Question

the star shown is constructed by extending each side of a regular pentagon. explain why the surrounding triangles are isosceles and congruent.
the base angles of each triangle in the star are each

since they are the

angles of a regular pentagon. since the base angles of each triangle are congruent, each triangle is

all of the triangles each have a side length in common, the side between the two base angles. so, they are all congruent by

(type an integer or a decimal. do not include the degree symbol in your answer.)

Explanation:

Step 1: Properties of a regular pentagon

The interior angles of a regular pentagon are equal. The measure of each interior angle of a regular pentagon is calculated as follows:
The formula for the measure of an interior angle of a regular polygon is \(\frac{(n - 2)\times180^{\circ}}{n}\), where \(n\) is the number of sides. For a pentagon, \(n = 5\).

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The base angles of each triangle in the star are equal. Let's consider the angles around a vertex of the pentagon. The angle adjacent to the interior angle of the pentagon (in the triangle) is \(180^{\circ}-108^{\circ}=72^{\circ}\). Since the base angles of each triangle are congruent (because all the triangles have a side - length in common, the side between the two base angles, so they are congruent by \(SAS\) (Side - Angle - Side) criterion).

Step 2: Isosceles triangle property

In a triangle, if two angles are equal, then the sides opposite to them are equal. Each triangle has two equal base angles. So, each triangle is isosceles.

Step 3: Congruence of triangles

All of the triangles each have a side - length in common (the side between the two base angles). Also, we have already established that the base angles are equal. By \(SAS\) (Side - Angle - Side) congruence criterion, all the triangles are congruent.

Answer:

The base angles of each triangle in the star are equal since they are the interior angles of a regular pentagon. Since the base angles of each triangle are congruent, each triangle is isosceles. All of the triangles each have a side - length in common (the side between the two base angles). So, they are all congruent by \(SAS\) (Side - Angle - Side) criterion.