Class 9 Maths Chapter 8 Exercise 8.1 Solutions
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.]
