Question

proving lines parallel
find the measure of the indicated angle that makes lines u and v parallel.
1)
2)
3)
4)
5)
6)
7)
8)

Explanation:

Step1: Use angle - related properties

For parallel lines \(u\) and \(v\), we use properties such as supplementary angles (\(a + b=180^{\circ}\) when they are same - side interior angles), alternate interior angles (\(a = b\)), and corresponding angles (\(a = b\)).

1. For the first case, since \(130^{\circ}\) and the adjacent angle (let's call it \(x\)) are supplementary (\(x+130^{\circ}=180^{\circ}\)), \(x = 180^{\circ}-130^{\circ}=50^{\circ}\). And if \(u\parallel v\), the unknown angle (let's call it \(y\)) is equal to \(x\) (alternate interior angles).
2. Given an angle of \(128^{\circ}\) on line \(u\), its supplementary angle (for the case of same - side interior angles and parallel lines) is \(180^{\circ}-128^{\circ}=52^{\circ}\).
3. Using the alternate interior angles property, if one of the alternate interior angles is \(66^{\circ}\), the unknown angle is \(66^{\circ}\).
4. By the alternate interior angles property, if one of the alternate interior angles is \(80^{\circ}\), the unknown angle is \(80^{\circ}\).
5. Using the corresponding angles property (since \(v\) has a right - angle - related \(90^{\circ}\) - like situation, and for parallel lines \(u\) and \(v\)), the unknown angle is \(90^{\circ}\).
6. By the alternate interior angles property, if one of the alternate interior angles is \(81^{\circ}\), the unknown angle is \(81^{\circ}\).
7. Using the corresponding angles property, if an angle on \(v\) is \(53^{\circ}\), the unknown angle is \(53^{\circ}\).
8. By the alternate interior angles property, if one of the alternate interior angles is \(58^{\circ}\), the unknown angle is \(58^{\circ}\).

Answer:

1. \(50^{\circ}\)
2. \(52^{\circ}\)
3. \(66^{\circ}\)
4. \(80^{\circ}\)
5. \(90^{\circ}\)
6. \(81^{\circ}\)
7. \(53^{\circ}\)
8. \(58^{\circ}\)