ABCDE is a regular pentagon and bisector of ∠ BAE meets CD at M. If bisector of ∠ BCD meets AM at P, find ∠ CPM.

Question

ABCDE is a regular pentagon and bisector of &#x2220; BAE meets CD at M. If bisector of &#x2220; BCD meets AM at P, find &#x2220; CPM.

Philoid · Accepted Answer

Given: ABCDE is a regular pentagon. The bisector of ∠ BAE meets CD at M and bisector of ∠ BCD meets AM at P.
To find: The measure of ∠ CPM.
Proof: Each angle of a pentagon =

=

= 108°
In quadrilateral ABCM,
∠ BAM + ∠ B + ∠ C + ∠ CMP = (2n - 4) right angles
= (2 × 4 - 4) right angles
= 4 right angles
= 4 × 90°
= 360°
⇒

+108° +108° +∠ CMP = 360°
⇒ 270° +∠ CMP = 360°
⇒ ∠ CMP = 360° -270°
⇒ ∠ CMP = 90°
∠ PCM =

∠ BCM =

× 108° = 54°
In Δ CPM, ∠ CPM + ∠ PCM + ∠ CMP = 180° [Sum of the three angles of a triangle is 180° ]
⇒ ∠ CPM + 54° + 90° = 180°
⇒ ∠ CPM + 144° = 180°
Þ ∠ CPM = 180° - 144° = 36°