There are several things we can see by investigating the given figure.
There are markers on two segments of the figure. These markers indicate that these segments are congruent, so the triangle they span is isosceles. Let's keep this in mind.
The measure of two angles are given on the figure. These are two angles of the large triangle.
The second observation will help us find the measure of ∠ 1 and ∠ 2 together. Using the given measure, m∠ 5=66^(∘), the first observation will help us find the measure of ∠ 1.
Finding m∠ 1
First let's concentrate on the isosceles triangle that the markers indicate.
According to the Isosceles Triangle Theorem, angles ∠ 5 and ∠ 1 are congruent, and hence have the same measure. Since the measure of ∠ 5 is given, this tells us the measure of ∠ 1.
m∠ 1=m∠ 5 m∠ 5=66^(∘)
⟹ m∠ 1=66^(∘)
The measure of ∠ 1 is 66^(∘).
Finding m∠ 2
Let's concentrate now on the large triangle.
According to the Triangle Angle-Sum Theorem, the angle measures of this green triangle add to 180^(∘). The Angle Addition Postulate guarantees that this is true even if an angle is split in two, like we have in this case.
m∠ 6+m∠ 5+m∠ 2+m∠ 1=180^(∘)
We can substitute the known measures in this equation to get the measure of ∠ 2.
