The image provided shows two intersecting lines, with angles labeled as $\angle 1$, $\angle 2$, and a given angle of 122°. You are asked to find the measures of $m\angle 1$ and $m\angle 2$ and identify the theorem used.

Step 1: Identify the Angle Relationships

• The given angle is 122°.
• $\angle 1$ and the 122° angle form a pair of vertical angles, which are always congruent.

Thus, $m\angle 1 = 122^\circ$, and this is justified by the Vertical Angles Congruence Theorem.

Step 2: Find $m\angle 2$

• $\angle 1$ and $\angle 2$ are linear pairs, meaning they are adjacent angles that sum to 180°. This can be justified by the Linear Pair Postulate.
• Therefore, $m\angle 2 = 180^\circ - m\angle 1 = 180^\circ - 122^\circ = 58^\circ$.

Final Answer:

• $m\angle 1 = 122^\circ$ by the Vertical Angles Congruence Theorem.
• $m\angle 2 = 58^\circ$ by the Linear Pair Postulate.

Would you like any more details, or do you have any questions about the solution?

Related Questions:

1. What is the definition of vertical angles?
2. Can linear pairs ever have congruent angles?
3. What theorem or postulate justifies that two angles in a linear pair sum to 180°?
4. How do corresponding angles behave when two lines are parallel?
5. What is the difference between alternate interior and alternate exterior angles?

Tip:

Always look for relationships like vertical angles and linear pairs when dealing with intersecting lines to quickly determine angle measures.