Question

which lines can be proved to be parallel from each given statement? name the postulate or theorem that justifies your answer. write none if there is not enough information to prove that two lines are parallel. 8. ∠1 ≅ ∠5 9. ∠7 ≅ ∠15 10. ∠10 ≅ ∠16 11. ∠1 ≅ ∠12 12. ∠4 ≅ ∠10 13. ∠12 and ∠15 are supplementary. 14. ∠7 and ∠14 are supplementary. 15. m∠5 = (2x - 3)° and m∠15 = (2x - 3)° 16. m∠4 = (3x + 50)° and m∠8 = (130 - 3x)°

Explanation:

8. $\angle1\cong\angle5$

- **Step1: Identify the relationship between the angles**
$\angle1$ and $\angle5$ are corresponding angles.
- **Step2: Apply the corresponding - angles postulate**
If two lines are cut by a transversal and the corresponding angles are congruent, then the two lines are parallel. Here, lines \(a\) and \(b\) are cut by transversal \(c\). So, \(a\parallel b\) by the Corresponding Angles Postulate.

9. $\angle7\cong\angle15$

- **Step1: Identify the relationship between the angles**
$\angle7$ and $\angle15$ are alternate - exterior angles.
- **Step2: Apply the alternate - exterior angles theorem**
If two lines are cut by a transversal and the alternate - exterior angles are congruent, then the two lines are parallel. Lines \(c\) and \(d\) are cut by transversal \(a\). So, \(c\parallel d\) by the Alternate Exterior Angles Theorem.

10. $\angle10\cong\angle16$

- **Step1: Identify the relationship between the angles**
$\angle10$ and $\angle16$ are alternate - interior angles.
- **Step2: Apply the alternate - interior angles theorem**
If two lines are cut by a transversal and the alternate - interior angles are congruent, then the two lines are parallel. Lines \(a\) and \(b\) are cut by transversal \(d\). So, \(a\parallel b\) by the Alternate Interior Angles Theorem.

11. $\angle1\cong\angle12$

- **Step1: Analyze the position of the angles**
$\angle1$ and $\angle12$ are not in a position (corresponding, alternate - interior, alternate - exterior, same - side interior) that would allow us to use a parallel - line postulate or theorem. So, there is not enough information.

12. $\angle4\cong\angle10$

- **Step1: Identify the relationship between the angles**
$\angle4$ and $\angle10$ are alternate - interior angles.
- **Step2: Apply the alternate - interior angles theorem**
If two lines are cut by a transversal and the alternate - interior angles are congruent, then the two lines are parallel. Lines \(c\) and \(d\) are cut by transversal \(b\). So, \(c\parallel d\) by the Alternate Interior Angles Theorem.

13. $\angle12$ and $\angle15$ are supplementary

- **Step1: Identify the relationship between the angles**
$\angle12$ and $\angle15$ are same - side interior angles.
- **Step2: Apply the same - side interior angles theorem**
If two lines are cut by a transversal and the same - side interior angles are supplementary, then the two lines are parallel. Lines \(c\) and \(d\) are cut by transversal \(a\). So, \(c\parallel d\) by the Same - Side Interior Angles Theorem.

14. $\angle7$ and $\angle14$ are supplementary

- **Step1: Analyze the position of the angles**
$\angle7$ and $\angle14$ are not in a position (corresponding, alternate - interior, alternate - exterior, same - side interior) that would allow us to use a parallel - line postulate or theorem. So, there is not enough information.

15. $m\angle5=(2x - 3)^{\circ}$ and $m\angle15=(2x - 3)^{\circ}$

- **Step1: Use the given angle measures**
Since \(m\angle5=m\angle15\), \(\angle5\cong\angle15\).
- **Step2: Identify the relationship between the angles**
$\angle5$ and $\angle15$ are corresponding angles.
- **Step3: Apply the corresponding angles postulate**
Lines \(c\) and \(d\) are cut by transversal \(a\). So, \(c\parallel d\) by the Corresponding Angles Postulate.

16. $m\angle4=(3x + 50)^{\circ}$ and $m\angle8=(130-3x)^{\circ}$

- **Step1: Find the sum of the angle measures**
\(m\angle4 + m\angle8=(3x + 50)+(130-3x)=180^{\circ}\).
- **Step2: Identify the relationship between the angles**
$\angle4$ and $\angle8$ are same - side exterior angles.
- **Step3: Apply the same - side exterior angles converse**
Lines \(a\) and \(b\) are cut by transversal \(c\). So, \(a\parallel b\) (since if two lines are cut by a transversal and the same - side exterior angles are supplementary, the lines are parallel).

Answer:

1. \(a\parallel b\), Corresponding Angles Postulate
2. \(c\parallel d\), Alternate Exterior Angles Theorem
3. \(a\parallel b\), Alternate Interior Angles Theorem
4. None
5. \(c\parallel d\), Alternate Interior Angles Theorem
6. \(c\parallel d\), Same - Side Interior Angles Theorem
7. None
8. \(c\parallel d\), Corresponding Angles Postulate
9. \(a\parallel b\), Same - Side Exterior Angles (converse)