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

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

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

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.

ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal

BD (see Fig. 8.21). Show that

(i) ΔAPB ≅ ΔCQD

(ii) AP = CQ

ABCD is a rectangle and P, Q, R, and S are mid-points of the sides AB, BC, CD, and DA

respectively. Show that the quadrilateral PQRS is a rhombus.

Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect

each other.

ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and

parallel to BC intersects AC at D. Show that

(i) D is the mid-point of AC

(ii) MD ⊥ AC

(iii) CM = MA = (1/2)AB

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