Question

1. when constructing the roof of a house, the top angle must measure 90 degrees. if the other interior angle measures 45 degrees, what is the measure of the exterior angle the roof makes with the gutter?
2. a building makes a 90 - degree angle with the ground. the man standing on top of that building threw an apple at the bottom of the tree. if we know that the exterior angle from where the apple fell to the tree measures 122 degrees, what was the angle at which the apple was thrown?

Explanation:

Step1: Find the third interior angle of the triangle (for problem 9)

The sum of interior angles of a triangle is \(180^{\circ}\). Let the third interior angle be \(y\). Given two interior angles: \(90^{\circ}\) and \(45^{\circ}\). Then \(y = 180-(90 + 45)=45^{\circ}\). The exterior angle \(x\) and the interior angle \(y\) are supplementary (they form a linear - pair). So \(x=180 - 45\).

Step2: Use the exterior - angle theorem (for problem 10)

The exterior - angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. Let the angle at which the apple was thrown be \(x\). We know one interior angle is \(90^{\circ}\) and the exterior angle is \(122^{\circ}\). Then \(x=122 - 90\).

Answer:

For problem 9: \(135^{\circ}\). For problem 10: \(32^{\circ}\).