Diagonal AC of a parallelogram ABCD bisects ∠A (see the given figure). Show that i) it bisects ∠C also, ii) ABCD is a rhombus (2024)

FAQs

What is the diagonal AC of a parallelogram ABCD bisects? ›

Solution: Given: The diagonal AC of a parallelogram ABCD bisects ∠A. We can use alternate interior angles property to show that the diagonal AC bisects ∠C and by showing all sides are equal, it can be proved ABCD is a rhombus.

In geometry, a rhombus is a quadrilateral that has all equal sides, with opposite sides parallel to each other. The quadrilateral ABCD is a rhombus, with AB = BC = CD = AD. AB is parallel to CD (AB||CD), and BC is parallel to AD (BC||CD).

Answer and Explanation: A rhombus is a parallelogram with four congruent sides. However, this questions is asking us about diagonals. A parallelogram will be a rhombus if its diagonals bisect pairs of opposite angles, but not each other, making choice 2 wrong.

∠A=45∘But ∠A+∠B=180∘ (Sum of consecutive angles)∴∠B=180∘−∠A=180∘−45∘=135∘

The diagonals of a rhombus bisect each other at right angles.

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. If — AB ≅ — CD and — BC ≅ — DA , then ABCD is a parallelogram. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Say that the rectangle had side lengths of length e and f. Then the side lengths of quadrilateral ABCD, by the Pythagorean Theorem, are √(e2)2+(f2)2. Since all sides of this quadrilateral are equal then quadrilateral ABCD must be a rhombus.

The diagonal lines of a parallelogram will bisect each other at the midpoint of each line. This means that each line will be cut in half by the other line.

The diagonals of a parallelogram bisect each other. Each diagonal of a parallelogram bisects it into two congruent triangles. If one pair of opposite sides of a quadrilateral is equal and parallel, then the quadrilateral is a parallelogram.

Do the diagonals of a parallelogram bisect each other at right angles? ›

In a special parallelogram as square, rhombus etc. diagonals bisect each other at right angle but in general parallelogram, diagonals of a parallelogram only bisects each other.

2. If the Diagonals of a quadrilateral bisect each other,it is a parallelogram.

In a rhombus ABCD show that diagonal AC bisects ∠A as well as ∠C and diagonal BD bisects ∠B as well as ∠D. ABCD is a rhombus.

If the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.

Summary: If ABCD is a rectangle in which diagonal AC bisects ∠A as well as ∠C, we have proved that ABCD is a square and diagonal BD bisects ∠B as well as ∠D.

Answer: The statement "The diagonals of a parallelogram bisect each other" is true. A parallelogram is a four-sided figure with opposite sides that are parallel. The diagonals of a parallelogram are the line segments that connect opposite corners of the parallelogram.

In a parallelogram if two diagonals are equal, then the parallelogram is a rectangle.So, ABCD is a rectangle.It is known that each interior angle of a rectangle is 90 degree. Since ABC is an interior angle of rectangle ABCD, so measure of angle ABC is 90 degree. Q.