We are asked to write an indirect proof for Theorem $5 - 10$ . Let's use the notation of the diagram where the theorem is stated.

Let's highlight the

hypothesis

and the

conclusion

of the theorem.

If m ∠ J > m ∠ K then K L > J L .

Step $1$

To start an indirect proof, we assume that the

conclusion

of the theorem is not true.

Assumption : K L \leq J L .

Step $2$

The second step of an indirect proof is to investigate any consequences of our assumption until we find something that contradicts the

hypothesis

of the theorem. Let's split the assumption into two parts.

Assumption : Either K L = J L or K L < J L .

Now we investigate the two cases separately.

Case $1$ : $K L = J L$

Let's look at what we know about triangle $△ J K L$ if $K L = J L .$

Claim	Explanation
$K L = J L$	Assumption
$∠ J ≅ ∠ K$	Isosceles Triangle Theorem
$m ∠ J = m ∠ K$	Congruent angles have the same measure

This contradicts the $hypothesis$ that $m ∠ J > m ∠ K .$

Case $2$ : $K L < J L$

Let's now look at what we know about triangle $△ J K L$ if $K L < J L .$

Claim	Explanation
$K L < J L$	Assumption
$m ∠ J < m ∠ K$	Theorem $5 - 9$

This contradicts the $hypothesis$ that $m ∠ J > m ∠ K .$

Step $3$

Since our assumption that

K L \leq J L

contradicts the

hypothesis

that

m ∠ J > m ∠ K,

this assumption must be false. This proves that if one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.

Extra

Let's combine Theorem

5 - 9

and

5 - 10

Notice that Theorem $5 - 9$ and Theorem $5 - 10$ are closely related.

Theorem $5 - 9$ says that in a triangle the angle opposite to a longer side is greater. This is proven in the book.
Theorem $5 - 10$ says that in a triangle the side opposite to a greater angle is longer. This is what we have proven now.

Together they say that in a triangle greater angle corresponds to longer side.

Step 1

Step 2

Case 1: KL=JL

Exercise

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