Question

ABCDE is a regular pentagon and the bisector of angle BAE meets CD at M and the bisector of angle BCD meets AM at P . Find angle CPM

Answer 1

Answer:

Step-by-step explanation:

total angle of pentagon = 540°

each angle = 108°

in quadrilateral ABCP,

angle PAB = 54°

angle ABC = 108°

angle BCP = 54°

angle CPA = x°

angle PAB +angle ABC +angle BCP +angle CPA = 360° (angle sum property of quadrilateral)

54° + 108° + 54° + x° = 360°

x° = 144°

angle CPM = 180° - x° (linear pair)

angle CPM = 36°

Answer 2

Answer:

The value of angle CPM is 36°.

Step-by-step explanation:

Given ABCDE is a regular pentagon.

• In a regular pentagon, the sum of the interior angles is equal to 540°.
• All the sides are equal and all the angles are of equal measure, = 108°.

Referring to the diagram shown, in the quadrilateral ABCP, $\angle PAB$ and $\angle BCP$ are the bisected polygon angles.

Therefore, $\angle PAB = 54^o$, $\angle BCP = 54^o$ and $\angle ABC = 108^o$

Sum of angles in a quadrilateral is 360°.

$\therefore \angle PAB +\angle BCP +\angle ABC +\angle CPA = 360^o$

$\implies 54 +\54 +108 +\angle CPA = 360$

$\implies 216 +\angle CPA = 360$

$\implies \angle CPA = 360-216 =144^o$

The points CPM makes a triangle.

And the sum of angles in a triangle is equal to 180°.

Angle CMP is a right angle and angle PCM is a bisected angle.

$\angle PCM+\angle CMP+\angle CPM=180^o$

$\implies 54+90+\angle CPM=180$

$\implies 144+\angle CPM=180$

$\implies \angle CPM=180-144=36$

Therefore, the value of angle CPM is 36°.