a parallelogram are equal, then show that it is a rectangle.

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Q.3. Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

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Q.4. Show that the diagonals of a square are equal and bisect each other at right angles.

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

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Q.6. Diagonal AC of a parallelogram ABCD bisects ∠ A (see Figure). Show that

(i) it bisects ∠ C also,

(ii) ABCD is a rhombus.

Q.7. ABCD is a rhombus. Show that diagonal AC bisects ∠ A as well as ∠ C and diagonal BD bisects ∠ B as well as ∠ D.

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Q.8. ABCD is a rectangle in which diagonal AC bisects ∠A as well as ∠C. Show that:

(i) ABCD is a square

(ii) diagonal BD bisects ∠B as well as ∠D.

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Q.9. In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Figure). Show that:

(i) Δ APD ≅ Δ CQB

(ii) AP = CQ

(iii) Δ AQB ≅ Δ CPD

(iv) AQ = CP

(v) APCQ is a parallelogram

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

(i) Δ APB ≅ Δ CQD

(ii) AP = CQ

Q.11. In Δ ABC and Δ DEF, AB = DE, AB || DE, BC = EF and BC || EF. Vertices A, B and C are joined to vertices D, E and F respectively (see Figure). Show that

(i) quadrilateral ABED is a parallelogram

(ii) quadrilateral BEFC is a parallelogram

(iii) AD || CF and AD = CF

(iv) quadrilateral ACFD is a parallelogram

(v) AC = DF

(vi) Δ ABC ≅ Δ DEF.

Q.12. ABCD is a trapezium in which AB || CD and AD = BC (see Figure). Show that

(i) ∠ A = ∠ B

(ii) ∠ C = ∠ D

(iii) Δ ABC ≅ Δ BAD

(iv) diagonal AC = diagonal BD

[Hint: Extend AB and draw a line through C parallel to DA intersecting AB produced at E.]

NCERT Solutions For Class 9 Maths Chapter 8, Quadrilaterals (All Exercises)