Proofs Involving the Triangle Inequality Theorem — Practice Geometry Questions

By Allen Ma, Amber Kuang

In geometry, the triangle inequality theorem states that when you add the lengths of any two sides of a triangle, their sum will be greater that the length of the third side.

By using the triangle inequality theorem and the exterior angle theorem, you should have no trouble completing the inequality proof in the following practice question.

Practice questions

Complete the following proof by adding the missing statement or reason.

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Given:

image1.png

Prove: ET > TV

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  1. What is the statement for Reason 2?

  2. What is the reason for Statement 3?

  3. What is the missing angle in Statement 4?

  4. What is the reason for Statement 5?

  5. What is the reason for Statement 6?

Answers and explanations

  1. image3.png

    A bisector divides an angle into two congruent angles.

  2. The exterior angle of a triangle is equal to the sum of the two nonadjacent interior angles.

    image4.png

    are the two nonadjacent interior angles of

    image5.png

    The exterior angle of a triangle is equal to the sum of the two nonadjacent interior angles of the triangle; therefore,

    image6.png

  3. image7.png

    You found that

    image8.png

    The whole is greater than its parts, which means that

    image9.png

  4. Transitive property

    Because

    image10.png

    the transitive property finds that

    image11.png

  5. In a triangle, the longest side is opposite the largest angle.

    image12.png

    You discovered that

    image13.png

    In a triangle, the longest side is opposite the largest angle, so ET > TV.