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
a. m<5=40
b. m<2=140
c. <5 and <2 are supplementary angles
d. <5 and <2 are same side interior angles
e. a || b
a. Given
b. Given
c. Definition of Supplementary Angles
d. Definition of Same Side Interior Angles
e. Converse of Alternate Interior Angles Theorem
To identify the reasons for each statement in the flow proof, let's carefully analyze the given information and the steps in the proof.
Given:
m<5 = 40
m<2 = 140
Proof:
a. Statement: m<5 = 40
Reason: Given information
b. Statement: m<2 = 140
Reason: Given information
c. Statement: <5 and <2 are supplementary angles
Reason: Two angles are supplementary if the sum of their measures is 180 degrees. In this case, since m<5 + m<2 = 40 + 140 = 180, <5 and <2 are supplementary angles.
d. Statement: <5 and <2 are same side interior angles
Reason: Two angles are same side interior angles if they are on the same side of a transversal and between the two parallel lines. Since a∥b is what we need to prove, it suggests that we are working with parallel lines and a transversal. Given that <5 and <2 are supplementary angles (from step c), and supplementary angles are on opposite sides of the transversal, it follows that <5 and <2 are same side interior angles.
e. Statement: a ∥ b
Reason: From step d, we conclude that <5 and <2 are same side interior angles. In a parallel line scenario, the presence of same side interior angles implies that the lines are parallel. Therefore, we can assert that a is parallel to b.
In summary, the reasons for each step in the flow proof are as follows:
a. Given information.
b. Given information.
c. Definition of supplementary angles.
d. Definition of same side interior angles.
e. Conclusion based on the properties of same side interior angles.
The reasons for each statement in the flow proof are as follows:
a. m<5 = 40 (Given)
Reason: This statement is given in the problem.
b. m<2 = 140 (Given)
Reason: This statement is also given in the problem.
c. <5 and <2 are supplementary angles
Reason: When the measures of two angles add up to 180 degrees, they are called supplementary angles.
d. <5 and <2 are same side interior angles
Reason: Same side interior angles are formed when two parallel lines are intersected by a transversal, and they occupy the same position on either side of the transversal.
e. a || b
Reason: Since <5 and <2 are both supplementary and same side interior angles, it indicates that lines a and b are parallel (by the Converse of the Same Side Interior Angles Theorem). Hence, "a || b" can be concluded.