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 definition of congruent. we also know that, by definition, ∠3 and ∠1 are a linear pair so they are supplementary by the definition of supplementary angles. 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 angle - pair relationship

Given \(w\parallel x\) and \(y\) is a transversal. \(\angle1\) and \(\angle5\) are corresponding angles. Since \(w\parallel x\), by the corresponding - angles postulate, \(\angle1\cong\angle5\), so \(m\angle1 = m\angle5\).

Step2: Use linear - pair property

\(\angle3\) and \(\angle1\) are a linear pair. By the definition of a linear pair, \(m\angle3 + m\angle1=180^{\circ}\) (linear - pair postulate).

Step3: Substitute

Since \(m\angle1 = m\angle5\), we substitute \(m\angle5\) for \(m\angle1\) in the equation \(m\angle3 + m\angle1 = 180^{\circ}\). We get \(m\angle3 + m\angle5=180^{\circ}\). By the definition of supplementary angles, \(\angle3\) and \(\angle5\) are supplementary.

Answer:

The first blank is "corresponding - angles postulate", the second blank is "linear - pair postulate", and the third blank is "substitution property of equality".