f 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, the following steps can be followed:
Proof:
Statements -
1. ∠3 ≅ ∠6 (Given)
2. m∠3 = m∠6 (Congruence Postulate)
3. ∠3 and ∠4 are a linear pair (Definition of linear pair)
4. m∠3 + m∠4 = 180° (Linear Pair Property)
5. m∠6 + m∠4 = 180° (Linear Pair Property)
6. ∠4 and ∠6 are congruent (Definition of congruent angles)
7. ∠4 and ∠6 are same-side interior angles (Definition of same-side interior angles)
8. The horizontal lines in the figure are parallel (Converse of same-side alternate angles theorem)
Reasons -
1. Given: It is given that ∠3 ≅ ∠6.
2. Congruence Postulate: When two angles are congruent, their measures are equal.
3. Definition of linear pair: A linear pair consists of two adjacent angles whose sum is 180 degrees.
4. Linear Pair Property: The sum of the angles in a linear pair is 180 degrees.
5. Linear Pair Property: The sum of the angles in a linear pair is 180 degrees.
6. Definition of congruent angles: Congruent angles have equal measures.
7. Definition of same-side interior angles: Same-side interior angles are two interior angles that lie on the same side of the transversal and on the same side of the two lines being intersected.
8. Converse of same-side alternate angles theorem: If two lines are cut by a transversal in such a way that same-side interior angles are congruent, then the lines are parallel.
By applying the Converse of same-side alternate angles theorem (step 8), we can conclude that the horizontal lines in the figure are parallel.