Question

1. make sense and persevere what are the measures of the angles in the right triangle formed by the two regular pentagons shown?

Explanation:

Step1: Calculate the interior angle of a regular pentagon

The formula for the interior angle of a regular polygon is \(\theta=\frac{(n - 2)\times180^{\circ}}{n}\), where \(n = 5\) (for a pentagon).

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Step2: Find the non - right angle in the right - triangle

Let \(x\) be the non - right angle in the right - triangle. We know that \(108^{\circ}=90^{\circ}+x\) (by the angle relationship in the figure). Solving for \(x\):

$$x = 108^{\circ}-90^{\circ}=18^{\circ}$$

Step3: Find the third angle in the right - triangle

In a right - triangle, one angle is \(90^{\circ}\), we found one non - right angle \(x = 18^{\circ}\). Using the angle sum property of a triangle (\(A + B+C=180^{\circ}\), where \(A = 90^{\circ}\), \(B = 18^{\circ}\)), let the third angle be \(y\).

$$y=180^{\circ}-(90^{\circ}+18^{\circ})=72^{\circ}$$

Answer:

The measures of the angles in the right - triangle are \(90^{\circ}\), \(18^{\circ}\), and \(72^{\circ}\)