Let's work through the image you provided and solve the problems for the numbered angles.

Problem 25:

• You are given that $m\angle 1 = 65^\circ$.
• Angles $\angle 1$ and $\angle 2$ form a pair of vertical angles. Vertical angles are always congruent, meaning they have the same measure.

Thus, $m\angle 2 = m\angle 1 = 65^\circ$.

Problem 26:

• You are given that $m\angle 2 = 67^\circ$.
• Angles $\angle 1$ and $\angle 2$ are supplementary angles because they are on a straight line. Supplementary angles sum up to $180^\circ$.

To find $m\angle 1$: $m\angle 1 = 180^\circ - m\angle 2 = 180^\circ - 67^\circ = 113^\circ$

Thus, $m\angle 1 = 113^\circ$.

Problem 27:

• You are given that $m\angle 3 = 38^\circ$.
• $\angle 3$ and $\angle 4$ are complementary angles because $\angle 4$ is a right angle (denoted by the square symbol), meaning the two angles sum up to $90^\circ$.

To find $m\angle 4$: $m\angle 4 = 90^\circ - m\angle 3 = 90^\circ - 38^\circ = 52^\circ$

Thus, $m\angle 4 = 52^\circ$.

Summary of Answers:

• Problem 25: $m\angle 2 = 65^\circ$
• Problem 26: $m\angle 1 = 113^\circ$
• Problem 27: $m\angle 4 = 52^\circ$

Would you like more details on any of these calculations, or do you have further questions?

Here are 5 related questions to enhance your understanding:

1. What is the relationship between vertical angles?
2. How do we identify complementary and supplementary angles?
3. Can two angles be both complementary and supplementary? Why or why not?
4. If two angles are supplementary and one is 45 degrees, what is the measure of the other angle?
5. How would you solve for an unknown angle if given the relationship between three or more angles?

Tip: Remember, complementary angles add up to 90 degrees, while supplementary angles add up to 180 degrees!