Match the justifications for each step in order to complete the proof. Note that lines p and r are parallel.
Given: <1 =~ <5
Prove: p || r
<4=~ <1
-1=~<5
p || r
<4=~<5
Match:
Alternate Interior Angles converse
vertical angles
Given
Transitive property
Given - Given
<1 =~ <5 - Given
Alternate Interior Angles converse - <4 =~ <1
Transitive property - <4 =~ <5
vertical angles - -1 =~ <5
p || r - p || r
Given: <1 =~ <5
To prove: p || r
Proof:
1. <4 =~ <1 (Given)
2. -1 =~ <5 (Given)
3. p || r (Alternate Interior Angles converse)
4. <4 =~ <5 (Transitive property)
Therefore, we have successfully proven that p || r using the given justifications.
To complete the proof, let's match the justifications for each step:
1. Given: <1 =~ <5
Explanation: This is given in the problem statement.
2. <4 =~ <1
Justification: Alternate Interior Angles converse. This is applicable when two parallel lines are intersected by a transversal, and the corresponding angles on the opposite sides of the transversal are congruent.
3. -1 =~ <5
Justification: Vertical angles. Vertical angles are formed when two lines intersect, and they are congruent.
4. p || r
Justification: Given. This is stated in the problem.
5. <4 =~ <5
Justification: Transitive property. If two angles are congruent to the same angle, then they are congruent to each other.
By matching these justifications to the steps, we have completed the proof that p || r.