We are given a regular polygon with its apothem and radii drawn. We want to find the measure of each numbered angle.
Let's find the measures one at a time.
Angle 7
The radii divide the regular hexagon into six congruent isosceles triangles. Since corresponding angles of congruent figures are congruent, the vertex angles of the isosceles triangles formed by the radii are congruent. Moreover, since a full turn measures 360^(∘), we can divide 360 by 6 to obtain their measures.
360/6=60^(∘)
The vertex angles of the isosceles triangles formed by the radii measure 60^(∘) each.
Therefore, the measure of angle 7 is 60^(∘).
Angle 8
The apothem bisects the vertex angle of the isosceles triangle formed by the radii. Since we know that the measure of this angle is 60^(∘), we can divide 60 by 2 to obtain the measure of angle 8.
60/2=30^(∘)
The measure of angle 8 is 30^(∘).
Angle 9
As previously mentioned, the apothem bisects the vertex angle of the isosceles triangle formed by the radii. We can see in the diagram that the apothem makes a right angle with the side of the hexagon. Therefore, we know the measures of two of the three interior angles of a right triangle. The missing angle is angle 9.
To find the measure of angle 9 we will use the Triangle Angle Sum Theorem. This theorem states that the sum of the three interior angles of a triangle add to 180^(∘).
90+30+m∠9=180^(∘)
⇕
m∠9=60^(∘)
The measure of angle 9 is 60^(∘).
