Question

If the diagonal of a quadrilateral bisects the opposite angles as shown, then the given quadrilateral is a _______.

• A. Kite
• B. Rectangle
• C. Square
• D. Rhombus

Answer 1

The correct option is D Rhombus

Given, in quadrilateral ABCD
AB || CD
and, BC || DA

So, it is a parallelogram.
We know that opposite angles of a parallelogram are equal.
$\therefore \angle DAB=\angle BCD$

Also, the diagonal AC bisects the angles $\angle DABand\angle BCD$
$\therefore \angle DAC=\angle CAB=\angle BCA=\angle ACD$

Therefore, the two triangles $△ABCand△CDA$ are isoceles triangle.
Hence, AB = BC and CD = DA

Also, $△ABS\cong △CDA\text{by ASA criteria}$
$\angle CAB=\angle DAC$
side AC (common side)
$\angle BCA=\angle ACD$

Therefore,
sides AB = BC = CD = DA (corresponding parts)
The given quadrilateral is a rhombus (by definition)