# Proving Inequalities in a Triangle

The Exterior Angle Inequality Theorem which states that “ The measure of the exterior angle is greater than the measure of either remote interior angle” and the Hinge Theorem which states that “ If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is greater than the included angle of the second, then the third side of the first triangle is longer than the third side of the second” and we have also the Converse of the Hinge Theorem.

### Exercise I

Direction: Complete the proofs by supplying the missing statements and reasons.

Given: HE = OM

M<EHM>m<OMH

Prove: EM > OH

Statements | Reasons |

1. HE = OM | Given |

2. HM = MH | Reflexive Property |

3. m<EHM > m<OMH | Given |

4. EM > OH | Triangle Inequality 2 |

### Exercise II

Direction: Give the reasons of the following statement.

Given: A is the midpoint of EN, <1 = <2, <3 > <4

Prove: CN > BE

Statements | Reasons |

1. <1 = <2 | Given |

2. Triangle BAC is isosceles | Definition of Isosceles Triangle |

3. EF = EM | Legs of isosceles triangle are congruent |

4. Ais the midpoint of EN | Given |

5. EA = NA | Definition of midpoint |

6. <3 > <4 | Given |

7. CN = BE | Hinge Theorem |

**Exercise III**

Direction: Write the statement or reason in the two-column proof.

Given: <EFM = <EMF

FA > MA

Prove: m<FEA > m<MEA

Statements | Reasons |

1. <EFM = EMF | Given |

2. Triangle FEM is isosceles | Definition of Isosceles trangle |

3. EF = EM | Legs of isosceles triangles are congruent |

4. AE = EA | Transitive Property |

5. FA > MA | Given |

6. M<FEA > m<MEA | Converse of Hinge Theorem |

Guide Questions:

Write your answer on the space provided based on the activity above.

1. What postulate/theorem did you use to prove the inequalities in a triangle?

__The sum of two sides of a triangle is always greater than the third side. A polygon bounded by three line-segments is known as the Triangle. It is the smallest possible polygon. A triangle has three sides, three vertices, and three interior angles.__

**V. Reflection**

### Guide Questions

1. What have you learned from the lesson?

__The triangle inequality theorem describes the relationship between the three sides of a triangle. According to this theorem, for any triangle, the sum of lengths of two sides is always greater than the third side.__

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