Question

given: w ∥ x and y is a transversal. prove: ∠3 and ∠5 are supplementary. use the drop-down menus to complete the proof. given that w ∥ x and y is a transversal, we know that ∠1 ≅ ∠5 by the drop-down. therefore, m∠1 = m ∠5 by the definition of congruent. we also know that, by definition, ∠3 and ∠1 are a linear pair so they are supplementary by the drop-down. by the drop-down, m∠3 + m ∠1 = 180. now we can substitute m∠5 for m∠1 to get m∠3 + m∠5 = 180. therefore, by the definition of supplementary angles, ∠3 and ∠5 are supplementary.

Explanation:

Step1: Identify the first property

When two parallel lines are cut by a transversal, corresponding angles are congruent. So \( \angle 1 \cong \angle 5 \) by the Corresponding Angles Postulate.

Step2: Identify the second property

A linear pair of angles is supplementary by the Linear Pair Postulate (which states that if two angles form a linear pair, then they are supplementary).

Step3: Identify the third property

The equation \( m\angle 3 + m\angle 1 = 180 \) comes from the definition of supplementary angles (which is what the Linear Pair Postulate is based on, but more directly, the definition of supplementary angles says that if two angles are supplementary, their measures add up to 180, and since \( \angle 3 \) and \( \angle 1 \) are a linear pair (hence supplementary), we use the definition of supplementary angles here (or the Linear Pair Postulate which is a specific case of supplementary angles for linear pairs). But in the proof flow, after stating they are a linear pair (so supplementary), the reason for \( m\angle 3 + m\angle 1 = 180 \) is the definition of supplementary angles (or Linear Pair Postulate, but the definition is more direct here).

But to fill the blanks:

1. First drop - down: Corresponding Angles Postulate (because \( \angle 1 \) and \( \angle 5 \) are corresponding angles when \( w\parallel x \) and \( y \) is a transversal)
2. Second drop - down: Linear Pair Postulate (because \( \angle 3 \) and \( \angle 1 \) form a linear pair, so they are supplementary by the Linear Pair Postulate)
3. Third drop - down: Substitution Property (Wait, no. Wait, the step is "By the [property], \( m\angle 3 + m\angle 1 = 180 \)". Since \( \angle 3 \) and \( \angle 1 \) are a linear pair, the reason is the Linear Pair Postulate (which says that linear pairs are supplementary, i.e., their measures sum to 180). Then, when we substitute \( m\angle 5 \) for \( m\angle 1 \), that's substitution. But let's re - examine:

Wait, the proof steps:

• Given \( w\parallel x \), \( y \) transversal, so \( \angle 1\cong\angle 5 \) (Corresponding Angles Postulate)
• \( m\angle 1 = m\angle 5 \) (def of congruent)
• \( \angle 3 \) and \( \angle 1 \) are linear pair, so they are supplementary (Linear Pair Postulate)
• By the definition of supplementary angles (since supplementary angles have a sum of 180), \( m\angle 3 + m\angle 1 = 180 \)
• Substitute \( m\angle 5 \) for \( m\angle 1 \): \( m\angle 3 + m\angle 5 = 180 \)
• Hence, \( \angle 3 \) and \( \angle 5 \) are supplementary (def of supplementary)

So the first blank: Corresponding Angles Postulate

Second blank: Linear Pair Postulate

Third blank: Definition of Supplementary Angles (or Linear Pair Postulate, but the definition of supplementary angles is that if two angles are supplementary, their measures add up to 180, and since \( \angle 3 \) and \( \angle 1 \) are supplementary (from linear pair postulate), then \( m\angle 3 + m\angle 1 = 180 \) by definition of supplementary angles)

Answer:

1. First drop - down: Corresponding Angles Postulate
2. Second drop - down: Linear Pair Postulate
3. Third drop - down: Definition of Supplementary Angles