5.6 Indirect Proof and 5.6 Indirect Proof and Inequalities in Two Inequalities in Two

trianglestriangles

Indirect proof starts by Assuming the opposite of the truth.

Statement: In a right triangle, the triangle does not have more than one right angle.

What would you assume if you want to prove the statement correct by an indirect proof?

Indirect proof starts by Assuming the opposite of the truth.

Statement: In a right triangle, the triangle does not have more than one right angle.

What would you assume if you want to prove the statement correct by an indirect proof?

Assume the triangle has two right angles.

Proving by an Indirect Method

Assume that triangle ABC does have more then one right angle. So measure of angles A and B are both 90 degrees.

Thus,But the sum of the three angles in a triangle equal 180

degrees, So, angle C equal 0 degrees, which is impossible. Angles

in a triangle must be greater then 0. Therefore, the assumption is wrong.

Thus, In a right triangle, the triangle does not have more then one right angle.

180BmAm

Proving by an Indirect Method

Assume that triangle ABC does have more then one right angle. So measure of angles A and B are both 90 degrees.

But the sum of the three angles in a triangle equal 180 degrees,

Thus,

So, angle C equal 0 degrees, which is impossible. Angles in a triangle must be greater then 0. Therefore, the assumption is wrong.

Thus, In a right triangle, the triangle does not have more then one right angle.

180BmAm

180CmBmAm

Proving by an Indirect Method

Assume that triangle ABC does have more then one right angle. So measure of angles A and B are both 90 degrees.

But the sum of the three angles in a triangle equal 180 degrees,

Thus,

So, angle C equal 0 degrees, which is impossible. Angles in a triangle must be greater then 0. Therefore, the assumption is wrong.

Thus, In a right triangle, the triangle does not have more then one right angle.

180BmAm

180CmBmAm

Proving by an Indirect Method

Assume that triangle ABC does have more then one right angle. So measure of angles A and B are both 90 degrees.

But the sum of the three angles in a triangle equal 180 degrees,

Thus,

So, angle C equal 0 degrees, which is impossible. Angles in a triangle must be greater then 0. Therefore, the assumption is wrong.

Thus, In a right triangle, the triangle does not have more then one right angle.

180BmAm

180CmBmAm

Proving by an Indirect Method

Assume that triangle ABC does have more then one right angle. So measure of angles A and B are both 90 degrees.

But the sum of the three angles in a triangle equal 180 degrees,

Thus,

So, angle C equal 0 degrees, which is impossible. Angles in a triangle must be greater then 0. Therefore, the assumption is wrong.

Thus, In a right triangle, the triangle does not have more then one right angle.

180BmAm

180CmBmAm

Hinge Theorem

If you have two sides of two different triangles congruent, then the set of sides with the larger angle between them has the larger side across from it.

A

B C X Y

Z

Hinge Theorem

Which side is larger?

A

B C X Y

Z

XYZmABCm

XYBC

ZYAB

Converse

The converse of the hinge theorem is also true: If the two sides of one triangle are congruent to two sides of another triangle, and the third side of the first triangle is greater than the third side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second triangle.

In some textbooks, the theorem and its converse are written as the SAS Inequality Theorem and the SSS Inequality Theorem respectively.

http://en.wikipedia.org/wiki/Hinge_theorem

Hinge Theorem

Which angle is larger ?

A

B C X Y

Z

XZAC

XYBC

ZYAB

YB

Here is a link to see how this works

5-5 The Hinge Theorem - Mr. Self - Cleveland High School

What are the possible measurements of Angle C

Use an inequality

A

B C

EFBC

DFAC

D

E F58

1115

HomeworkHomework

Page 305-306Page 305-306

##8 – 28 even8 – 28 even

HomeworkHomework

Page 305 – 306 Page 305 – 306

##7 – 19 odd,7 – 19 odd,

27,2927,29