2.2 Triangle Theorems

Triangle Angle Sum Theorem:

The sum of the measures of the angles of a triangle is equal to 180°.

So: mA + mB + mC = 180^o

Note : We will be using some new symbols to be used for representing angle measurement variables.
"Alpha" = α, "Beta" = β, "Gama" = γ, "Delta" = δ, "Epsilon" = ε, "Zeta" = ζ, "Eta" = η

Example A

Question:

Does the Triangle Sum Theorem always work?

Solution:

Yes, it does!

You can use the Geogebra app to verify for yourself.

Step:
1. Create Polygon (Triangle) ABC
2. Identify Measure of Angle ABC (α= )
3. Identify Measure of Angle BCA (β= )
4. Identify Measure of Angle CAB (γ= )
5. Using

click on

if needed
- Press "="
- Click

then

              - Type "α+β+γ" then press "Enter"

      You now have a Triangle with the sum
       of the angles = 180^o

You can move the points A, B, & C to see if the sum always is = 180^o

Press

to run the Geogebra build simulation.

Try it for yourself in the Sandbox

Watch the video:

Quick Question...

To Answer:

Click either "True" or "False"
Use the given "keypad" to enter your numerical answer

To use the app:

Clickto begin, click for full screen mode
Clickto check your answer.
Clickto goto the next problem
Clickto stop the app

Triangle Angle Sum Theorem Corollaries

(A corollary to a theorem is a statement that follows directly from that theorem. )

The acute angles of a right triangle are complementary.
The measure of each angle of an equiangular triangle is 60°.
A triangle can have at most one right or one obtuse angle.

Exterior Angle Theorem:

The measure of each exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

So:     mABE = mBAC + mACB

          and

          mACD = mBAC + mCBA

Example B

Question:

Does the Exterior Angle Theorem always work?

Solution:

Yes, it does!

You can use the Geogebra app to verify for yourself.

Step:
1. Create Polygon (Triangle) ABC
2. Create Line BC
3. Create Point D, to the left of Point C
4. Create Point E, to the right of Point B
5. Identify Measure of Angle ABC (α= )
6. Identify Measure of Angle BCA (β= )
7. Identify Measure of Angle CAB (γ= )
8. Identify Measure of Angle ACD (δ= )
9. Identify Measure of Angle EBA (ε= )
5. Use "Input" to do the following:
              - Type "=α+γ" then press "Enter"
              - Type "=β+γ" then press "Enter"

      You will now see that:
       (Angle CAB) + (Angle ABC) = (Angle ACD)
                                           &
       (Angle CAB) + (Angle ACB) = (Angle ABE)

You can move the points A, B, & C to see if these sums are always equal

Press

to run the Geogebra build simulation.

Try it for yourself in the Sandbox

Watch the video:

Quick Question...

To Answer:

Click either "True" or "False"
Use the given "keypad" to enter your numerical answer

To use the app:

Clickto begin, click for full screen mode
Clickto check your answer.
Clickto goto the next problem
Clickto stop the app

The Relationship Between Sides and Angles in a Triangle

If one side of a triangle is longer than another side, then the angle
opposite the first side is larger than the angle opposite the second side.
If one angle of a triangle is larger than another angle, then the side
opposite the first angle is longer than the side opposite the second angle.

Example C

Problem: Is this Triangle Relationship always true?	Solution: Yes, it is! You can use the Geogebra app to verify for yourself.	Watch the video:

Quick Question...

To Answer:

Click either "True" or "False"
Use the given "keypad" to enter your numerical answer

To use the app:

Clickto begin, click for full screen mode
Clickto check your answer.
Clickto goto the next problem
Clickto stop the app

Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

AND...

If you have or choose any 2-sides of a triangle, then:

Note: This idea is the BIG IDEA for triangles. If you have three segments of
different lengths and the Triangle Inequality Theorem does not work
then you can not make a triangle

Example D

Problem:

Is the Triangle Inequality Theorem always true

Solution:

Yes, it is!

You can use the Geogebra app to verify for yourself.

Watch the video:

Quick Question...

To Answer:

Click either "True" or "False"
Use the given "keypad" to enter your numerical answer

To use the app:

Clickto begin, click for full screen mode
Clickto check your answer.
Clickto goto the next problem
Clickto stop the app

Try this!

Given the segments on the left:

Try to make 3-triangles, 1-blue, 1-green & 1-red triangle.

Can you make make 3-triangle using any colors (sides) you want?

Triangle Angle Bisector Theorem

If a line bisects an angle of a triangle, then it divides the opposite sides proportionally to the other two sides of the triangle

Example E

Problem:

Is this Triangle Angle Bisector Theorem always true?

Solution:

Yes, it is!

You can use the Geogebra app to verify for yourself.

Watch the video:

Quick Question...

To Answer:

Click either "True" or "False"
Use the given "keypad" to enter your numerical answer

To use the app:

Clickto begin, click for full screen mode
Clickto check your answer.
Clickto goto the next problem
Clickto stop the app

Practice Problems

A. In ΔABC, A = 51^o and B = 15^o , what is the measure of C ?
B. In the figure on the right, if ABE = 125^o and A = 15^o , what are the measures of B & C ?
C. In ΔABC, if the following is true: A > B > C, put the sides in order of length from least to greatest.
D. If possible, use Geogebra to create ΔABC, where = 7, = 8, and = 9?
E If possible, use Geogebra to create ΔABC, where = 10, = 20, and = 30?
F. Use the Triangle Angle Bisector Theorem to verify that ΔABC, on the right, is bisected by.

G. Use the Triangle Angle Bisector Theorem with ΔABC, on the right, to find the length of if = 3.81, = 3.96, = 2.43

Assignment:

1. In ΔABC, A = 107^o and B = 49^o , what is the measure of C ?
2. ΔABC is an Obtuse Isosceles Triangle with B = 17^o what could the measures of A & C be?
3. Is it possible to create an Equilateral Triangle with an angle measure of 45^o ? If possible, use Geogebra to create this triangle.
4. Is it possible to create a Right Triangle with an angle measure of 45^o ? If possible, use Geogebra to create this triangle.
5 Is it possible to create an Isosceles Triangle with both base angle measures of 85^o ? If possible, use Geogebra to create this triangle.
6. If ACD = 123^o and B = 55^o , what are the measures of A & C ?	Figure for problems 6-9:
7. If B = 60^o and C = 60^o , what are the measures of A & ABE?
8. If B = 74^o and A = 27^o , what is the measure of ACD?
9. Use Geogebra to create the figure on the right, where C = 49^o and ABE = 113^o . Find all of the remaining angle measures.
10. In ΔABC, <, < .Use Geogebra to create the triangle, Then list the the angles in order from least to greatest.
11. In ΔABC, if ≤, ≤, .is it possible for A < B?
12. If possible, use Geogebra to create ΔABC, where = 9, = 5, and = 3?

13. In ΔABC, where = 16 and = 23 what is the largest possible length that could be to make a triangle?

14. In ΔABC, where = 16 and = 23 what is the smallest possible length that could be to make a triangle?
15. Use the Triangle Angle Bisector Theorem with ΔABC, on the right, to find the length of if = 4.26, = 5.41, = 2.11
16. Use the Triangle Angle Bisector Theorem with ΔABC, on the right, to find the length of if = 2.59, = 4.51, = 3.14
17. Use the Triangle Angle Bisector Theorem with ΔABC, on the right, to find the length of if = 3.81, = 3.96, = 2.43
18. Use the Triangle Angle Bisector Theorem with ΔABC, on the right, to find the length of if = 5.16, = 3.9, = 3.53
19. Use the Triangle Angle Bisector Theorem with ΔABC, on the right, to find the length of if = 4.45, = 4.65, = 1.83
20. Use the Triangle Angle Bisector Theorem to verify that ΔABC, on the right, is bisected by, if = 4.95, = 5.7, = 4.84, and= 5.57