Question

1. determine the values of a, b, c, and d

Explanation:

Step1: Identify polygon type

This is a regular decagon (10-sided polygon), so all sides are equal, and all internal angles are equal.

Step2: Calculate internal angle

The formula for internal angle of an n-sided polygon is $\frac{(n-2)\times180^\circ}{n}$. For $n=10$:
$\frac{(10-2)\times180^\circ}{10} = 144^\circ$

Step3: Analyze congruent triangles

All the triangles formed by the lines from vertices are isosceles, with sides equal to the decagon's sides. For triangle with angle $a$: base angles are equal. The internal angle of the decagon is split, so $a = 180^\circ - 2\times\frac{144^\circ}{2} = 36^\circ$

Step4: Find angle b

Angle $b$ is an internal angle of the decagon minus angle $a$:
$b = 144^\circ - 36^\circ = 108^\circ$

Step5: Find angle c

Angle $c$ is equal to angle $a$ (congruent isosceles triangles):
$c = 36^\circ$

Step6: Find angle d

Angle $d$ is an internal angle of the decagon:
$d = 144^\circ$

Answer:

$a=36^\circ$, $b=108^\circ$, $c=36^\circ$, $d=144^\circ$