The flow proof shows the statements that lead to the conclusion that a∥b
in the diagram. The reasons for each statement in the flow proof are missing. Identify the reasons for each of the 5 statements shown in the flow proof. In your response, label the steps a-e.
Given m<5=40, m<2=140
Prove a || b
m<5=40 ->
a. _ <5 and <2 are supplementary angles ->
m<2=140. -> c. _ a || b
b. _ <5 and <2 are same side interior angles -> e. _
d. _
a. <5 and <2 are supplementary angles (Given)
b. <5 and <2 are same side interior angles (Given)
c. <2 = 140 (Given)
d. <5 = 40 (Given)
e. a || b (If two angles are supplementary and same side interior angles, then the lines containing those angles are parallel)
a. <5 and <2 are supplementary angles - Reason: Given
b. <5 and <2 are same side interior angles - Reason: Definition of same side interior angles
c. m<2=140 - Reason: Given
d. _ - Reason: Not provided in the given information
e. a || b - Reason: Definition of parallel lines
To determine the reasons for each statement in the given flow proof, let's analyze the information provided step by step:
Given:
m∠5 = 40° (Step a)
m∠2 = 140° (Step c)
Reasons for each statement:
Step a: <5 and <2 are supplementary angles
Supplementary angles are defined as two angles whose sum is 180 degrees. In this case, <5 and <2 are given to be supplementary angles because when you add their measures, the sum is 180 degrees. Therefore, the reason for this statement is that <5 and <2 are supplementary angles.
Step c: <5 and <2 are same side interior angles
Same side interior angles are a pair of angles formed when two lines are intersected by a transversal in a way that the angles are on the same side of the transversal and between the two lines. In the given diagram, <5 and <2 are formed in this manner since they both lie on the same side of the transversal and are between the lines a and b. Hence, the reason for this statement is that <5 and <2 are same side interior angles.
Step b: <5 and <2 are same side interior angles
This statement is a repetition of Step c, which already establishes that <5 and <2 are same side interior angles. Therefore, the reason for this statement remains the same as in Step c.
Step d: <5 and <2 are same side interior angles
This statement is a repetition of Steps c and b, again reaffirming that <5 and <2 are same side interior angles. Hence, the reason for this statement remains the same as in Steps c and b.
Step e: a || b
To prove that lines a and b are parallel, we need to establish the reason for this conclusion. Based on the previous steps, we can determine that if two interior angles on the same side of a transversal are supplementary, then the lines they intersect are parallel. Since <5 and <2 are supplementary angles as shown in Step a, we can conclude that lines a and b are parallel. Therefore, the reason for this statement is that two interior angles (<5 and <2) on the same side of the transversal are supplementary, indicating that lines a and b are parallel.
In summary, the reasons for each statement in the flow proof are as follows:
a. <5 and <2 are supplementary angles.
b. <5 and <2 are same side interior angles.
c. <5 and <2 are same side interior angles.
d. <5 and <2 are same side interior angles.
e. Two interior angles on the same side of a transversal are supplementary, indicating that lines a and b are parallel.