2. A diagonal of a parallelogram divides it into two congruent triangles. 5. Diagonals of a rectangle bisect each other and are equal and vice-versa.

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1 QURILTERLS 1. Sum of the angles of a quadrilateral is diagonal of a parallelogram divides it into two congruent triangles. 3. In a parallelogram, (i) opposite sides are equal (ii) opposite angles are equal (iii) diagonals bisect each other 4. quadrilateral is a parallelogram, if (i) opposite sides are equal or (ii) opposite angles are equal or (iii) diagonals bisect each other or (iv) a pair of opposite sides is equal and parallel 5. iagonals of a rectangle bisect each other and are equal and vice-versa. 6. iagonals of a rhombus bisect each other at right angles and vice-versa. 7. iagonals of a square bisect each other at right angles and are equal, and viceversa. 8. The line-segment joining the mid-points of any two sides of a triangle is parallel to the third side and is half of it. 9. line through the mid-point of a side of a triangle parallel to another side bisects the third side. 10. The quadrilateral formed by joining the mid-points of the sides of a quadrilateral, in order, is a parallelogram.

2 EXERISE 1 1. The angles of quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral. nswer: s you know angle sum of a quadrilateral = 360 SO, 3x + 5x + 9x + 13x = x = 360 x = 12 Hence, angles are: 36, 60, 108, If the diagonals of a parallelogram are equal, then show that it is a rectangle. nswer: In the following parallelogram both diagonals are equal: So, Hence, = = = =90 s all are right angles so the parallelogram is a rectangle. 3. Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

3 O nswer: In the given quadrilateral diagonals and bisect each other at right angle. We have to prove that === In O & O O=O (O is the midpoint) O=O (common side) O = O (Right angle) So, O O So, = Similarly === can be proved which means that is a rhombus. 4. Show that the diagonals of a square are equal and bisect each other at right angles. nswer: In the figure given above let us assume that = 90 So, O = O = 45 Hence, O O=O (Sides opposite equal angles are equal) Similarly O=O=O can be proved This gives the proof of diagonals of square being equal. 5. Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square. nswer: Using the same figure, If O=O Then O = O = 45 (ngles opposite to equal sides are equal) So, all angles of the quadrilateral are right angles making it a square. 6. iagonal of a parallelogram bisects. Show that

4 (i) it bisects also, (ii) is a rhombus. nswer: is a parallelogram where diagonal bisects In & = (diagonal is bisecting the angle) = (ommon side) = (parallel sides are equal in a parallelogram) Hence, So, = This proves that bisects as well Now let us assume another diagonal intersecting on O. s it is a parallelogram so will bisect and vice versa In O & O O = O (opposite angles are equal in parallelogram so their halves will be equal) O=O O=O Hence, O O So, O = O = 90 s diagonals are intersecting at right angles so it is a rhombus

5 P Q 7. In parallelogram, two points P and Q are taken on diagonal such that P = Q. Show that: (i) P Q (ii) P = Q (iii) Q P (iv) Q = P (v) PQ is a parallelogram nswer: In P & Q P=Q (given) = (opposite sides are equal) P = Q (opposite angles halves are equal) Hence, P Q So, P=Q Proved In Q & P = (Opposite sides are equal) P=Q (given) Q = P ( opposite angles halves are equal) Hence, Q P So, Q=P Proved P = Q (corresponding angles of congruent triangles P & Q) In QP & QP PQ = QP (from previous proof) P=Q (given) PQ=PQ (common Side) So, QP QP So, QP = QP

6 With equal opposite angles and equal opposite sides it is proved that PQ is a parallelogram 8. is a parallelogram and P and Q are perpendiculars from vertices and on diagonal. Show that (i) P Q (ii) P = Q P Q nswer: In P & Q P = Q (alternate angles of transversal ) = P = Q (right angles) Hence, P Q So, P=Q 9. In and EF, = E, E, = EF and EF. Vertices, and are joined to vertices, E and F respectively. Show that E F

7 (i) quadrilateral E is a parallelogram (ii) quadrilateral EF is a parallelogram (iii) F and = F (iv) quadrilateral F is a parallelogram (v) = F (vi) EF. nswer: In & EF =E (given) =EF (given) = EF ( E & EF) Hence, EF In quadrilateral E = E E So, E is a parallelogram (opposite sides are equal and parallel) So, E (1) Similarly quadrilateral F can be proven to be a parallelogram So, E F (2) From equations (1) & (2) It is proved that F So, =F Similarly =F and F can be proved 10. is a trapezium in which and =. Show that (i) = (ii) = (iii) (iv) diagonal = diagonal [Hint : Extend and draw a line through parallel to intersecting produced at E.] E

8 nswer: In E E= (Opposite sides are equal in parallelogram) = (given) So, =E E = E E + = 180 (angles on the same side of a straight line) E + = 180 (adjacent angles of parallelogram are complementary) Substituting E = E it is clear that = Now, + = 180 (adjacent angles of parallelogram) nd, + = 180 ( adjacent angles of a parallelogram) s =, so it is clear that = In & = (common side) = (given) = Hence, EXERISE 2 1. is a quadrilateral in which P, Q, R and S are mid-points of the sides,, and. is a diagonal. Show that : 1 (i) SR and SR = 2 (ii) PQ = SR (iii) PQRS is a parallelogram. R T S P Q nswer: Let us extend the line SR to T so that T is parallel to S In SR & RT R=R (R is the mid point of side ) RS = TRS (Opposite angles)

9 SR = RT (alternate angle of transversal ST when T) Hence, SR RT So, SR=RT ST= (Opposite sides of parallelogram) 1 So, SR= 2 s SR is touching the mid points of and so as per mid point theorem SR Similarly PQ can be proven which will prove that PQRS is a parallelogram. 2. is a rhombus and P, Q, R and S are the mid-points of the sides,, and respectively. Show that the quadrilateral PQRS is a rectangle. S R P Q nswer: Following the method used in the previous question it can be proved that PQRS is a parallelogram. To prove it to be a rectangle we need to prove that S = R = Q = P = 90 In SR, RQ, QP & PS S=R=Q=P=R=Q=P=S (ll sides of rhombus are equal and PQRS are midpoints) SR = RS = RQ = QR = QP = PQ = PS = SP So, SR RQ QP PS So, SR = RQ = QP = PS = 90 Hence, SR + RS = 90 Or, SR = RS = RQ = QR = QP = PQ = PS = SP = 45 s, SP + PSR + SR = 180 PSR = 180 ( ) = 90 Similarly S = R = Q = P = 90 Hence, PQRS is a rectangle.

10 3. is a trapezium in which, is a diagonal and E is the mid-point of. line is drawn through E parallel to intersecting at F. Show that F is the mid-point of. E G F nswer: In G=G parallel line to the base originating from mid point of second side will intersect at the midpoint of the third side. EF So, EF So, In EG E is the mid point of So, G is the mid point of Now, in GF G is the mid point of So, F will be mid point of ( Mid point theorem) 4. In a parallelogram, E and F are the mid-points of sides and respectively. Show that the line segments F and E trisect the diagonal. nswer: In E & F = (Opposite sides are equal in parallelogram) F=E (Half of opposite sides of parallelogram) E = F (Opposite angles are equal) So, E F Hence, E=F In quadrilateral EF E F & E=F E=F So, E F So, EF is a parallelogram.

11 E P Q F In Q PE Q (proved earlier by proving E F) E is the mid point of So, P is the mid point of Q So, P=PQ P FQ P F is the mid point of So, PQ=Q So, P=PQ=Q proved 5. Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other. nswer: is a quadrilateral in which P, Q, R, & S are mid points of,, & In SR is touching mid points of and So, SR Similarly following can be proved PQ QR PS So, PQRS is a parallelogram. PR and QS are diagonals of the parallelogram PQRS, so they will bisect each other.

12 R S Q P 6. is a triangle right angled at. line through the mid-point M of hypotenuse and parallel to intersects at. Show that (i) is the mid-point of (ii) M 1 (iii) M = M = 2 M nswer: M M is the mid point of

13 So, is the mid point of (Mid point theorem) = M = 90 (alternate angle to transversal M) Now in M & M = M=M M = M So, M M (SS Theorem) So, M=M 1 M= 2 1 So, M=M= 2 Finish Line & eyond