Question

in $\triangle mno$, $m\angle m = 56^\circ$ and $m\angle n = 94^\circ$. which statement about the sides of $\triangle mno$ must be true?
answer
\bigcirc om < mn < no
\bigcirc mn < om < no
\bigcirc no < mn < om
\bigcirc no < om < mn
\bigcirc om < no < mn
\bigcirc mn < no < om

Explanation:

Step1: Find angle O

In $\triangle MNO$, the sum of angles is $180^\circ$. So $m\angle O = 180^\circ - m\angle M - m\angle N = 180 - 56 - 94 = 30^\circ$.

Step2: Relate angles to sides

In a triangle, the larger the angle, the longer the opposite side. Angles: $m\angle N = 94^\circ$ (largest), $m\angle M = 56^\circ$, $m\angle O = 30^\circ$ (smallest). Opposite sides: $OM$ (opposite $\angle N$), $ON$ (opposite $\angle M$), $MN$ (opposite $\angle O$). So side lengths: $MN < ON < OM$? Wait, no: opposite $\angle M$ (56°) is $ON$, opposite $\angle N$ (94°) is $OM$, opposite $\angle O$ (30°) is $MN$. So angle order: $\angle O < \angle M < \angle N$, so side order: $MN < ON < OM$? Wait, no, let's recheck. Wait, angle $N$ is 94°, angle $M$ is 56°, angle $O$ is 30°. So opposite sides: $OM$ (opposite $N$), $ON$ (opposite $M$), $MN$ (opposite $O$). So since $\angle O < \angle M < \angle N$, then $MN < ON < OM$? Wait, no, the side opposite the larger angle is longer. So $\angle N$ (94°) is largest, so $OM$ (opposite $N$) is longest. $\angle M$ (56°) is next, so $ON$ (opposite $M$) is next. $\angle O$ (30°) is smallest, so $MN$ (opposite $O$) is shortest. So order: $MN < ON < OM$? Wait, but the options: let's check the options. Wait, the options have $NO < OM < MN$? Wait, maybe I mixed up the labels. Wait, triangle $MNO$: vertices M, N, O. So sides: $MN$, $NO$, $OM$. Let's list angles and opposite sides:

• $\angle M$ (at M) is 56°, opposite side is $NO$.
• $\angle N$ (at N) is 94°, opposite side is $OM$.
• $\angle O$ (at O) is 30°, opposite side is $MN$.

So angle sizes: $\angle O = 30^\circ < \angle M = 56^\circ < \angle N = 94^\circ$. Therefore, opposite sides: $MN$ (opposite $\angle O$) < $NO$ (opposite $\angle M$) < $OM$ (opposite $\angle N$). Wait, but the options: one of the options is $NO < OM < MN$? No, wait the options: let's look at the options again. Wait the options include "NO < OM < MN"? No, the options are:

1. $OM < MN < NO$
2. $MN < OM < NO$
3. $NO < MN < OM$
4. $NO < OM < MN$
5. $OM < NO < MN$
6. $MN < NO < OM$

Wait, let's recalculate angle O: 180 - 56 - 94 = 30°, correct. So angles: O=30°, M=56°, N=94°. So opposite sides:

• Opposite O (30°): MN
• Opposite M (56°): NO
• Opposite N (94°): OM

So since 30° < 56° < 94°, then MN (opposite 30°) < NO (opposite 56°) < OM (opposite 94°). So the side order is $MN < NO < OM$, which is the last option: $MN < NO < OM$. Wait, but let's check the options. The last option (bottom right) is $MN < NO < OM$? Wait, the options as per the image: the last option (fourth row, third column?) Wait, the options are:

First row:

• OM < MN < NO
• MN < OM < NO
• NO < MN < OM

Second row:

• NO < OM < MN
• OM < NO < MN
• MN < NO < OM

Ah, the last option (second row, third column) is $MN < NO < OM$. So that's the correct one. Because angle O (30°) < angle M (56°) < angle N (94°), so side MN (opposite O) < side NO (opposite M) < side OM (opposite N).

Answer:

MN < NO < OM (the last option in the second row, third column)