Question

1. diagram of intersecting lines with angles labeled 79°, 103°, and variables x, y

Explanation:

Step1: Find y (supplementary angles)

Angles on a straight line sum to \(180^\circ\). So \(y + 103^\circ = 180^\circ\), thus \(y = 180^\circ - 103^\circ = 77^\circ\)? Wait, no, wait. Wait, actually, vertical angles or supplementary? Wait, let's re-examine. Wait, the angle with \(103^\circ\) and \(y\): wait, no, the angle adjacent to \(103^\circ\) and \(y\)? Wait, no, first, let's find \(x\) and \(y\). Wait, maybe first find \(y\) as supplementary to \(103^\circ\)? Wait, no, actually, the angle opposite to \(103^\circ\) would be equal, but wait, the angle with \(79^\circ\), \(x\), and \(y\)? Wait, no, let's see: the two lines are intersecting, so vertical angles. Wait, first, let's find \(y\). The angle \(103^\circ\) and \(y + 79^\circ\)? Wait, no, maybe I made a mistake. Wait, let's start over.

First, find \(y\): the angle \(103^\circ\) and the angle containing \(79^\circ\) and \(y\) are supplementary? Wait, no, the straight line: the angle with \(103^\circ\) and the angle adjacent to it (which is \(y + 79^\circ\))? Wait, no, actually, when two lines intersect, adjacent angles are supplementary. Wait, the angle of \(103^\circ\) and the angle that is \(79^\circ + x\)? No, wait, let's look at the diagram. There are two intersecting lines, creating vertical angles. Let's denote the intersection point. So, one angle is \(103^\circ\), its vertical angle would be equal, but the other angles: the angle with \(79^\circ\), \(x\), and \(y\). Wait, maybe first find \(y\) as equal to \(180^\circ - 103^\circ - 79^\circ + x\)? No, that's confusing. Wait, let's use the fact that the sum of angles around a point is \(360^\circ\), but actually, for a straight line, it's \(180^\circ\). Wait, the angle \(103^\circ\) and the angle opposite to it (let's say angle A) are equal. Then, the angle adjacent to \(103^\circ\) is \(180^\circ - 103^\circ = 77^\circ\)? Wait, no, maybe the angle with \(79^\circ\) and \(x\) is vertical to \(103^\circ\)? No, that can't be. Wait, maybe the correct approach is:

First, find \(y\): since \(y\) and \(103^\circ\) are supplementary? Wait, no, \(y + 79^\circ + x = 180^\circ\)? No, maybe I need to find \(x\) first. Wait, the angle \(103^\circ\) and the angle that is \(79^\circ + x\) are vertical angles? Wait, no, vertical angles are equal. Wait, if two lines intersect, vertical angles are equal. So, the angle opposite to \(103^\circ\) is \(103^\circ\), and the angle opposite to \(79^\circ + x\) is \(y\). Wait, no, let's see: the sum of angles on a straight line is \(180^\circ\). So, \(103^\circ + (79^\circ + x) = 180^\circ\)? Wait, no, that would be if they are adjacent. Wait, maybe the angle \(103^\circ\) and the angle composed of \(79^\circ\) and \(x\) are supplementary. So:

\(103^\circ + 79^\circ + x = 180^\circ\)

Wait, no, that would be three angles, but actually, two angles: \(103^\circ\) and ( \(79^\circ + x\) ) are supplementary. So:

\(103 + 79 + x = 180\)

Wait, \(103 + 79 = 182\), which is more than 180, so that can't be. So I must have misidentified the angles.

Wait, maybe the angle \(y\) is vertical to \(103^\circ\)? No, \(y\) looks smaller. Wait, maybe the angle adjacent to \(103^\circ\) is \(y\), so \(y + 103^\circ = 180^\circ\), so \(y = 77^\circ\). Then, the other angle: \(79^\circ + x + y = 180^\circ\)? Wait, \(79 + x + 77 = 180\), so \(x = 180 - 79 - 77 = 24^\circ\). Wait, that makes sense. Let's check:

- \(y\) and \(103^\circ\) are supplementary: \(y = 180 - 103 = 77^\circ\)
- Then, the straight line with \(79^\circ\), \(x\), and \(y\): \(79 + x + 77 = 180\) → \(x = 180 - 79 - 77 = 24^\circ\)

Yes, that works. So:

Step1: Calculate y (supplementary to 103°)

\(y = 180^\circ - 103^\circ = 77^\circ\)

Step2: Calculate x (sum of angles on straight line)

\(x + 79^\circ + y = 180^\circ\) → \(x = 180^\circ - 79^\circ - y\)
Substitute \(y = 77^\circ\):
\(x = 180 - 79 - 77 = 24^\circ\)

Answer:

\(x = 24^\circ\), \(y = 77^\circ\) (assuming we need both; if only one, clarify, but based on diagram, likely x and y. If the question was to find x and y, this is the solution.)