Question

lesson 7 - 1 parallel lines cut by a transversal practice and problem solving: a/b use the figure at the right for exercises 1 - 6. 1. name both pairs of alternate interior angles. name the corresponding angle to ∠3. name the relationship between ∠1 and ∠5. 4. name the relationship between ∠2 and ∠3. 5. name an interior angle that is supplementary to ∠7. 6. name an exterior angle that is supplementary to ∠5. use the figure at the right for problems 7 - 10. line mp || line qs. find the angle measures. m∠krq when m∠knm = 146°. 8. m∠qrn when m∠mnr = 52°. if m∠rnp=(8x + 63)° and m∠nrs = 5x°, find the following angle measures. 9. m∠rnp = 10. m∠nrs = in the figure at the right, there are no parallel lines. use the figure for problems 11 - 14. 11. name both pairs of alternate exterior angles. 12. name the corresponding angle to ∠4. 13. name the relationship between ∠3 and ∠6. 14. are there any supplementary angles? if so, name two pairs. if not, explain why not.

Explanation:

Step1: Recall angle - pair relationships for parallel lines cut by a transversal

For parallel lines \(a\) and \(b\) cut by a transversal \(c\):

• Alternate - interior angles are non - adjacent angles between the two parallel lines and on opposite sides of the transversal. In the first figure, the pairs of alternate - interior angles are \(\angle2\) and \(\angle7\), \(\angle3\) and \(\angle6\).
• Corresponding angles are in the same relative position with respect to the parallel lines and the transversal. The corresponding angle to \(\angle3\) is \(\angle7\).
• The relationship between \(\angle1\) and \(\angle5\) is corresponding angles.
• \(\angle2\) and \(\angle3\) are adjacent angles that form a linear pair, so they are supplementary.
• An interior angle supplementary to \(\angle7\) is \(\angle3\) (since they are a linear pair).
• An exterior angle supplementary to \(\angle5\) is \(\angle1\) (since they are a linear pair).

Step2: Use properties of parallel lines to find angle measures

For parallel lines \(MP\parallel QS\) cut by a transversal:

• If \(\angle KNM = 146^{\circ}\), and \(\angle KNM\) and \(\angle KRQ\) are corresponding angles, then \(m\angle KRQ=146^{\circ}\).
• If \(\angle MNR = 52^{\circ}\), and \(\angle MNR\) and \(\angle QRN\) are supplementary (linear pair), then \(m\angle QRN = 180 - 52=128^{\circ}\).
• Since \(\angle RNP\) and \(\angle NRS\) are alternate - interior angles for \(MP\parallel QS\), we have \(8x + 63=5x\) (this is incorrect in the setup, they should be supplementary for non - parallel case. For parallel lines, \(8x + 63+5x=180\)).

$$13x=180 - 63$$

$$13x = 117$$

$$x = 9$$

Then \(m\angle RNP=8x + 63=8\times9+63=72 + 63=135^{\circ}\) and \(m\angle NRS=5x=5\times9 = 45^{\circ}\).

For the non - parallel lines:

• Alternate exterior angles are non - adjacent angles outside the two lines and on opposite sides of the transversal. The pairs of alternate exterior angles are \(\angle1\) and \(\angle8\), \(\angle2\) and \(\angle7\).
• The corresponding angle to \(\angle4\) is \(\angle8\).
• The relationship between \(\angle3\) and \(\angle6\) is alternate interior angles.
• Supplementary angles exist. For example, \(\angle1\) and \(\angle2\) form a linear pair, and \(\angle3\) and \(\angle4\) form a linear pair.

Answer:

1. \(\angle2\) and \(\angle7\), \(\angle3\) and \(\angle6\)
2. \(\angle7\)
3. Corresponding angles
4. Supplementary (linear pair)
5. \(\angle3\)
6. \(\angle1\)
7. \(146^{\circ}\)
8. \(128^{\circ}\)
9. \(135^{\circ}\)
10. \(45^{\circ}\)
11. \(\angle1\) and \(\angle8\), \(\angle2\) and \(\angle7\)
12. \(\angle8\)
13. Alternate interior angles
14. Yes, for example, \(\angle1\) and \(\angle2\), \(\angle3\) and \(\angle4\)