Question

segment ac is a diagonal of regular polygon abcde. what does ∠acd equal?
select one:
a. 36
b. 72
c. 108
d. 5

the sum of the angles of a triangle is
select one:
a. 60
b. 360
c. 90
d. 180

Explanation:

Step1: Find polygon interior angle

First, calculate the measure of each interior angle of regular pentagon ABCDE. The formula for an interior angle of an n-sided regular polygon is $\frac{(n-2)\times180^\circ}{n}$. For $n=5$:
$$\frac{(5-2)\times180^\circ}{5}=108^\circ$$
So $\angle BCD = 108^\circ$.

Step2: Analyze isosceles triangle ABC

In regular pentagon ABCDE, $AB=BC$, so $\triangle ABC$ is isosceles with $\angle ABC=108^\circ$. Calculate base angles:
$$\angle ACB=\frac{180^\circ - 108^\circ}{2}=36^\circ$$

Step3: Calculate $\angle ACD$

Subtract $\angle ACB$ from $\angle BCD$:
$$\angle ACD=\angle BCD - \angle ACB=108^\circ - 36^\circ=72^\circ$$

Step4: Identify triangle angle sum

The fundamental geometric property states the sum of angles in any triangle is a fixed value.

Answer:

1. b. 72
2. d. 180