ABCD is a parallelogram and line segments AX, CY bisects the angles A and C respectively. Show that AX II CY.

Prove that the opposite angles of an isosceles trapezium are supplementary.

Show that the bisectors of angles of a parallelogram enclose a rectangle.

In the figure, ABCD is a parallelogram. E and F are the mid-points of sides AB and CD respectively. Show that the line segments AF and EC trisect the diagonal BD.

In D ABC, D, E and F are respectively the mid-points of sides AB, BC and CA (see Fig.). Show that Δ ABC is divided into four congruent triangles by joining D, E and F.

In the given figure ABCD is a parallelogram and E is the mid-point of AD. A line through D, drawn parallel to EB, meets AB produced at F and BC at L Prove that
(i) AF = 2DC
(ii) DF = 2DL

l, m and n are three parallel lines intersected by transversals p and q such that l, m and n cut off equal intercepts AB and BC on p (see Fig.). Show that l, m and n cut off equal intercepts DE and EF on q also.

There was four plants in Suraj's fields. Suraj named their bases of P, Q, R, S. He joined PQ, QR, RS and SP. His teacher told him that the quadrilateral PQRS was a parallelogram. He asked him to find the measure of all the angles of the parallelogram,
provided that the measure of anyone interior angle of PQRS. To obtain a technique and hence to solve the problem, he worked hard and spent much time.
(i) Obtain all the angles of the paralellogram PQRS if ㄥR=80°.
(ii) Which mathematical concept is used in the above problem?
(iii) Which value was depicted by Suraj on such a problem

Ankush prepare a poster in the form of parallelogram, as in figure.
(i) If ㄥA=(5x + 7)° and LB = (3x- 3)°, find all the angles of a parallelogram ABCD.
(ii) Which mathemetical concept is used in this question?
(iii) By writing a slogan on poster which value is depicted by Ankush?

Rani has a photo-frame without a photo in the shape of a triangle with sides a, b, c in length. She wants to find the perimeter of a triangle formed by joining the mid-points of the sides of the photoframe. She could not understand how to overcome this problem. She shares this problem with her classmate Renu. Renu helps her and the required perimeter is computed.
(i) Find the perimeter of the triangle formed by joining the mid-points of the frame.
(ii) Which mathemetical concept is used in the above problem?
(iii) Which value is depicted between Rani and Renu?