Download 5.6 Indirect Proof and Inequalities in Two triangles

5.6 Indirect Proof and
Inequalities in Two triangles
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. mA  mB  180
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.
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.
mA  mB  180
But the sum of the three angles in a triangle equal 180
degrees,
Thus, mA  mB  mC  180
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.
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.
mA  mB  180
But the sum of the three angles in a triangle equal 180
degrees,
Thus, mA  mB  mC  180
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.
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.
mA  mB  180
But the sum of the three angles in a triangle equal 180
degrees,
Thus, mA  mB  mC  180
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.
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.
mA  mB  180
But the sum of the three angles in a triangle equal 180
degrees,
Thus, mA  mB  mC  180
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.
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.
Z
A
B
C
X
Y
Hinge Theorem
Which side is larger?
AB  ZY
BC  XY
mABC  mXYZ
Z
A
B
C
X
Y
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 ?
B  Y
AB  ZY
BC  XY
AC  XZ
Z
A
B
C
X
Y
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
AC  DF
BC  EF
A
D
11
15
58
B
C
E
F
Homework
Page 305-306
# 8 – 28 even
Homework
Page 305 – 306
# 7 – 19 odd,
27,29