Looking at the image, I can see four problems where we need to find x so that lines a and b are parallel, and state which converse theorem was used.
Let me work through each problem:
Problem 1:
- We have two angles: (10x - 23)° and 137°
- These are alternate interior angles
- For lines to be parallel, alternate interior angles must be equal
- So: 10x - 23 = 137
- Add 23 to both sides: 10x = 160
- Divide by 10: x = 16
- Converse used: Alternate Interior Angles Converse
Problem 2:
- We have angles: 74° and (9x + 11)°
- These are corresponding angles
- For lines to be parallel, corresponding angles must be equal
- So: 9x + 11 = 74
- Subtract 11 from both sides: 9x = 63
- Divide by 9: x = 7
- Converse used: Corresponding Angles Converse
Problem 3:
- We have angles: 96° and (6x - 30)°
- These are consecutive interior angles (same-side interior angles)
- For lines to be parallel, consecutive interior angles must be supplementary (add to 180°)
- So: 96 + (6x - 30) = 180
- Simplify: 6x + 66 = 180
- Subtract 66 from both sides: 6x = 114
- Divide by 6: x = 19
- Converse used: Consecutive Interior Angles Converse
Problem 4:
- We have angles: 33° and (7x + 5)°
- These are alternate exterior angles
- For lines to be parallel, alternate exterior angles must be equal
- So: 7x + 5 = 33
- Subtract 5 from both sides: 7x = 28
- Divide by 7: x = 4
- Converse used: Alternate Exterior Angles Converse
Final Answer:
1. x = 16, Alternate Interior Angles Converse
2. x = 7, Corresponding Angles Converse
3. x = 19, Consecutive Interior Angles Converse
4. x = 4, Alternate Exterior Angles Converse
Parent Tip: Review the logic above to help your child master the concept of geometry proving lines parallel worksheet.
