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The apothem makes a right angle with the side of the polygon and bisects the vertex angle of the isosceles triangle formed by the radii.
m∠RCY=45^(∘), m∠RCZ=22.5^(∘), ∠ZRC=67.5^(∘)
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.
360^(∘)/8=45^(∘) The vertex angles of the isosceles triangles formed by the radii measure 45^(∘) each.
Therefore, m∠RCY is 45^(∘).
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^(∘).
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^(∘).