Question 1177180: Given: Segment NK is a median of triangle JMN; IN>NM Prove: m<1>m<2
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! **1. Given Information**
* NK is a median of triangle JMN. This means that NK bisects side JM, so JK = KM.
* IN > NM
**2. Triangle Inequality Theorem**
In any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
**3. Applying the Triangle Inequality Theorem to Triangle INK**
* IN + NK > IK
**4. Applying the Triangle Inequality Theorem to Triangle NMK**
* NM + NK > MK
**5. Using the Given Information and Steps 3 and 4**
* Since IN > NM (given), and NK is common to both triangles INK and NMK, we can conclude that:
IN + NK > NM + NK
* This further implies that IK > MK
**6. Relating IK and MK to JK and KM**
* We know that JK = KM (because NK is a median).
* Therefore, IK > JK
**7. Hinge Theorem**
The Hinge Theorem states that if two triangles have two congruent sides, then the triangle with the larger included angle has the longer third side.
**8. Applying the Hinge Theorem**
* In triangles JNK and KNM:
* JN = NM (given)
* NK = NK (common side)
* IK > JK (from step 6)
* Therefore, by the Hinge Theorem, m∠1 > m∠2.
**Conclusion**
We have successfully proven that m∠1 > m∠2 using the given information, the Triangle Inequality Theorem, and the Hinge Theorem.
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