If two lines are cut by a transversal in such a way that alternate interior angles
are congruent, then the lines are parallel.
Given: Use the lines and angles numbering system shown on page 156 in your online textbook.
Alternate interior angles ∠3 and ∠6 are congruent ( ∠3 ≅ ∠6 ).
Prove: The horizontal lines in the figure are parallel.
Proof:
Statements -
1. ∠3≅∠6
2. m∠3 = m∠6
3. ∠3 and ∠4 are a _______ pair.
4. m∠3 + m∠4 = 180°
5. m∠6 + m∠4 = 180°
6. ∠4 and ∠6 are ________.
7. ∠4 and ∠6 are same-side ______ angles.
8. The horizontal lines in the figure are _______.
Reasons -
1. Given
2. ______ Congruence Postulate (think it’s angle)
3. Definition of ____ pair.
4. _____ Pair Property ( think linear)
5. _____ Property.
6. Definition of _____ angles
7. ____ of same-side _____ angles
8. Converse of same-side alternate angles theorem.
Please help ! I’m open to learning and any help is appreciated !
To prove that the horizontal lines in the figure are parallel, we need to use the information given and the properties and theorems related to angles and parallel lines.
Proof:
Statements -
1. ∠3≅∠6 (Given)
2. m∠3 = m∠6 (Congruence Postulate)
3. ∠3 and ∠4 are a pair of alternate interior angles.
4. m∠3 + m∠4 = 180° (Linear Pair Property)
5. m∠6 + m∠4 = 180° (Linear Pair Property)
6. ∠4 and ∠6 are congruent. (Vertical Angles Theorem)
7. ∠4 and ∠6 are same-side interior angles.
8. The horizontal lines in the figure are parallel.
Reasons -
1. Given.
2. Angles that are congruent have equal measures.
3. The angles ∠3 and ∠4 are formed on opposite sides of the transversal and between the two lines, thus they are alternate interior angles.
4. If two angles form a linear pair, then their measures add up to 180°.
5. Same reason as statement 4, but using angles ∠6 and ∠4.
6. Vertical angles are congruent.
7. Same-side interior angles are angles that are on the same side of the transversal and inside the two lines.
8. If alternate interior angles are congruent, then the lines are parallel. This is the converse of the same-side alternate angles theorem.
By using the properties and theorems mentioned above, we have proven that the horizontal lines in the figure are parallel.