Question

practice & problem solving
what is the value of x in each figure?
11.
12.
13.
14.

1. make sense and persevere during a storm,

a tree is blown against a building so that it
forms a triangle with remote interior angles
of 90° and 52°. what is the measure of the
corresponding exterior angle formed by the
leaning tree?

Explanation:

Step1: Use the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). For problem 11:
\(x + 50+74=180\)
\(x=180-(50 + 74)\)
\(x = 56\)

Step2: Use the exterior - angle theorem

The exterior - angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.
For problem 12: \(x=72 + 63\), so \(x = 135\)
For problem 13: First, find the non - adjacent interior angle to the \(152^{\circ}\) exterior angle. Let \(y\) be the non - adjacent interior angle. Then \(y=180 - 152=28^{\circ}\). Then, using the exterior - angle theorem for the other exterior angle (\(103^{\circ}\)), \(x+28 = 103\), so \(x=103 - 28=75\)
For problem 14: First, find the non - adjacent interior angle to the \(66^{\circ}\) exterior angle. Let \(z\) be the non - adjacent interior angle. Then \(z = 180-66 = 114^{\circ}\). Using the triangle - angle sum theorem in the small triangle: \(x+40+114=180\), \(x=180-(40 + 114)\), \(x = 26\)

Step3: Use the exterior - angle theorem for problem 15

Given two remote interior angles \(90^{\circ}\) and \(52^{\circ}\). By the exterior - angle theorem, the measure of the exterior angle \(E\) is \(E=90 + 52\)

Answer:

1. \(x = 56\)
2. \(x = 135\)
3. \(x = 75\)
4. \(x = 26\)
5. The measure of the exterior angle is \(142^{\circ}\)