We are asked to find the measure of eight angles, but we do not have to do it in the order of the numbering. We will find the measures in the following order.
m∠ 5, m∠ 7, m∠ 1, m∠ 8, m∠ 6, m∠ 4, m∠ 3, m∠ 2
This is not the only order these angle measures can be found.
Even if you use a different order, your angle measures should match the measures we get.
Finding m∠ 5 and m∠ 7
Notice that angles ∠ 5 and ∠ 7 are in a special position to the angle with measure 110 given on the figure. These three angles are formed by two intersecting lines.
Angle ∠ 5 and the angle with the given measure are vertical angles, so their measure is the same.
m∠ 5=110^(∘)
Angles ∠ 5 and ∠ 7 together form a straight angle, so their measures add to 180^(∘).
The measure of ∠ 5 is 110^(∘), the measure of ∠ 7 is 70^(∘).
Let's add these measures to the figure.
Finding m∠ 1 and m∠ 8
Let's focus now on the triangle with angles ∠ 1 and ∠ 8. The markers on the figure indicate that these angles are congruent, so their measure is the same.
m∠ 1=m∠ 8
Notice that the measure of one of the exterior angles of this triangle is also given on the figure.
The Exterior Angle Theorem tells a relationship between these angles.
The measure of the exterior angle
is the sum of the measures
of the two remote interior angles.
This lets us set up and solve an equation for m∠ 1.
Since ∠ 1 and ∠ 8 are congruent, this means that the measure of both of these angles is 65^(∘).
Let's add these measures to the figure.
Finding m∠ 6
To find the measure of ∠ 6, let's focus on the triangle with this angle as one of its angles.
Notice that by now we know the measure of the other two angles.
The Triangle Angle-Sum Theorem tells a relationship between these angles.
The sum of the measures of the
interior angles of a triangle is180^(∘).
This lets us set up and solve an equation for m∠ 6.
The measure of ∠ 6 is 45^(∘). Let's add this measure to the figure.
Finding m∠ 4
To find the measure of ∠ 4, let's focus on the triangle with this angle as one of its angles.
Notice that by now we know the measure of the other two angles.
The measure of ∠ 3 is 95^(∘). Let's add this measure on the figure.
Finding m∠ 2
Finally, to find the measure of ∠ 2, let's focus on the triangle with this angle as one of its angles.
Notice that by now we know the measure of the other two angles.
