Exercise 16 Page 352

We want to prove the given statement using an indirect proof.

To prove a statement using an indirect proof, we need to follow three steps.

1. Assume temporarily that the conclusion we want to prove is false and therefore its opposite is true.
2. Perform a logical reasoning until we reach a contradiction.
3. Conclude that the initial statement must be true since the assumption led to a contradiction.We want to prove that YZ is greater than 4. Therefore, let's temporarily assume that YZ is less than or equal to 4. YZ ≤ 4 We are given that XY= 4. We will now add XY to both sides of the inequality by using the Addition Property of Inequality.
   YZ ≤ 4

   AddIneq

   LHS+XY≤RHS+XY

   YZ + XY ≤ 4 + XY

   Substitute

   XY= 4

   YZ + XY ≤ 4 + 4

   AddTerms

   Add terms

   YZ + XY ≤ 8
   We have that the sum of the lengths of the sides YZ and XY is less than or equal to 8. On the other hand, by the Triangle Inequality Theorem, we know that the sum of two arbitrary side lengths of a triangle is greater than the length of the third side. We can apply this theorem to △ XYZ. XY+YZ>XZ We are also given that XZ= 8. Let's now substitute this length into the triangle inequality.
   XY+YZ>XZ

   Substitute

   XZ= 8

   XY+YZ> 8

   CommutativePropAdd

   Commutative Property of Addition

   YZ + XY>8
   We obtained that the same sum as before is greater than 8!

   Temporary Assumption Result	Triangle Inequality Result
   YZ + XY ≤ 8	YZ + XY>8

   Note that the inequalities in the table contradict each other. Therefore, assuming that YZ≤ 4 contradicts a result obtained by the Triangle Inequality Theorem. This means that the temporary assumption cannot be true. This proves that YZ > 4.