5-5 Indirect Proof

Quick Review

In an indirect proof, you first assume temporarily the opposite of what you want to prove. Then you show that this temporary assumption leads to a contradiction.

Example

Which two statements contradict each other?

1. The perimeter of Δ A B C cap delta eh b c  is 14.
2. Δ A B C cap delta eh b c  is isosceles.
3. The side lengths of Δ A B C cap delta eh b c  are 3, 5, and 6.

An isosceles triangle can have a perimeter of 14.

The perimeter of a triangle with side lengths 3, 5, and 6 is 14.

An isosceles triangle must have two sides of equal length. Statements II and III contradict each other.

Exercises

Write a convincing argument that uses indirect reasoning.

26. The product of two numbers is even. Show that at least one of the numbers must be even.
27. Two lines in the same plane are not parallel. Show that a third line in the plane must intersect at least one of the two lines.
28. Show that a triangle can have at most one obtuse angle.
29. Show that an equilateral triangle cannot have an obtuse angle.
30. The sum of three integers is greater than 9. Show that one of the integers must be greater than 3.

5-6 and 5-7 Inequalities in Triangles

Quick Review

For any triangle,

• the measure of an exterior angle is greater than the measure of each of its remote interior angles
• if two sides are not congruent, then the larger angle lies opposite the longer side
• if two angles are not congruent, then the longer side lies opposite the larger angle
• the sum of any two side lengths is greater than the third

The Hinge Theorem states that if two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger included angle.

Example

Which is greater, BC or AD?

B A ¯ ≅ C D ¯ b eh bar , approximately equal to , c d bar  and B D ¯ ≅ D B ¯ , b d bar , approximately equal to . d b bar , comma  so Δ A B D cap delta eh b d  and Δ C D B cap delta c d b  have two pairs of congruent corresponding sides. Since 60 > 45 , 60 greater than 45 , comma  you know B C > A D b c greater than eh d  by the Hinge Theorem.

Exercises

31. In Δ R S T , m ∠ R = 70 cap delta r s t comma m angle , r equals 70  and m ∠ S = 80 . m angle , s equals 80 , .  List the sides of Δ R S T cap delta r s t  in order from shortest to longest.

Is it possible for a triangle to have sides with the given lengths? Explain.

32. 5 in., 8 in., 15 in.
33. 10 cm, 12 cm, 20 cm
34. The lengths of two sides of a triangle are 12 ft and 13 ft. Find the range of possible lengths for the third side.

Use the figure below. Complete each statement with > , < , greater than comma less than comma  or = . equals .

35. m ∠ B A D m ∠ A B D m angle b eh d begin box ,
36. m ∠ C B D m ∠ B C D m angle c b d begin box ,
37. m ∠ A B D m ∠ C B D m angle eh b d begin box ,

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