To solve for the measures of the numbered angles in the given diagram, we can use several geometric properties related to parallel lines, such as corresponding angles, alternate interior angles, and supplementary angles.

Steps:

1. Corresponding Angles: When two parallel lines are intersected by a transversal, corresponding angles are equal.
2. Alternate Interior Angles: These angles, formed on opposite sides of the transversal between the parallel lines, are equal.
3. Linear Pair: Adjacent angles on the same line sum to 180° (supplementary angles).

Let's use this approach to find the measure of each angle, given that the lines $m$ and $n$ are parallel, and the diagram gives us two known angles: $\angle 5 = 55^\circ$ and $\angle 8 = 96^\circ$.

Angle Measures:

• $\mathbf{m\angle 1}$: This is a corresponding angle to $\angle 5$. Since corresponding angles are congruent, $m\angle 1 = 55^\circ$.

• $\mathbf{m\angle 2}$: This is an alternate interior angle to $\angle 8$, so $m\angle 2 = 96^\circ$.

• $\mathbf{m\angle 3}$: This angle forms a linear pair with $\angle 1$, so $m\angle 3 = 180^\circ - 55^\circ = 125^\circ$.

• $\mathbf{m\angle 4}$: This is a vertical angle to $\angle 3$, so $m\angle 4 = 125^\circ$.

• $\mathbf{m\angle 5}$: Given in the diagram as $55^\circ$.

• $\mathbf{m\angle 6}$: This is a corresponding angle to $\angle 8$, so $m\angle 6 = 96^\circ$.

• $\mathbf{m\angle 7}$: This angle forms a linear pair with $\angle 6$, so $m\angle 7 = 180^\circ - 96^\circ = 84^\circ$.

• $\mathbf{m\angle 8}$: Given in the diagram as $96^\circ$.

Final Answers:

• $m\angle 1 = 55^\circ$
• $m\angle 2 = 96^\circ$
• $m\angle 3 = 125^\circ$
• $m\angle 4 = 125^\circ$
• $m\angle 5 = 55^\circ$
• $m\angle 6 = 96^\circ$
• $m\angle 7 = 84^\circ$
• $m\angle 8 = 96^\circ$

Would you like further explanations on these calculations or more details about geometric properties?

Here are 5 related questions to extend your learning:

1. How can you determine the measure of other angles using alternate exterior angles?
2. What property relates vertical angles and their measures?
3. What happens to angle measures if the transversal is perpendicular to the parallel lines?
4. How do you prove that two angles are supplementary or complementary in a diagram like this?
5. What is the difference between corresponding angles and alternate interior angles?

Tip: Remember, angles that form a linear pair always add up to 180°.