To find the sides of triangle MNP in ascending order, we need to determine the angles of the triangle and then use the fact that the sides opposite larger angles are longer.
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We know:
- \( m \angle M = 64^\circ \)
- \( m \angle N = 48^\circ \)
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To find \( m \angle P \): \[ m \angle P = 180^\circ - m \angle M - m \angle N = 180^\circ - 64^\circ - 48^\circ = 68^\circ \]
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Now we have the angles of the triangle:
- \( m \angle M = 64^\circ \)
- \( m \angle N = 48^\circ \)
- \( m \angle P = 68^\circ \)
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Now we arrange the angles in order from smallest to largest:
- \( m \angle N = 48^\circ \) (smallest)
- \( m \angle M = 64^\circ \)
- \( m \angle P = 68^\circ \) (largest)
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According to the properties of triangles, the side lengths opposite these angles are also in the same order:
- Side opposite \( m \angle N \) (let's call it side \( np \))
- Side opposite \( m \angle M \) (let's call it side \( mp \))
- Side opposite \( m \angle P \) (let's call it side \( mn \))
Thus, the sides of triangle MNP in ascending order, based on the angle measurements, will be:
- Side \( np \) (opposite angle 48°)
- Side \( mp \) (opposite angle 64°)
- Side \( mn \) (opposite angle 68°)
In summary, if we denote the sides as follows:
- \( a \): opposite \( m \angle M \) (64°)
- \( b \): opposite \( m \angle N \) (48°)
- \( c \): opposite \( m \angle P \) (68°)
The sides in ascending order will be:
- Side opposite \( N \) (shortest)
- Side opposite \( M \)
- Side opposite \( P \) (longest)
So, the answer is:\( NP < MP < MN \) (where \( NP \) is opposite \( N \), \( MP \) is opposite \( M \), \( MN \) is opposite \( P \)).