Question

For questions 1 and 2, use the diagram below.

Answer 1

1) it's just 360 divided by 8 sectors =  45 degrees (answer B)
2) (180-45)/2 = 67.50 (answer D is the nearest)

Answer 2

Answer:  The correct options are (1). B, (2). D.

Step-by-step explanation:  We are given a figure with a regular octagon where the radius and an apothem are drawn.

(1) We are to find the measure of ∠1.

The radius of the regular octagon divides it into 8 congruent isosceles triangles, where the sum of the vertex angles of each of the triangles is 360°.

If, 'β' denotes the measure of each vertex angle, then we must have

$8\beta=360^\circ\\\\\Rightarrow \beta=\dfrac{360^\circ}{8}\\\\\Rightarrow \beta=45^\circ.$

Since ∠1 is also a vertex angle of one of the isosceles triangles.

Therefore,

m∠1 = β

⇒m∠1 = 45°.

Thus, option (B) is correct.

(2) We are to find the measure of ∠2.

The measures of the two base angles opposite to the equal sides of an isosceles triangle are equal.

So, from the isosceles triangle with one of the base angles as ∠2 that

$45^\circ+m\angle 2+m\angle 2=180^\circ\\\\\Rightarrow 45^\circ+2\times m\angle 2=180^\circ\\\\\Rightarrow 2\times m\angle 2=135^\circ\\\\\Rightarrow m\angle 2=67.5^\circ.$

Thus, option (D) is correct.

The correct options are (1). B, (2). D.