Question

parallel lines cut by a transversal
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find the measure of the angle indicated in bold.
1)
2)
3)
4)
5)
6)
date 9 nov 20 period 1
x + 137
x + 57
11x + 9
10x + 20
12x + 15
13x + 5
x + 83
x + 113
jane sicle interio
x + 113
x + 83
x + 113 + x + 83 = 180
196 + 2x + 180
196
196
2x -16 x = -8
x + 56
x + 146

Explanation:

Step1: Analyze Problem 1

The two angles are same - side interior angles (since parallel lines cut by transversal), so they are supplementary. Thus, \((x + 137)+(x + 57)=180\).
Simplify: \(2x+194 = 180\), \(2x=180 - 194=- 14\), \(x=-7\).
Now find the angle \(x + 137=-7 + 137 = 130\) or \(x + 57=-7+57 = 50\)? Wait, no, same - side interior angles sum to 180. Wait, maybe I misidentified. Wait, the two angles are on the same side of transversal, above and below parallel lines. Wait, actually, if the lines are parallel, same - side interior angles are supplementary. So \((x + 137)+(x + 57)=180\), \(2x=180 - 194=-14\), \(x = - 7\). Then the angle \(x + 137=130\), \(x + 57 = 50\), and \(130+50 = 180\), correct.

Step2: Analyze Problem 2

The two angles are corresponding angles (since parallel lines cut by transversal), so they are equal. Thus, \(11x + 9=10x + 20\).
Subtract \(10x\) and 9: \(x=20 - 9 = 11\).
Then the angle is \(11\times11 + 9=121 + 9 = 130\) or \(10\times11+20 = 130\), correct.

Step3: Analyze Problem 3

The two angles are alternate - exterior or alternate - interior? Wait, they are corresponding angles (since parallel lines cut by transversal), so \(12x + 15=13x + 5\).
Subtract \(12x\) and 5: \(x = 10\).
Then the angle \(12\times10+15 = 135\), \(13\times10 + 5=135\), correct.

Step4: Analyze Problem 4

The two angles are same - side interior angles? Wait, no, they are consecutive interior angles? Wait, no, they are on the same side of transversal, below parallel lines. Wait, actually, they are supplementary (same - side interior angles). So \((x + 83)+(x + 113)=180\).
Simplify: \(2x+196 = 180\), \(2x=-16\), \(x=-8\).
Then the angle \(x + 83=-8 + 83 = 75\), \(x + 113=-8+113 = 105\), and \(75 + 105=180\), correct.

Step5: Analyze Problem 5

From the hand - written note, the two angles \(x + 113\) and \(x + 83\) are same - side interior angles (parallel lines cut by transversal), so \((x + 113)+(x + 83)=180\).
Simplify: \(2x+196 = 180\), \(2x=-16\), \(x=-8\).
Then \(x + 113=-8 + 113 = 105\), \(x + 83=-8 + 83 = 75\), and \(105+75 = 180\), correct.

Step6: Analyze Problem 6

The two angles are same - side interior angles? Wait, \(x + 56\) and \(x + 146\) are on parallel lines cut by transversal, same - side interior angles? Wait, no, if they are same - side interior, they should be supplementary. So \((x + 56)+(x + 146)=180\).
Simplify: \(2x+202 = 180\), \(2x=-22\), \(x=-11\).
Then \(x + 56=-11 + 56 = 45\), \(x + 146=-11+146 = 135\), and \(45 + 135=180\), correct.

Answer:

1. The measure of the bold angle is \(\boldsymbol{130^\circ}\) (or \(50^\circ\), but based on calculation \(x=-7\), \(x + 137 = 130\))
2. The measure of the bold angle is \(\boldsymbol{130^\circ}\)
3. The measure of the bold angle is \(\boldsymbol{135^\circ}\)
4. The measure of the bold angle is \(\boldsymbol{75^\circ}\) (or \(105^\circ\), \(x=-8\), \(x + 83 = 75\))
5. The measure of the bold angle is \(\boldsymbol{105^\circ}\) (or \(75^\circ\), \(x=-8\), \(x + 113 = 105\))
6. The measure of the bold angle is \(\boldsymbol{45^\circ}\) (or \(135^\circ\), \(x=-11\), \(x + 56 = 45\))