Question

1. in the figure below, lines p and q are parallel. the measure of $\angle3 = 126^\circ$, what is the measure of all of the angles shown? figure of parallel lines p, q cut by transversal t, with angles 1 - 8 labeled

Explanation:

Step1: Identify Vertical Angles and Linear Pairs

Vertical angles are equal, and linear pairs sum to \(180^\circ\). For \(\angle 3 = 126^\circ\):
- \(\angle 1\) (vertical to \(\angle 3\)? No, \(\angle 1\) and \(\angle 3\) are linear pair? Wait, \(\angle 3\) and \(\angle 1\): no, \(\angle 3\) and \(\angle 2\) are linear pair? Wait, line \(p\) and transversal \(t\): \(\angle 3\) and \(\angle 1\) are vertical? Wait, no, let's correct. \(\angle 3\) and \(\angle 1\): no, \(\angle 3\) and \(\angle 2\) are adjacent (linear pair), so \(\angle 3 + \angle 2 = 180^\circ\). \(\angle 3 = 126^\circ\), so \(\angle 2 = 180 - 126 = 54^\circ\). \(\angle 1\) is vertical to \(\angle 3\)? No, \(\angle 1\) is vertical to \(\angle 3\)? Wait, \(\angle 1\) and \(\angle 3\): no, \(\angle 1\) and \(\angle 3\) are vertical? Wait, the intersection of \(p\) and \(t\): \(\angle 1\) and \(\angle 3\) are vertical? Wait, no, \(\angle 1\) and \(\angle 3\) are vertical? Wait, \(\angle 1\) and \(\angle 3\) are vertical angles? Wait, no, \(\angle 1\) and \(\angle 3\) are vertical? Wait, the four angles at \(p\) and \(t\): \(\angle 1\) (top right), \(\angle 2\) (bottom right), \(\angle 3\) (bottom left), \(\angle 4\) (top left). So \(\angle 1\) and \(\angle 3\) are vertical? No, \(\angle 1\) and \(\angle 3\) are vertical? Wait, \(\angle 1\) and \(\angle 3\) are vertical angles? Wait, no, \(\angle 1\) and \(\angle 3\) are vertical? Wait, \(\angle 1\) and \(\angle 3\) are vertical? Wait, \(\angle 1\) (top right) and \(\angle 3\) (bottom left) are vertical? No, vertical angles are opposite each other. So \(\angle 1\) (top right) and \(\angle 3\) (bottom left) are vertical? Wait, no, \(\angle 1\) (top right) and \(\angle 3\) (bottom left) are vertical? Wait, \(\angle 1\) and \(\angle 3\) are vertical angles, so \(\angle 1 = \angle 3 = 126^\circ\)? No, that can't be, because \(\angle 1\) and \(\angle 2\) are linear pair. Wait, I messed up. Let's start over.

At the intersection of \(p\) and \(t\):
- \(\angle 3\) and \(\angle 2\) are adjacent (linear pair), so \(\angle 3 + \angle 2 = 180^\circ\). Given \(\angle 3 = 126^\circ\), so \(\angle 2 = 180 - 126 = 54^\circ\).
- \(\angle 1\) is vertical to \(\angle 3\)? No, \(\angle 1\) is vertical to \(\angle 3\)? Wait, \(\angle 1\) (top right) and \(\angle 3\) (bottom left) are vertical? No, \(\angle 1\) (top right) and \(\angle 3\) (bottom left) are vertical? Wait, no, vertical angles are opposite: \(\angle 1\) (top right) and \(\angle 3\) (bottom left) are vertical? Wait, no, \(\angle 1\) (top right) and \(\angle 3\) (bottom left) are vertical? Wait, \(\angle 1\) and \(\angle 3\) are vertical, so \(\angle 1 = \angle 3 = 126^\circ\)? But \(\angle 1\) and \(\angle 2\) are linear pair, so \(\angle 1 + \angle 2 = 180\), which would be \(126 + 54 = 180\), correct. \(\angle 4\) is vertical to \(\angle 2\), so \(\angle 4 = \angle 2 = 54^\circ\).

Step2: Use Parallel Lines (Corresponding Angles)

Lines \(p \parallel q\), transversal \(t\). Corresponding angles are equal. So:
- \(\angle 3\) and \(\angle 7\) are corresponding? Wait, \(\angle 3\) (on \(p\), below transversal) and \(\angle 7\) (on \(q\), below transversal) are corresponding, so \(\angle 7 = \angle 3 = 126^\circ\).
- \(\angle 2\) and \(\angle 6\) are corresponding, so \(\angle 6 = \angle 2 = 54^\circ\).
- \(\angle 1\) and \(\angle 5\) are corresponding, so \(\angle 5 = \angle 1 = 126^\circ\).
- \(\angle 4\) and \(\angle 8\) are corresponding, so \(\angle 8 = \angle 4 = 54^\circ\).

Step3: Vertical Angles for \(q\) and \(t\)

- \(\angle 5\) and \(\angle 7\) are linear pair? No, \(\angle 5\) and \(\angle 6\) are linear pair, so \(\angle 5 + \angle 6 = 180\) (126 + 54 = 180, correct). \(\angle 5\) and \(\angle 7\) are vertical? No, \(\angle 5\) (top right on \(q\)) and \(\angle 7\) (bottom left on \(q\)) are vertical, so \(\angle 7 = \angle 5 = 126^\circ\) (matches earlier). \(\angle 6\) and \(\angle 8\) are vertical, so \(\angle 8 = \angle 6 = 54^\circ\) (matches earlier).

Step4: Summarize All Angles

- \(\angle 1 = 126^\circ\) (vertical to \(\angle 3\) or corresponding to \(\angle 5\))
- \(\angle 2 = 54^\circ\) (linear pair with \(\angle 3\))
- \(\angle 3 = 126^\circ\) (given)
- \(\angle 4 = 54^\circ\) (vertical to \(\angle 2\))
- \(\angle 5 = 126^\circ\) (corresponding to \(\angle 1\))
- \(\angle 6 = 54^\circ\) (corresponding to \(\angle 2\))
- \(\angle 7 = 126^\circ\) (vertical to \(\angle 5\) or corresponding to \(\angle 3\))
- \(\angle 8 = 54^\circ\) (vertical to \(\angle 6\) or corresponding to \(\angle 4\))

Answer:

\(\angle 1 = 126^\circ\), \(\angle 2 = 54^\circ\), \(\angle 3 = 126^\circ\), \(\angle 4 = 54^\circ\), \(\angle 5 = 126^\circ\), \(\angle 6 = 54^\circ\), \(\angle 7 = 126^\circ\), \(\angle 8 = 54^\circ\)