Question

julian describes an angle in the triangle using these statements.
$overline{gh}$ is the adjacent side.
$overline{hk}$ is the opposite side.
$overline{gk}$ is the hypotenuse.
which angle(s) is julian describing?
options: $angle g$, $angle h$ and $angle k$, $angle h$, $angle k$

Explanation:

Step1: Recall Right Triangle Terms

In a right triangle, the hypotenuse is opposite the right angle, the adjacent side is next to the angle (non - hypotenuse), and the opposite side is across from the angle. The triangle here has a right angle at \(H\) (since \(\angle H\) is the right angle, as seen from the right - angle symbol).

Step2: Analyze for \(\angle K\)

- For \(\angle K\):
- The adjacent side to \(\angle K\) would be \(HK\)? No, wait. Wait, the adjacent side to an angle in a right triangle (other than the right angle) is the leg that forms the angle with the hypotenuse. Wait, the hypotenuse is \(GK\). The side \(GH\): let's see, for \(\angle K\), the adjacent side should be the leg that is part of \(\angle K\) (other than the hypotenuse). Wait, the right angle is at \(H\), so the triangle is \(\triangle GHK\) with right angle at \(H\). So sides: \(GH\) and \(HK\) are legs, \(GK\) is hypotenuse.
- For an angle (say \(\angle K\)):
- The adjacent side is the leg that is adjacent to \(\angle K\), which is \(HK\)? No, wait. Wait, in \(\triangle GHK\), right - angled at \(H\), the angles are \(\angle G\), \(\angle H = 90^{\circ}\), and \(\angle K\).
- For \(\angle K\):
- The adjacent side: the side that is part of \(\angle K\) and is not the hypotenuse. The sides forming \(\angle K\) are \(HK\) (one leg) and \(GK\) (hypotenuse). Wait, no. Wait, the two legs are \(GH\) and \(HK\), hypotenuse \(GK\).
- For \(\angle K\):
- The adjacent side: the leg that is adjacent to \(\angle K\) (i.e., the leg that is not opposite to \(\angle K\)). The side opposite to \(\angle K\) is \(GH\), and the adjacent side is \(HK\)? Wait, no, the problem states that \(GH\) is the adjacent side, \(HK\) is the opposite side, and \(GK\) is the hypotenuse.
- Let's match the definitions with the angle. If \(GH\) is adjacent, \(HK\) is opposite, and \(GK\) is hypotenuse, then the angle for which \(GH\) is adjacent, \(HK\) is opposite must be \(\angle K\). Because:
- For \(\angle K\):
- Adjacent side: the side that is next to \(\angle K\) (forms the angle with the hypotenuse). The side \(HK\) is one leg, \(GH\) is the other leg. Wait, if we consider the angle \(\angle K\), the adjacent side (the leg that is adjacent to \(\angle K\)) is \(HK\)? No, wait, the adjacent side to \(\angle K\) should be the leg that is part of \(\angle K\) and is not the opposite side. Wait, the opposite side to \(\angle K\) is \(GH\) (since in a right - angled triangle at \(H\), the side opposite \(\angle K\) is \(GH\)), the adjacent side to \(\angle K\) is \(HK\)? No, the problem says \(GH\) is adjacent, \(HK\) is opposite. So if \(GH\) is adjacent to the angle, and \(HK\) is opposite to the angle, and \(GK\) is hypotenuse, then the angle must be \(\angle K\). Because:
- For \(\angle K\):
- Opposite side: \(GH\) (wait, no, earlier mistake). Wait, no, in \(\triangle GHK\), right - angled at \(H\), the side opposite \(\angle K\) is \(GH\), the side adjacent to \(\angle K\) is \(HK\)? But the problem says \(GH\) is adjacent, \(HK\) is opposite. So maybe I got the angle wrong. Wait, let's re - express.
- Wait, maybe the angle is \(\angle K\). Let's check the other angles:
- For \(\angle G\):
- Opposite side would be \(HK\), adjacent side would be \(GH\). But the problem says \(HK\) is opposite and \(GH\) is adjacent. Wait, if for an angle, opposite is \(HK\) and adjacent is \(GH\), then the angle is \(\angle K\). Because:
- In \(\angle K\), the opposite side is \(GH\) (wait, no, I'm confused). Wait, let's use the definitions:
- Adjacent side: the side that is adjacent (next to) the angle, forming the angle with the hypotenuse.
- Opposite side: the side that is opposite (across from) the angle.
- Hypotenuse: the side opposite the right angle.
- The right angle is at \(H\), so hypotenuse is \(GK\).
- For angle \(\angle K\):
- The sides:
- Adjacent: \(HK\)? No, wait, the two legs are \(GH\) and \(HK\). The angle \(\angle K\) is formed by \(HK\) (one leg) and \(GK\) (hypotenuse). So the adjacent side to \(\angle K\) is \(HK\), and the opposite side is \(GH\). But the problem says \(GH\) is adjacent and \(HK\) is opposite. Wait, that would mean the angle is \(\angle K\) if we reverse? No, maybe I mixed up adjacent and opposite. Wait, maybe the angle is \(\angle K\) because:
- If \(GH\) is adjacent to the angle, that means the angle is such that \(GH\) is next to it. So the angle that has \(GH\) as adjacent, \(HK\) as opposite, and \(GK\) as hypotenuse is \(\angle K\). Because for \(\angle K\), the adjacent side (the leg adjacent to \(\angle K\)) is \(HK\)? No, I think I made a mistake. Wait, let's check the other angles:
- \(\angle G\): For \(\angle G\), the adjacent side would be \(GH\), the opposite side would be \(HK\), and hypotenuse \(GK\). Oh! Wait, that's it. For \(\angle G\):
- Adjacent side: \(GH\) (since \(GH\) is the leg that forms \(\angle G\) with the hypotenuse \(GK\)).
- Opposite side: \(HK\) (since \(HK\) is across from \(\angle G\)).
- Hypotenuse: \(GK\) (opposite the right angle at \(H\)).
- So \(\angle G\) has \(GH\) as adjacent, \(HK\) as opposite, and \(GK\) as hypotenuse. Wait, but earlier I thought \(\angle K\), but now I see for \(\angle G\):
- The angle \(\angle G\) is in the right - angled triangle at \(H\). The sides: \(GH\) is adjacent to \(\angle G\) (because \(GH\) is one of the legs that makes up \(\angle G\) along with the hypotenuse \(GK\)), \(HK\) is opposite to \(\angle G\) (because \(HK\) is the leg that is not part of \(\angle G\) and is across from it), and \(GK\) is the hypotenuse.
- Wait, but the options: the first option is \(\angle G\), then \(\angle H\) (but \(\angle H\) is the right angle, so it doesn't have adjacent/opposite in the same way), \(\angle H\) and \(\angle K\) (no), \(\angle K\). Wait, I think I messed up. Let's start over.
- In a right - angled triangle (right - angled at \(H\)), the angles are \(\angle G\), \(\angle H = 90^{\circ}\), \(\angle K\).
- For an acute angle (either \(\angle G\) or \(\angle K\)):
- Adjacent side: the leg that is part of the angle (i.e., the leg that forms the angle with the hypotenuse).
- Opposite side: the leg that is not part of the angle (i.e., the leg across from the angle).
- Hypotenuse: the side opposite the right angle.
- Let's take \(\angle G\):
- The sides forming \(\angle G\) are \(GH\) (leg) and \(GK\) (hypotenuse). So the adjacent side to \(\angle G\) is \(GH\), the opposite side is \(HK\) (since \(HK\) is across from \(\angle G\)), and hypotenuse is \(GK\).
- Let's take \(\angle K\):
- The sides forming \(\angle K\) are \(HK\) (leg) and \(GK\) (hypotenuse). So the adjacent side to \(\angle K\) is \(HK\), the opposite side is \(GH\) (since \(GH\) is across from \(\angle K\)), and hypotenuse is \(GK\).
- The problem states: \(GH\) is the adjacent side, \(HK\) is the opposite side, \(GK\) is the hypotenuse. So this matches the description for \(\angle G\) (adjacent \(GH\), opposite \(HK\), hypotenuse \(GK\)). Wait, but earlier I thought \(\angle K\), but no, for \(\angle G\), adjacent is \(GH\), opposite is \(HK\), hypotenuse \(GK\). So the angle Julian is describing is \(\angle G\).

Answer:

\(\angle G\) (or the option corresponding to \(\angle G\) in the given choices, e.g., if the first option is \(\angle G\), then the answer is the option with \(\angle G\))