5.6 Indirect Proof and Inequalities in Two triangles
-
Upload
john-mckee -
Category
Documents
-
view
30 -
download
1
description
Transcript of 5.6 Indirect Proof and Inequalities in Two triangles
![Page 1: 5.6 Indirect Proof and Inequalities in Two triangles](https://reader036.fdocuments.net/reader036/viewer/2022082517/56812c2e550346895d90afa4/html5/thumbnails/1.jpg)
5.6 Indirect Proof and 5.6 Indirect Proof and Inequalities in Two Inequalities in Two
trianglestriangles
![Page 2: 5.6 Indirect Proof and Inequalities in Two triangles](https://reader036.fdocuments.net/reader036/viewer/2022082517/56812c2e550346895d90afa4/html5/thumbnails/2.jpg)
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?
![Page 3: 5.6 Indirect Proof and Inequalities in Two triangles](https://reader036.fdocuments.net/reader036/viewer/2022082517/56812c2e550346895d90afa4/html5/thumbnails/3.jpg)
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.
![Page 4: 5.6 Indirect Proof and Inequalities in Two triangles](https://reader036.fdocuments.net/reader036/viewer/2022082517/56812c2e550346895d90afa4/html5/thumbnails/4.jpg)
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
![Page 5: 5.6 Indirect Proof and Inequalities in Two triangles](https://reader036.fdocuments.net/reader036/viewer/2022082517/56812c2e550346895d90afa4/html5/thumbnails/5.jpg)
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
![Page 6: 5.6 Indirect Proof and Inequalities in Two triangles](https://reader036.fdocuments.net/reader036/viewer/2022082517/56812c2e550346895d90afa4/html5/thumbnails/6.jpg)
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
![Page 7: 5.6 Indirect Proof and Inequalities in Two triangles](https://reader036.fdocuments.net/reader036/viewer/2022082517/56812c2e550346895d90afa4/html5/thumbnails/7.jpg)
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
![Page 8: 5.6 Indirect Proof and Inequalities in Two triangles](https://reader036.fdocuments.net/reader036/viewer/2022082517/56812c2e550346895d90afa4/html5/thumbnails/8.jpg)
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
![Page 9: 5.6 Indirect Proof and Inequalities in Two triangles](https://reader036.fdocuments.net/reader036/viewer/2022082517/56812c2e550346895d90afa4/html5/thumbnails/9.jpg)
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
![Page 10: 5.6 Indirect Proof and Inequalities in Two triangles](https://reader036.fdocuments.net/reader036/viewer/2022082517/56812c2e550346895d90afa4/html5/thumbnails/10.jpg)
Hinge Theorem
Which side is larger?
A
B C X Y
Z
XYZmABCm
XYBC
ZYAB
![Page 11: 5.6 Indirect Proof and Inequalities in Two triangles](https://reader036.fdocuments.net/reader036/viewer/2022082517/56812c2e550346895d90afa4/html5/thumbnails/11.jpg)
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
![Page 12: 5.6 Indirect Proof and Inequalities in Two triangles](https://reader036.fdocuments.net/reader036/viewer/2022082517/56812c2e550346895d90afa4/html5/thumbnails/12.jpg)
Hinge Theorem
Which angle is larger ?
A
B C X Y
Z
XZAC
XYBC
ZYAB
YB
![Page 13: 5.6 Indirect Proof and Inequalities in Two triangles](https://reader036.fdocuments.net/reader036/viewer/2022082517/56812c2e550346895d90afa4/html5/thumbnails/13.jpg)
Here is a link to see how this works
5-5 The Hinge Theorem - Mr. Self - Cleveland High School
![Page 14: 5.6 Indirect Proof and Inequalities in Two triangles](https://reader036.fdocuments.net/reader036/viewer/2022082517/56812c2e550346895d90afa4/html5/thumbnails/14.jpg)
What are the possible measurements of Angle C
Use an inequality
A
B C
EFBC
DFAC
D
E F58
1115
![Page 15: 5.6 Indirect Proof and Inequalities in Two triangles](https://reader036.fdocuments.net/reader036/viewer/2022082517/56812c2e550346895d90afa4/html5/thumbnails/15.jpg)
HomeworkHomework
Page 305-306Page 305-306
##8 – 28 even8 – 28 even
![Page 16: 5.6 Indirect Proof and Inequalities in Two triangles](https://reader036.fdocuments.net/reader036/viewer/2022082517/56812c2e550346895d90afa4/html5/thumbnails/16.jpg)
HomeworkHomework
Page 305 – 306 Page 305 – 306
##7 – 19 odd,7 – 19 odd,
27,2927,29