Question

proofs continued

• prove relationships and theorems about angles and lines.

vocabulary

• corresponding angles theorem - if two parallel lines are cut by a transversal.
• alternate interior angles theorem - if two parallel lines are cut by a transversal.
• alternate exterior angles theorem - if two parallel lines are cut by a transversal.
• consecutive interior angles theorem - if two parallel lines are cut by a transversal.

practice
given: m || n
prove: ∠1 ≅ ∠3
statements reasons

1. m || n 1. given
2. ∠1 ≅ ∠2 2.
3. ∠2 ≅ ∠3 3.
4. ∠1 ≅ ∠3 4.

given: m || n and a || b
prove: ∠1 ≅ ∠3
statements reasons

1. m || n and a || b 1. given
2. ∠1 ≅ ∠2 2.
3. ∠2 ≅ ∠3 3.
4. ∠1 ≅ ∠3 4.

given: m || n
prove: m∠1 + m∠3 = 180°
statements reasons

1. m || n 1. given
2. m∠1 = m∠2 2.
3. m∠2 + m∠3 = 180° 3.
4. m∠1 + m∠3 = 180° 4.

given: m || n and a || b || c
prove: ∠1 ≅ ∠4
statements reasons

1. m || n and a || b || c 1. given
2. ∠1 ≅ ∠2 2.
3. ∠2 ≅ ∠3 3.
4. ∠3 ≅ ∠4 4.
5. ∠1 ≅ ∠4 5.

Explanation:

Step1: Recall corresponding - angles theorem

If two parallel lines are cut by a transversal, corresponding angles are congruent. In the first practice problem (left - hand side), since \(m\parallel n\), \(\angle1\) and \(\angle2\) are corresponding angles, so \(\angle1\cong\angle2\) (by the Corresponding Angles Theorem).

Step2: Recall vertical - angles theorem

Vertical angles are congruent. \(\angle2\) and \(\angle3\) are vertical angles, so \(\angle2\cong\angle3\) (by the Vertical Angles Theorem).

Step3: Use the transitive property of congruence

Since \(\angle1\cong\angle2\) and \(\angle2\cong\angle3\), then \(\angle1\cong\angle3\) (by the Transitive Property of Congruence).

For the second practice problem (top - right):

Step1: Recall corresponding - angles theorem

Given \(m\parallel n\) and \(a\parallel b\), \(\angle1\) and \(\angle2\) are corresponding angles, so \(\angle1\cong\angle2\) (by the Corresponding Angles Theorem).

Step2: Recall alternate - interior angles theorem

Since \(a\parallel b\), \(\angle2\) and \(\angle3\) are alternate - interior angles, so \(\angle2\cong\angle3\) (by the Alternate Interior Angles Theorem).

Step3: Recall vertical - angles theorem

\(\angle3\) and \(\angle4\) are vertical angles, so \(\angle3\cong\angle4\) (by the Vertical Angles Theorem).

Step4: Use the transitive property of congruence

Since \(\angle1\cong\angle2\), \(\angle2\cong\angle3\) and \(\angle3\cong\angle4\), then \(\angle1\cong\angle4\) (by the Transitive Property of Congruence).

For the third practice problem (bottom - left):

Step1: Recall corresponding - angles theorem

Given \(m\parallel n\), \(\angle1\) and \(\angle2\) are corresponding angles, so \(m\angle1 = m\angle2\) (by the Corresponding Angles Theorem).

Step2: Recall consecutive - interior angles theorem

\(\angle2\) and \(\angle3\) are consecutive interior angles. Since \(m\parallel n\), \(m\angle2 + m\angle3=180^{\circ}\) (by the Consecutive Interior Angles Theorem).

Step3: Substitute

Substitute \(m\angle1\) for \(m\angle2\) in the equation \(m\angle2 + m\angle3 = 180^{\circ}\), we get \(m\angle1 + m\angle3=180^{\circ}\).

Answer:

For the first problem:
Statements Reasons

1. \(m\parallel n\) 1. Given
2. \(\angle1\cong\angle2\) 2. Corresponding Angles Theorem
3. \(\angle2\cong\angle3\) 3. Vertical Angles Theorem
4. \(\angle1\cong\angle3\) 4. Transitive Property of Congruence

For the second problem:
Statements Reasons

1. \(m\parallel n\) and \(a\parallel b\) 1. Given
2. \(\angle1\cong\angle2\) 2. Corresponding Angles Theorem
3. \(\angle2\cong\angle3\) 3. Alternate Interior Angles Theorem
4. \(\angle3\cong\angle4\) 4. Vertical Angles Theorem
5. \(\angle1\cong\angle4\) 5. Transitive Property of Congruence

For the third problem:
Statements Reasons

1. \(m\parallel n\) 1. Given
2. \(m\angle1 = m\angle2\) 2. Corresponding Angles Theorem
3. \(m\angle2 + m\angle3 = 180^{\circ}\) 3. Consecutive Interior Angles Theorem
4. \(m\angle1 + m\angle3 = 180^{\circ}\) 4. Substitution Property