Given: l || n
Prove: <4=~<6
Match the justifications for each step in order to prove the Alternate Interior Angles Theorem. Note that lines I and n are parallel.
<6=~<4
<4=~<2
<2=~<6
l || n
Match:
Corresponding angels
Vertical angles
Transitive Property
Givin
- Corresponding angles: <4 and <6 are alternate interior angles formed by the parallel lines l and n.
- Vertical angles: <4 and <2 are vertical angles.
- Transitive Property: If <4 and <2 are congruent, and <2 and <6 are congruent, then <4 and <6 must also be congruent.
- Given: The initial statement that lines l and n are parallel.
Given: l || n
Prove: <4 =~ <6
Step 1: Given
Step 2: Corresponding angles (Alternate Interior Angles Theorem) - This justification states that when a transversal intersects two parallel lines, the interior angles on the same side of the transversal and not adjacent to the lines have equal measures. Since <4 and <6 are alternate interior angles on the lines l and n, they are congruent.
Step 3: Transitive Property - This justification states that if two angles are congruent to the same angle, then they are congruent to each other. Therefore, since <4 =~ <2 and <2 =~ <6, we can conclude that <4 =~ <6.
Step 4: Given (l || n) - This justification simply restates the given information that the lines l and n are parallel.
Therefore, we have proven that <4 =~ <6, using the Alternate Interior Angles Theorem.
To prove the Alternate Interior Angles Theorem, we need to show that <4 and <6 are congruent. Here's how we can do it step-by-step:
Step 1: Given that line l is parallel to line n, we can use the fact that corresponding angles formed by a transversal intersecting parallel lines are congruent.
Step 2: By the Corresponding Angles Theorem, since <4 and <2 are corresponding angles, we can conclude that <4 and <2 are congruent.
Step 3: Using the Transitive Property of congruence, we can say that if <4 is congruent to <2, and <2 is congruent to <6 (which we will prove next), then <4 must be congruent to <6.
Step 4: By the Vertical Angles Theorem, we know that <2 and <6 are vertical angles. Vertical angles are always congruent, so we can conclude that <2 is congruent to <6.
Therefore, by using the Corresponding Angles Theorem, the Transitive Property, and the Vertical Angles Theorem, we have proven that <4 is congruent to <6, which completes the proof of the Alternate Interior Angles Theorem.