How do you prove that diagonals of a parallelogram bisect each other?

How do you prove that diagonals of a parallelogram bisect each other?

Theorem: The diagonals of a parallelogram bisect each other. Proof: Given ABCD, let the diagonals AC and BD intersect at E, we must prove that AE ∼ = CE and BE ∼ = DE. The converse is also true: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Why do diagonals of parallelogram bisect each other?

But this may or may not be true because only the opposite angles are in the parallelogram are true which may or may not be equal to 90∘. Hence, the diagonals of a parallelogram bisect each other but not necessarily at right angles.

How do you prove that diagonals bisect each other in a rectangle?

1 Answer

1. AC and OB are diagonals. In the figure let the intersecting point of OB and AC be P. To show that diagonals bisect each other we have to prove that OP = PB.
2. OP = OB. Similarly we can prove that PC = PA. Thus diagonals bisect each other in a rectangle .
3. ∴ The diagonals of a rectangle bisects each other and equal .

Do diagonals bisect angles in a parallelogram?

The opposite angles of a parallelogram are equal. The opposite sides of a parallelogram are equal. The diagonals of a parallelogram bisect each other.

Are diagonals congruent in a parallelogram?

The diagonals of a parallelogram are sometimes congruent. The diagonals of a rhombus are always perpendicular. The consecutive angles of a parallelogram are never complementary. A square is always a rhombus.

Does a diagonal of a parallelogram bisect a pair of opposite angles if so how many do?

If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus. To prove a square, you must prove it is both a rectangle and a rhombus. If a quadrilateral is a kite, then its diagonals are perpendicular.

Do diagonals bisect angles in parallelogram?

All of the properties of a parallelogram apply (the ones that matter here are parallel sides, opposite angles are congruent, and consecutive angles are supplementary). All sides are congruent by definition. The diagonals bisect the angles. The diagonals are perpendicular bisectors of each other.

What parallelogram has diagonals bisect both pairs of opposite angles?

rhombus
If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.