Congruency of Triangles

Q. If $ABCD$ is a parallelogram and $AP=CQ$, show that $AC$ and $PQ$ bisect each other.

Q.
Points P and Q have been taken on opposite sides AB and CD respectively of a parallelogram ABCD such that AP = CQ. Which of the following is correct?

1. AC and PQ bisect each other.
2. AC = PQ
3. AC = CD
4. None of the above

Q. A diagonal of a parallelogram divides the parallelogram into two congruent triangles.

1. False
2. True

Q.

If a, b, c are all non zero and a+b+c=0, prove that

a²/bc + b²/ca + c²/ab =3

Q.

In the given figure, ABCD is a quadrilateral in which AB = AD and BC = DC. Prove that (i) AC bisects $\angle Aand\angle C$, (ii) BE = DE, (iii) $\angle ABC=\angle ADC$.

Q.

In the adjoining figure, $ABCD$ is a square and $\Delta EDC$ is an equilateral triangle. Prove that (i) $AE=BE,$ (ii) $\angle DAE={15}^{\circ }$.

Q.

In the adjoining figure, ABCD is a parallelogram in which AB is produced to E so that BE = AB. Prove that ED bisects BC.

Q.

In the adjoining figure, ABCD is a parallelogram and E is the midpoint of side BC. If DE and AB when produced meet at F, prove that AF = 2AB.

Q. The triangle formed by joining the mid-points of an equilateral triangle is triangle.

1. an equilateral
2. an isosceles
3. a scalene

Q.
Points P and Q have been taken on opposite sides AB and CD respectively of a parallelogram ABCD such that AP = CQ. Which of the following is correct?

1. AC = CD
2. None of the above
3. AC and PQ bisect each other.
4. AC = PQ