In parallelogram ABCD, the diagonal AC bisects ∠A. Which of the following must also be true?

Class 9QuadrilateralsGeometry — Quadrilaterals

AAC bisects ∠C as well
BABCD is a square
CABCD is a rectangle
DAC bisects ∠B and ∠D

Step-by-Step Solution

If AC bisects ∠A, then alternate-angle equalities force adjacent sides equal, making ABCD a rhombus; in a rhombus a diagonal bisects both angles it joins, so AC also bisects ∠C. (A rhombus need not be a square or a rectangle.)

Answer: A) AC bisects ∠C as well

Key idea: Special parallelograms

**Rectangle** (parallelogram + all angles 90°):
- All properties of a parallelogram.
- Diagonals are EQUAL (in addition to bisecting each other).

**Rhombus** (parallelogram + all sides equal):
- All properties of a parallelogram.
- Diagonals bisect each other AT RIGHT ANGLES.
- Diagonals also bisect the angles of the rhombus.

**Square** (rectangle + rhombus):
- All properties of both.
- Diagonals: equal, bisect at right angles, AND bisect angles.

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