Question

Mara is building a set of shelves supported by a diagonal brace, as shown in the diagram. Answer the questions to decide whether all three shelves are parallel. 1. If ∠2 measures 120°, what is the measure of ∠1? Explain how you found the measure of ∠1. (The image) 2. Think of the top and middle shelves as two lines cut by a transversal. What type of angles are ∠1 and ∠5? 3. Use the relationship between ∠1 and ∠5 to decide whether the top and middle shelves are parallel. 4. Is the bottom shelf parallel to the top shelf? Explain. Write your answer in the space below. 5. Is the bottom shelf parallel to the middle shelf? Explain.

Answer 1

Parallel Lines

1. Angles ∠1 and ∠2 are linear because they form a line. The sum of their measures is 180°: ∠1 + ∠2 = 180°.

Given m∠2 = 120°, then

m∠1 = 180° - m∠2 = 180° - 120° = 60°

Thus:

m∠1 = 60°

2. Angles ∠1 and ∠5 are called corresponding pair of angles.

3. We have found that m∠1 = 60° and we are given that m∠5 = 60°.

If corresponding angles are congruent, then the lines must be parallel.

4. Given that m∠5 = 60° and ∠7 and ∠5 are linear, thus m∠7 = 120°.

Being ∠7 and ∠11 corresponding angles and having the same measure, the middle and the bottom shelf are parallel.