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
[tex]8\beta=360^\circ\\\\\Rightarrow \beta=\dfrac{360^\circ}{8}\\\\\Rightarrow \beta=45^\circ.[/tex]
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
[tex]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.[/tex]
Thus, option (D) is correct.
The correct options are (1). B, (2). D.
