Question

ABCD is a parallelogram as shown in the figure. If AB = 2AD and P is midpoint of AB, then what is the value of \[\angle CPD\] ?
(a). 90°
(b). 60°
(c). 45°
(d). 135°

Answer 1

Hint: Use the property of the parallelogram that the adjacent angles are supplementary. Then find the angle CPD in terms of the adjacent angles and find its value.

Complete step-by-step answer:

ABCD is a parallelogram, hence, the adjacent angles of the parallelogram are supplementary, that is, the sum of angles is 180°.

\[\angle A + \angle B = 180^\circ ............(1)\]

It is given that AB = 2 AD, then we have:

\[AD = \dfrac{{AB}}{2}..............(2)\]

It is given that P is the midpoint of AB. Then, we have the following:

\[AP = \dfrac{{AB}}{2}..........(3)\]

From equation (2) and equation (3), we have:

\[AD = AP\]

Hence, the triangle APD is an isosceles triangle.

The sum of the angles of the triangle APD is 180°.

\[\angle APD + \angle ADP + \angle A = 180^\circ \]

\[2\angle APD + \angle A = 180^\circ \]

Solve for the angle APD to get as follows:

\[\angle APD = 90^\circ - \dfrac{{\angle A}}{2}............(4)\]

The opposite sides of the parallelogram are equal. Hence, AD = BC, then we have:

\[BP = BC\]

Then, the triangle BPC is also an isosceles triangle.

The sum of the angles of the triangle BPC is 180°.

\[\angle BPC + \angle BCP + \angle B = 180^\circ \]

\[2\angle BPC + \angle B = 180^\circ \]

Solve for the angle BPC to get as follows:

\[\angle BPC = 90^\circ - \dfrac{{\angle B}}{2}............(5)\]

The angle on a straight line add up to 180°, then we have on line APB as follows:

\[\angle BPC + \angle CPD + \angle APD = 180^\circ \]

Using equations (4) and (5), we get:

\[90^\circ - \dfrac{{\angle B}}{2} + \angle CPD + 90^\circ - \dfrac{{\angle A}}{2} = 180^\circ \]

Simplifying, we have:

\[180^\circ - \dfrac{{\angle B + \angle A}}{2} + \angle CPD = 180^\circ \]

Canceling 180° on both sides, we have:

\[\angle CPD = \dfrac{{\angle B + \angle A}}{2}\]

From equation (1), we have:

\[\angle CPD = \dfrac{{180^\circ }}{2}\]

\[\angle CPD = 90^\circ \]

Hence, the correct answer is option (a).

Note: You can also find the angles CDP and DCP in terms of the angle D and angle C respectively and then find the angle CPD using the properties of the triangle.