Question

in $\delta hij$, $m\angle h = 92^\circ$ and $m\angle i = 20^\circ$. which statement about the sides of $\delta hij$ must be true?

answer
$\circ$ $jh > hi > ij$ $\circ$ $hi > jh > ij$
$\circ$ $jh > ij > hi$ $\circ$ $ij > jh > hi$
$\circ$ $ij > hi > jh$ $\circ$ $hi > ij > jh$

Explanation:

Step1: Find the third angle

In a triangle, the sum of interior angles is \(180^\circ\). Given \(m\angle H = 92^\circ\) and \(m\angle I = 20^\circ\), we find \(m\angle J\) as:

$$ m\angle J=180^\circ - 92^\circ - 20^\circ = 68^\circ $$

Step2: Relate angles to sides

In a triangle, the larger the angle, the longer the side opposite to it.

• Opposite \(\angle H = 92^\circ\) is side \(IJ\).
• Opposite \(\angle J = 68^\circ\) is side \(HI\).
• Opposite \(\angle I = 20^\circ\) is side \(JH\).

Since \(92^\circ>68^\circ>20^\circ\) (i.e., \(\angle H>\angle J>\angle I\)), the sides opposite follow \(IJ > HI > JH\). Wait, no, wait: Wait, opposite \(\angle H\) (92°) is \(IJ\), opposite \(\angle J\) (68°) is \(HI\), opposite \(\angle I\) (20°) is \(JH\). So angle order: \(\angle H(92^\circ)>\angle J(68^\circ)>\angle I(20^\circ)\), so side order (opposite) should be \(IJ > HI > JH\)? Wait no, wait: Wait, let's label the triangle properly. In \(\triangle HIJ\), vertices are \(H\), \(I\), \(J\). So side opposite \(\angle H\) is \(IJ\) (connects \(I\) and \(J\)), side opposite \(\angle I\) is \(JH\) (connects \(J\) and \(H\)), side opposite \(\angle J\) is \(HI\) (connects \(H\) and \(I\)). So angle measures: \(\angle H = 92^\circ\), \(\angle I = 20^\circ\), \(\angle J = 68^\circ\). So angle size: \(\angle H > \angle J > \angle I\). Therefore, side opposite: \(IJ\) (opposite \(\angle H\)) > \(HI\) (opposite \(\angle J\)) > \(JH\) (opposite \(\angle I\)). Wait, but let's check the options. Wait, maybe I mixed up. Wait, let's re-express:

Wait, angle at \(H\): \(\angle H\), so side opposite is \(IJ\) (between \(I\) and \(J\)). Angle at \(I\): \(\angle I\), side opposite is \(JH\) (between \(J\) and \(H\)). Angle at \(J\): \(\angle J\), side opposite is \(HI\) (between \(H\) and \(I\)). So angles: \(\angle H = 92^\circ\), \(\angle J = 68^\circ\), \(\angle I = 20^\circ\). So \(\angle H > \angle J > \angle I\), so sides opposite: \(IJ > HI > JH\). Wait, but let's check the options. Wait, the options include "IJ > HI > JH" as one of them. Wait, but let's confirm again.

Wait, maybe I made a mistake. Let's re-calculate the third angle: \(180 - 92 - 20 = 68\), correct. So angles: 92°, 68°, 20°. So the largest angle is 92° (at H), so the side opposite (IJ) is the longest. Then next angle is 68° (at J), so side opposite (HI) is next. Then smallest angle 20° (at I), side opposite (JH) is shortest. So the order of sides from longest to shortest is IJ > HI > JH. So the option "IJ > HI > JH" is correct.

Wait, but let's check the options again. The options are:

1. JH > HI > IJ

1. HI > JH > IJ

1. JH > IJ > HI

1. IJ > JH > HI

1. IJ > HI > JH

1. HI > IJ > JH

So the correct one is "IJ > HI > JH", which is the fifth option (if we count the first row first column as 1, first row second as 2, second row first as 3, second row second as 4, third row first as 5, third row second as 6). Wait, the third row first option is "IJ > HI > JH", so that's the correct one.

Answer:

IJ > HI > JH (the option with "IJ > HI > JH")