Question

find each angle measure:
1.
m∠1
2.
m∠hjk
3.
m∠3
4.
m∠4
5.
m∠rst

1. in the figure below, lines p and q are parallel. the measure of ∠3 = 126°. what is the measure of all of the angles shown?

Explanation:

Step1: Vertical angles

Vertical angles are equal. For problem 1, since the angle given is \(67^{\circ}\) and \(\angle1\) is vertical to it, \(m\angle1 = 67^{\circ}\).

Step2: Supplementary angles

Supplementary angles add up to \(180^{\circ}\). For problem 2, if the adjacent angle is \(142^{\circ}\), then \(m\angle HJK=180 - 142=38^{\circ}\). But wait, no, actually, if we assume parallel lines and transversal, for problem 2, if the angle adjacent to \(\angle HJK\) is \(142^{\circ}\), then \(m\angle HJK = 180- 142=38^{\circ}\) is wrong. Wait, no, actually, if we consider the linear - pair. Wait, no, for problem 2, if we assume parallel lines and transversal, the angle adjacent to \(142^{\circ}\) (linear - pair) is \(180 - 142 = 38^{\circ}\), but if \(\angle HJK\) and the \(38^{\circ}\) angle are alternate interior angles (assuming parallel lines), no, wait, no. Wait, for problem 2, if we consider the linear - pair with \(142^{\circ}\), the adjacent angle is \(180-142 = 38^{\circ}\), but if we assume the lines are parallel and using the property of consecutive interior angles (no, wait, no). Wait, actually, for problem 2, if we consider the angle adjacent to \(142^{\circ}\) (linear - pair) is \(38^{\circ}\), but if \(\angle HJK\) and the \(38^{\circ}\) angle are vertical angles (no). Wait, no, for problem 2, if we assume the two lines are parallel and the transversal, then \(\angle HJK\) and the angle adjacent to \(142^{\circ}\) (linear - pair: \(180 - 142=38^{\circ}\)) are alternate interior angles. No, wait, no. Wait, actually, for problem 2, if we consider the angle \(\angle HJK\) and the \(142^{\circ}\) angle: if the two lines are parallel, then \(\angle HJK=180 - 142=38^{\circ}\) (consecutive interior angles). But wait, no, consecutive interior angles sum to \(180^{\circ}\). Wait, no, if two parallel lines are cut by a transversal, consecutive interior angles are supplementary. So if one is \(142^{\circ}\), the other (consecutive interior) is \(180 - 142 = 38^{\circ}\). But if \(\angle HJK\) is that angle, then \(m\angle HJK = 38^{\circ}\). But the user wrote \(113\) which is wrong. Wait, no, let's start over.

Problem 1

Since \(\angle1\) and the \(67^{\circ}\) angle are vertical angles (opposite angles formed by two intersecting lines), by the vertical - angles theorem \(m\angle1=67^{\circ}\).

Problem 2

If we assume the two lines are parallel and the transversal. The angle adjacent to \(142^{\circ}\) (linear - pair) is \(180 - 142=38^{\circ}\). But if \(\angle HJK\) and the \(38^{\circ}\) angle are alternate interior angles (assuming parallel lines and transversal), no, wait, no. Wait, actually, if we consider the two parallel lines and transversal, \(\angle HJK\) and the \(142^{\circ}\) angle are consecutive interior angles. By the consecutive - interior - angles theorem (if two parallel lines are cut by a transversal, consecutive interior angles are supplementary), \(m\angle HJK=180 - 142 = 38^{\circ}\). But maybe there was a mis - reading of the figure. If we assume another property. Wait, no, vertical angles: no.

Problem 3

If we assume parallel lines and transversal, the angle adjacent to \(111^{\circ}\) (linear - pair) is \(180 - 111=69^{\circ}\). If \(\angle3\) and the \(69^{\circ}\) angle are alternate interior angles (because of parallel lines and transversal), then \(m\angle3 = 69^{\circ}\).

Problem 4

If we assume the two angles \(\angle4\) and the angle adjacent to it (assuming perpendicular - like, but no, if we assume the two lines are parallel and transversal and \(\angle4\) and its adjacent angle form a right - angle (maybe a mis - drawn figure). Wait, no, if we assume that \(\angle4\) and the angle below it (assuming parallel lines and transversal and the lines are perpendicular - like). Wait, no, if we use the property of parallel lines and transversal, if \(\angle4\) and the angle adjacent to it (let's say \(x\)) and assume that the sum of \(\angle4\) and \(x\) is \(180^{\circ}\) (linear - pair). But if we assume that \(\angle4\) is \(90^{\circ}\) (maybe the figure implies perpendicular, but in terms of parallel - line properties, if there is a right - angle symbol (implied).

Problem 5

The angle adjacent to \(138^{\circ}\) (linear - pair) is \(180 - 138 = 42^{\circ}\). If \(\angle RST\) and the \(42^{\circ}\) angle are vertical angles (opposite angles formed by two intersecting lines), then \(m\angle RST=42^{\circ}\).

Problem 6

- **For \(\angle1\) and \(\angle3\)**:
- \(\angle1\) and \(\angle3\) are vertical angles. Given \(m\angle3 = 126^{\circ}\), by the vertical - angles theorem \(m\angle1=126^{\circ}\).
- \(\angle2\) and \(\angle3\) form a linear - pair. So \(m\angle2=180 - 126=54^{\circ}\).
- \(\angle4\) and \(\angle2\) are vertical angles. So \(m\angle4 = 54^{\circ}\).
- **For \(\angle5\) and \(\angle7\)**:
- Since \(p\parallel q\) and \(t\) is a transversal, \(\angle5\) and \(\angle3\) are corresponding angles. So \(m\angle5 = 126^{\circ}\).
- \(\angle7\) and \(\angle5\) are vertical angles. So \(m\angle7 = 126^{\circ}\).
- \(\angle6\) and \(\angle5\) form a linear - pair. So \(m\angle6=180 - 126 = 54^{\circ}\).
- \(\angle8\) and \(\angle6\) are vertical angles. So \(m\angle8 = 54^{\circ}\).

Answer:

1. \(m\angle1 = 67^{\circ}\)
2. \(m\angle HJK = 38^{\circ}\) (assuming parallel lines and transversal, consecutive - interior - angles theorem)
3. \(m\angle3 = 69^{\circ}\) (assuming parallel lines and transversal, alternate - interior - angles theorem)
4. \(m\angle4 = 90^{\circ}\) (assuming perpendicular - like intersection implied by the figure)
5. \(m\angle RST = 42^{\circ}\) (vertical - angles theorem)
6. \(m\angle1 = 126^{\circ}\), \(m\angle2 = 54^{\circ}\), \(m\angle3 = 126^{\circ}\), \(m\angle4 = 54^{\circ}\), \(m\angle5 = 126^{\circ}\), \(m\angle6 = 54^{\circ}\), \(m\angle7 = 126^{\circ}\), \(m\angle8 = 54^{\circ}\)