Question

in $\triangle hij$, $ij = 8$, $jh = 11$, and $hi = 4$. which list has the angles of $\triangle hij$ in order from largest to smallest?\
\bigcirc $m\angle h, m\angle j, m\angle i$\
\bigcirc $m\angle i, m\angle h, m\angle j$\
\bigcirc $m\angle j, m\angle h, m\angle i$\
\bigcirc $m\angle j, m\angle i, m\angle h$\
\bigcirc $m\angle i, m\angle j, m\angle h$\
\bigcirc $m\angle h, m\angle i, m\angle j$

Explanation:

Step1: Recall the triangle angle - side relationship

In a triangle, the larger the side opposite an angle, the larger the angle. So we need to first identify the sides opposite each angle in \(\triangle HIJ\).

Let's label the triangle: In \(\triangle HIJ\), the vertices are \(H\), \(I\), and \(J\). The side opposite \(\angle H\) is \(IJ\), the side opposite \(\angle I\) is \(JH\), and the side opposite \(\angle J\) is \(HI\).

Step2: Identify the lengths of the sides opposite each angle

We are given: \(IJ = 8\), \(JH=11\), and \(HI = 4\).

• Side opposite \(\angle H\): \(IJ = 8\)
• Side opposite \(\angle I\): \(JH=11\)
• Side opposite \(\angle J\): \(HI = 4\)

Step3: Compare the lengths of the sides

Now, we compare the lengths of these sides: \(11>8 > 4\), that is \(JH>IJ>HI\).

Step4: Relate the side lengths to the angle measures

Since the larger the side opposite an angle, the larger the angle, we have:

• The angle opposite the longest side (\(JH = 11\)) is \(\angle I\), so \(m\angle I\) is the largest.
• The angle opposite the middle - length side (\(IJ = 8\)) is \(\angle H\), so \(m\angle H\) is the middle - sized angle.
• The angle opposite the shortest side (\(HI = 4\)) is \(\angle J\), so \(m\angle J\) is the smallest.

So the order of the angles from largest to smallest is \(m\angle I,m\angle H,m\angle J\).

Answer:

\(m\angle I,m\angle H,m\angle J\)