Monday, 25 Oct 2021

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

StatementsReasons
1. HE = OMGiven
2. HM = MHReflexive Property
3. m<EHM > m<OMHGiven
4. EM > OHTriangle 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

StatementsReasons
1. <1 = <2Given
2.  Triangle BAC is isoscelesDefinition of Isosceles Triangle
3. EF = EMLegs of isosceles triangle are congruent
4. Ais the midpoint of ENGiven
5. EA = NADefinition of midpoint
6. <3 > <4Given
7. CN = BEHinge Theorem

Exercise III

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

Given: <EFM = <EMF

FA > MA

Prove: m<FEA > m<MEA

StatementsReasons
1. <EFM = EMFGiven
2. Triangle FEM is isoscelesDefinition of Isosceles trangle
3. EF = EMLegs of isosceles triangles are congruent
4. AE = EATransitive Property
5. FA > MAGiven
6. M<FEA > m<MEAConverse 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.

Image: Pexels

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