a How many congruent triangles do the radii form? What is the angle measure of a full turn?
B
b Since the radii have the same length, the triangles formed by the radii are isosceles triangles.
A
a 45^(∘)
B
b 67.5^(∘)
Practice makes perfect
a We want to find the measures of the angles formed by two consecutive radii in a regular polygon with 8 sides. Since the radii are congruent and the sides of the octagon are also congruent, the 8 triangles formed are congruent.
Recall that corresponding angles of congruent figures are congruent. Therefore, the vertex angles of the isosceles triangles formed by the radii are congruent.
Since the measure of a full turn is 360^(∘), we can find the measure of an angle formed by two consecutive radii by dividing 360 by 8.
360/8= 45^(∘)
The measure of the angles formed by two consecutive radii is 45^(∘).
b As shown in Part A, by drawing the radii we form 8 congruent isosceles triangles. Therefore, the angles formed by two radii and a side of the polygon are all congruent.
Consider just one of these triangles, and remember that the angle formed by two radii measures 45^(∘). Let x be the measure of the missing congruent angles.
By the Triangle Angle Sum Theorem we know that the measures of the three angles of a triangle add to 180^(∘). We can use this theorem to write an equation in terms of x.
x+ x+ 45=180
Let's solve our equation!
