Question
Triangle KNM is shown.
Triangle K N M is shown. Angle M K N is 25 degrees. Angle K N M is 130 degrees. Angle N M K is 25 degrees.
What is true about the sides of KNM?
KN = NM
KN + NM = KM
KM = 2(NM)
KN = One-halfKM
Triangle K N M is shown. Angle M K N is 25 degrees. Angle K N M is 130 degrees. Angle N M K is 25 degrees.
What is true about the sides of KNM?
KN = NM
KN + NM = KM
KM = 2(NM)
KN = One-halfKM
Answers
GPT-4o mini
To analyze the triangle KNM given the angles, we first note the angle measures:
- Angle \( MKN = 25^\circ \)
- Angle \( KNM = 130^\circ \)
- Angle \( NKM = 25^\circ \)
From the angle measures, we see that triangle KNM has two angles, \( MKN \) and \( NKM \), that are equal (both are \( 25^\circ \)). This implies that triangle KNM is an isosceles triangle with sides opposite these equal angles being equal.
In triangle KNM:
- The sides opposite the equal angles are:
- \( KN \) is opposite angle \( NKM \) (also \( 25^\circ \))
- \( NM \) is opposite angle \( MKN \) (also \( 25^\circ \))
Thus, by the properties of isosceles triangles, we have:
\[
KN = NM
\]
This matches one of the statements provided.
Now evaluating the other statements:
1. **\( KN + NM = KM \)**: This is not true in general for triangle sides, as it suggests a linear relation that does not hold for triangle inequality.
2. **\( KM = 2(NM) \)**: This implies that \( KM \) is exactly twice \( NM \), which is also not supported by the angle measures or the triangle properties.
3. **\( KN = \frac{1}{2}KM \)**: There is no way to determine this directly from the angles without additional information about the actual lengths or ratios of the sides.
Therefore, the only statement that is conclusively true based on the properties of the triangle KNM is:
\[
KN = NM
\]
- Angle \( MKN = 25^\circ \)
- Angle \( KNM = 130^\circ \)
- Angle \( NKM = 25^\circ \)
From the angle measures, we see that triangle KNM has two angles, \( MKN \) and \( NKM \), that are equal (both are \( 25^\circ \)). This implies that triangle KNM is an isosceles triangle with sides opposite these equal angles being equal.
In triangle KNM:
- The sides opposite the equal angles are:
- \( KN \) is opposite angle \( NKM \) (also \( 25^\circ \))
- \( NM \) is opposite angle \( MKN \) (also \( 25^\circ \))
Thus, by the properties of isosceles triangles, we have:
\[
KN = NM
\]
This matches one of the statements provided.
Now evaluating the other statements:
1. **\( KN + NM = KM \)**: This is not true in general for triangle sides, as it suggests a linear relation that does not hold for triangle inequality.
2. **\( KM = 2(NM) \)**: This implies that \( KM \) is exactly twice \( NM \), which is also not supported by the angle measures or the triangle properties.
3. **\( KN = \frac{1}{2}KM \)**: There is no way to determine this directly from the angles without additional information about the actual lengths or ratios of the sides.
Therefore, the only statement that is conclusively true based on the properties of the triangle KNM is:
\[
KN = NM
\]