https://file.aiquickdraw.com/mathbot/file/1729093328717q6pbk50w.jpg

Let's go through the problems in the image.

### Question 2:
The diagram shows two parallel lines \(a\) and \(b\) intersected by a transversal \(t\), which creates various angles labeled 1 through 8.

We are asked to determine whether certain statements are **True** or **False**:

- **(a)** \(\angle 1 \cong \angle 5\)  
  **Answer:** **True**.  
  These are corresponding angles when line \(a\) is parallel to line \(b\), so they are congruent.

- **(b)** \(m\angle 2 = m\angle 8\)  
  **Answer:** **True**.  
  These are alternate exterior angles, and alternate exterior angles are equal when lines are parallel.

- **(c)** \(\angle 3\) and \(\angle 6\) are supplementary  
  **Answer:** **True**.  
  These are same-side interior angles (also known as consecutive interior angles), and they are supplementary (add up to \(180^\circ\)).

- **(d)** \(m\angle 4 + m\angle 7 = 180^\circ\)  
  **Answer:** **True**.  
  These are consecutive interior angles formed by the transversal, and they are supplementary.

### Question 3:
You are given a diagram with four lines (\(a\), \(b\), \(c\), and \(d\)) intersected by a transversal. The angles formed by the intersections are labeled with degree measures.

You are asked to determine which lines are parallel based on the given angle measures.

- **Line \(a\):** \(115^\circ\)
- **Line \(b\):** \(120^\circ\)
- **Line \(c\):** \(55^\circ\)
- **Line \(d\):** \(65^\circ\)

To determine parallelism, you check for corresponding or alternate angles that are equal or supplementary. 

- \(120^\circ\) (line \(b\)) and \(65^\circ\) (line \(d\)) are **not** supplementary (they do not add up to \(180^\circ\)). 
- However, lines \(a\) and \(c\) have corresponding angles of \(115^\circ\) and \(55^\circ\), respectively, which **do add up to 180°**. This means lines **\(a\) and \(c\)** must be parallel.

**Answer:** Line \(a\) is parallel to line \(c\).

Would you like more details or have any further questions on this?  
Here are 5 related questions to consider:
1. How do corresponding angles help identify parallel lines?
2. Can alternate interior angles be used to check parallelism?
3. Why are same-side interior angles always supplementary for parallel lines?
4. How can vertical angles help solve angle congruence problems in parallel lines?
5. What happens if the transversal is not perpendicular to the parallel lines?

**Tip:** Always remember that corresponding angles are congruent when lines are parallel, and consecutive interior angles are supplementary.

Given line a is parallel to line b, determine whether the following statements are True or False: 
(a) ∠1 ≅ ∠5
(b) m∠2 = m∠8
(c) ∠3 and ∠6 are supplementary
(d) m∠4 + m∠7 = 180°. 
Additionally, based on the angle measures provided, which pair of lines must be parallel?

Learn how to determine whether lines are parallel by identifying corresponding and supplementary angles. This problem covers key geometric concepts involving parallel lines, transversals, and angle relationships such as corresponding, alternate exterior, and supplementary angles.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation