Question

name: measures of interior and exterior angles of triangles september 15, 2025 score: use the figure to find the measure of the angle. explain your reasoning. (section 3.1) 1. ∠2 2. ∠6 3. ∠4 4. ∠1 complete the statement. explain your reasoning. (section 3.1) 5. if the measure of ∠1 = 123°, then the measure of ∠7 = 6. if the measure of ∠2 = 58°, then the measure of ∠5 = 7. if the measure of ∠5 = 119°, then the measure of ∠3 = 8. if the measure of ∠4 = 60°, then the measure of ∠6 = find the measures of the interior angles. (section 3.2) 9. 10. 11. find the measure of the exterior angle. (section 3.2) 12. 13. 14. park in a park, a bike path and a horse - riding path are parallel. in one part of the park, a hiking trail intersects the two paths. find the measures of ∠1 and ∠2. explain your reasoning. (section 3.1) 15. ladder a ladder leaning against a wall forms a triangle and exterior angles with the wall and the ground. what are the measures of the exterior angles? justify your answer. (section 3.2)

Explanation:

Step1: Identify angle - relationships

For parallel - line problems, use corresponding, vertical, and supplementary angle relationships. For triangle problems, use the angle - sum property of a triangle ($180^{\circ}$ for interior angles and exterior - angle property).

Step2: Solve for angles in parallel - line cases

1. If two parallel lines are cut by a transversal:

• Vertical angles are equal. For example, if we consider the first set of parallel lines with a transversal, $\angle1$ and $\angle3$ are vertical angles, $\angle2$ and $\angle4$ are vertical angles, $\angle5$ and $\angle7$ are vertical angles, $\angle6$ and $\angle8$ are vertical angles.
• Corresponding angles are equal. Alternate interior angles are equal. Alternate exterior angles are equal. Supplementary angles on the same side of the transversal add up to $180^{\circ}$.

1. For example, in question 5, if $\angle1 = 123^{\circ}$, and $\angle1$ and $\angle7$ are corresponding angles (assuming parallel lines), then $\angle7=123^{\circ}$.

Step3: Solve for angles in triangle cases

1. The sum of the interior angles of a triangle is $180^{\circ}$. So, for a triangle with angles $A$, $B$, and $C$, $A + B + C=180^{\circ}$.

• In question 9, if two angles of a triangle are $60^{\circ}$ and $60^{\circ}$, then the third angle $x$ is $180-(60 + 60)=60^{\circ}$.

1. The measure of an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. In question 12, if the two non - adjacent interior angles of a triangle are $55^{\circ}$ and $50^{\circ}$, then the exterior angle $b$ is $55 + 50=105^{\circ}$.

Answer:

1. Without seeing the full figure for the first set of parallel - line questions, assume standard angle - relationships. If $\angle7 = 82^{\circ}$ (from the figure), and $\angle2$ and $\angle7$ are alternate exterior angles (assuming parallel lines), then $\angle2 = 82^{\circ}$.
2. $\angle6$: If we assume parallel lines and use angle - relationships, if $\angle2 = 82^{\circ}$ and $\angle2$ and $\angle6$ are corresponding angles, then $\angle6 = 82^{\circ}$.
3. $\angle4$: If $\angle2$ and $\angle4$ are vertical angles, then $\angle4=\angle2 = 82^{\circ}$.
4. $\angle1$: If $\angle1$ and $\angle2$ are supplementary (linear pair), and $\angle2 = 82^{\circ}$, then $\angle1=180 - 82=98^{\circ}$.
5. $123^{\circ}$ (corresponding angles).
6. If $\angle2 = 58^{\circ}$, and $\angle2$ and $\angle5$ are alternate interior angles (assuming parallel lines), then $\angle5 = 58^{\circ}$.
7. If $\angle5 = 119^{\circ}$, and $\angle3$ and $\angle5$ are corresponding angles (assuming parallel lines), then $\angle3 = 119^{\circ}$.
8. If $\angle4 = 60^{\circ}$, and $\angle4$ and $\angle6$ are alternate interior angles (assuming parallel lines), then $\angle6 = 60^{\circ}$.
9. $60^{\circ}$ ($180-(60 + 60)$).
10. $x=180-(40 + 25)=115^{\circ}$.
11. In a right - isosceles triangle, $x = 45^{\circ}$ ($180-90 = 90$, and since it's isosceles, $x=\frac{90}{2}=45^{\circ}$).
12. $105^{\circ}$ ($55 + 50$).
13. First, use the exterior - angle property: $(z + 50)=(z + 10)+4z$. Solving for $z$:

• Expand: $z + 50=z + 10+4z$.
• Combine like terms: $z+50 = 5z + 10$.
• Subtract $z$ from both sides: $50 = 4z+10$.
• Subtract 10 from both sides: $40 = 4z$.
• Divide by 4: $z = 10$. Then the exterior angle $(z + 50)$ is $60^{\circ}$.

1. If the two paths are parallel and the hiking trail is a transversal, if one angle is $72^{\circ}$, $\angle1 = 72^{\circ}$ (corresponding angle) and $\angle2=180 - 72 = 108^{\circ}$ (supplementary to the corresponding angle).
2. Without specific angle measures given for the ladder - wall - ground triangle, assume the interior angles of the triangle formed by the ladder, wall, and ground. If the interior angles of the triangle are $x$ and $5x$ and the right - angle ($90^{\circ}$), then $x+5x+90 = 180$, $6x=90$, $x = 15^{\circ}$. The exterior angles: One exterior angle adjacent to $x$ is $180 - 15=165^{\circ}$, and the exterior angle adjacent to $5x$ is $180 - 75 = 105^{\circ}$.