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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.

Find m∠1 and m∠2. Tell which theorem can be used.

Learn how to find the measures of angles using the Vertical Angles Congruence Theorem and the Linear Pair Postulate. This problem involves understanding the relationships between vertical angles and adjacent angles that form a linear pair, suitable for students in Grades 8-10.

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