Exercise 8 Page 624

We are given a regular polygon with its apothem and radii drawn.

We will find the measures of ∠RCY, ∠RCZ, and ∠ZRC one at a time.

Finding m∠RCY

The radii divide the regular octagon into eight 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 8 to obtain their measures. 360^(∘)/8=45^(∘) The vertex angles of the isosceles triangles formed by the radii measure 45^(∘) each.

Therefore, m∠RCY is 45^(∘).

Finding m∠RCZ

The apothem bisects the vertex angle of the isosceles triangle formed by the radii. Since we know that the measure of the vertex angle is 45^(∘), we can divide 45 by 2 to obtain the measure of ∠RCZ. 45^(∘)/2=22.5^(∘)

The measure of ∠RCZ is 22.5^(∘).

Finding m∠ZRC

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 octagon. Therefore, we know the measures of two of the three interior angles of a right triangle. The missing angle is ∠ZRC.

To find the measure of ∠ZRC, we will use the Triangle Angle Sum Theorem. This theorem states that the sum of the three interior angles of a triangle is 180^(∘). 90^(∘)+22.5^(∘)+m∠ZRC=180^(∘) ⇕ m∠ZRC=67.5^(∘) The measure of ∠ZRC is 67.5^(∘).