Page 1 : Similarly, SQ TPR, So, PSQR is a parallelogram., , , , , , EXERCISE 8.1, 1. The angles of quadrilateral are in the ratio 3: 5 : 9 : 13. Find all the angles of the, quadrilateral., 2. If the diagonals of a parallelogram are equal, then show that it is a rectangle., 3. Show that if the diagonals of a quadrilateral bisect each other at right angles, then it, is a rhombus., 4. Show that the diagonals of a square are equal and bisect each other at right angles., 5. Show that if the diagonals of a quadrilateral are equal and bisect each other at right, angles, then it is a square. D, Cc, 6. Diagonal AC of a parallelogram ABCD bisects, Z A (see Fig. 8.19). Show that, (@) it bisects Z C also,, (ii) ABCD is arhombus., A B, 7. ABCD is a rhombus. Show that diagonal AC, bisects Z A as well as Z C and diagonal BD Fig. 8.19, bisects Z B as well as Z D., 8. ABCD is a rectangle in which diagonal AC bisects Z A as well as Z C. Show that:, (i) ABCD is a square (ii) diagonal BD bisects Z B as well as 2D., 2021-22, QUADRILATERALS 147, 9. In parallelogram ABCD, two points P and Q are A D, taken on diagonal BD such that DP = BQ ee, (see Fig. 8.20). Show that:, () AAPD=ACQB, (ii) AP=CQ, (iii) AAQB=ACPD pot, (iv) AQ=CP s Cc, (v) APCQisa parallelogram Fig. 8.20, 10. ABCD is a parallelogram and AP and CQ are D Cc, perpendiculars from vertices A and C on diagonal, BD (see Fig. 8.21). Show that, (@) AAPB=ACQD, (ii) AP=CQ, A B, Fig. 8.21, 11. InA ABC and ADEF, AB = DE, AB || DE, BC=EF A, and BC ll EF. Vertices A, B and C are joined to, vertices D, E and F respectively (see Fig. 8.22)., Show that B&, @ quadrilateral ABED is a parallelogram ., (i) quadrilateral BEFC is a parallelogram, (iii) AD Il CF and AD =CF, (iv) quadrilateral ACFD is a parallelogram, (v) AC=DF, (vi) AABC=A DEF., 12. ABCD is a trapezium in which AB || CD and A B Pe, , , , AD = BC (see Fig. 8.23). Show that, @ ZA=ZB, , Gi) ZC=ZD a, Gii) AABC=A BAD D Cc, (iv) diagonal AC = diagonal BD Fig. 8.23, , [Hint: Extend AB and draw a line through C, parallel to DA intersecting AB produced at E.]