The image contains a geometric problem where several angles are provided, and we are asked to prove that lines AB and CD are parallel.

Given information:

• ∠1 = 35°
• ∠2 = 35°
• ∠3 = 35°
• ∠4 = 145°

Task:

Prove that AB ∥ CD.

Proof:

1. Step 1: Analyze the angle pairs.

   • ∠1 and ∠4 are corresponding angles because AB and CD are cut by a transversal line.
   • Corresponding angles are equal if and only if the lines are parallel.

2. Step 2: Apply the given values.

   • From the problem, we know ∠1 = 35° and ∠4 = 145°.
   • Since these two angles are not equal, but supplementary (their sum is 180°), we can infer that AB ∥ CD.

Conclusion:

Since ∠1 and ∠4 are supplementary corresponding angles, lines AB and CD are parallel. Therefore, we have proven that AB ∥ CD.

Would you like further details or clarifications?

Here are 5 related questions to deepen your understanding:

1. What are corresponding angles in a transversal setup?
2. How do you prove lines are parallel using alternate interior angles?
3. Can supplementary angles always prove parallelism between two lines?
4. How do we identify transversal lines in geometric figures?
5. How does the sum of angles in a triangle relate to proving parallelism?

Tip: In geometry, always look for corresponding and alternate interior angles when working with parallel lines and transversals.