Question

1.
triangle with angles 59°, 76°, and angle 1.
$m\angle1 = \underline{\quad\quad}$

2.
triangle with angles 67°, 62°, and a line with angle 1.
$m\angle1 = \underline{\quad\quad}$

3.
triangle with angle 115°, a line with angle 152°, and angle 1.
$m\angle1 = \underline{\quad\quad}$

Explanation:

Step1: Recall Triangle Angle Sum

The sum of angles in a triangle is \(180^\circ\). For a triangle, \(m\angle1 + \text{other two angles} = 180^\circ\). For linear pairs, adjacent angles sum to \(180^\circ\).

Step2: Solve for \(m\angle1\) in Problem 1

Given angles \(59^\circ\) and \(76^\circ\).
\(m\angle1 = 180^\circ - 59^\circ - 76^\circ = 180^\circ - 135^\circ = 45^\circ\).

Step3: Solve for \(m\angle1\) in Problem 2

First, find the third angle of the triangle: \(180^\circ - 67^\circ - 62^\circ = 51^\circ\).
\(m\angle1\) and this angle form a linear pair, so \(m\angle1 = 180^\circ - 51^\circ = 129^\circ\).

Step4: Solve for \(m\angle1\) in Problem 3

First, find the adjacent angle to \(152^\circ\): \(180^\circ - 152^\circ = 28^\circ\).
The triangle has angles \(115^\circ\), \(28^\circ\), and \(m\angle1\).
\(m\angle1 = 180^\circ - 115^\circ - 28^\circ = 37^\circ\).

Answer:

1. \(45^\circ\)
2. \(129^\circ\)
3. \(37^\circ\)