1. **Problem 13:** Given a horizontal line intersecting a diagonal line with a 39° angle shown in the upper-left. - We need to find $m\angle 1$, $m\angle 2$, and $m\angle 3$. - Important rule: Vertical angles are equal, and angles on a straight line sum to 180°. 2. **Step 1:** Identify angles. - $m\angle 3 = 39^\circ$ (given, vertical angle to the 39° angle). 3. **Step 2:** Find $m\angle 1$. - $m\angle 1$ and 39° are supplementary (on a straight line), so $$m\angle 1 + 39^\circ = 180^\circ$$ - Subtract 39° from both sides: $$m\angle 1 = 180^\circ - 39^\circ = 141^\circ$$ 4. **Step 3:** Find $m\angle 2$. - $m\angle 2$ is vertical to $m\angle 1$, so $$m\angle 2 = m\angle 1 = 141^\circ$$ --- 1. **Problem 14:** A vertical line intersects a diagonal line with a 73° angle shown in the upper-left. - Find $m\angle 1$, $m\angle 2$, and $m\angle 3$. - Important rule: Angles around a point sum to 360°, and supplementary angles sum to 180°. 2. **Step 1:** Given $m\angle 3 = 73^\circ$ (vertical angle). 3. **Step 2:** Find $m\angle 1$. - $m\angle 1$ and 73° are supplementary: $$m\angle 1 + 73^\circ = 180^\circ$$ - Subtract 73°: $$m\angle 1 = 180^\circ - 73^\circ = 107^\circ$$ 4. **Step 3:** Find $m\angle 2$. - $m\angle 2$ is vertical to $m\angle 1$, so $$m\angle 2 = 107^\circ$$ --- 1. **Problem 15:** Rectangle-like figure with angles 57°, 19°, and 112° given. - Find $m\angle 1$, $m\angle 2$, and $m\angle 3$. - Important rule: Angles in a triangle sum to 180°. 2. **Step 1:** Find $m\angle 1$. - $m\angle 1$ is supplementary to 112°: $$m\angle 1 + 112^\circ = 180^\circ$$ - Subtract 112°: $$m\angle 1 = 68^\circ$$ 3. **Step 2:** Find $m\angle 2$. - $m\angle 2$ is given as 57°. 4. **Step 3:** Find $m\angle 3$. - Sum of angles in triangle: $$m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ$$ - Substitute known values: $$68^\circ + 57^\circ + m\angle 3 = 180^\circ$$ - Simplify: $$125^\circ + m\angle 3 = 180^\circ$$ - Subtract 125°: $$m\angle 3 = 55^\circ$$