1. First 10! Step 1: Draw a triangle and label the vertices ABC. Step 2:
Measure each angle with a protractor.
Step 3:
Calculate the sum of the angle measures.
Step 4:
Did your neighbor find the same thing?

3. A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem.

4. Example 1: Application
After an accident, the positions of cars are measured by law enforcement to investigate the collision. Use the diagram drawn from the information collected to find mXYZ.
mXYZ + mYZX + mZXY = 180°
Sum. Thm
Substitute 40 for mYZX and 62 for mZXY.
mXYZ = 180
mXYZ = 180
Simplify.
mXYZ = 78°
Subtract 102 from both sides.

5. Example 1: Application
After an accident, the positions of cars are measured by law enforcement to investigate the collision. Use the diagram drawn from the information collected to find mYWZ.
118°
Step 1 Find mWXY.
mYXZ + mWXY = 180°
Lin. Pair Thm. and  Add. Post.
62 + mWXY = 180
Substitute 62 for mYXZ.
mWXY = 118°
Subtract 62 from both sides.

6. Example 1: Application Continued
After an accident, the positions of cars are measured by law enforcement to investigate the collision. Use the diagram drawn from the information collected to find mYWZ.
118°
Step 2 Find mYWZ.
mYWX + mWXY + mXYW = 180°
Sum. Thm
Substitute 118 for mWXY and 12 for mXYW.
mYWX = 180
mYWX = 180
Simplify.
mYWX = 50°
Subtract 130 from both sides.

7. Let the acute angles be A and B, with mA = 63.7°.
Example 2
The measure of one of the acute angles in a right triangle is 63.7°. What is the measure of the other acute angle?
Let the acute angles be A and B, with mA = 63.7°.
mA + mB = 90°
Acute s of rt. are comp.
mB = 90
Substitute 63.7 for mA.
mB = 26.3°
Subtract 63.7 from both sides.

8. The interior is the set of all points inside the figure
The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure. It is formed by one side of the triangle and extension of an adjacent side.
Exterior
Interior

9. Each exterior angle has two remote interior angles
Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle.
4 is an exterior angle.
The remote interior angles of 4 are 1 and 2.
Exterior
Interior
3 is an interior angle.

10. Example 3: Applying the Exterior Angle Theorem
Find mB.
mA + mB = mBCD
Ext.  Thm.
Substitute 15 for mA, 2x + 3 for mB, and 5x – 60 for mBCD.
15 + 2x + 3 = 5x – 60
2x + 18 = 5x – 60
Simplify.
Subtract 2x and add 60 to both sides.
78 = 3x
26 = x
Divide by 3.
mB = 2x + 3 = 2(26) + 3 = 55°

11. Example 3 Find mACD. mACD = mA + mB 6z – 9 = 2z + 1 + 90
Ext.  Thm.
Substitute 6z – 9 for mACD, 2z + 1 for mA, and 90 for mB.
6z – 9 = 2z
6z – 9 = 2z + 91
Simplify.
Subtract 2z and add 9 to both sides.
4z = 100
z = 25
Divide by 4.
mACD = 6z – 9 = 6(25) – 9 = 141°

12. Example 4 Find mP and mT. P  T mP = mT 2x2 = 4x2 – 32
Third s Thm.
mP = mT
Def. of  s.
2x2 = 4x2 – 32
Substitute 2x2 for mP and 4x2 – 32 for mT.
–2x2 = –32
Subtract 4x2 from both sides.
x2 = 16
Divide both sides by -2.
So mP = 2x2 = 2(16) = 32°.
Since mP = mT, mT = 32°.

13. Geometric figures are congruent if they are the same size and shape
Geometric figures are congruent if they are the same size and shape. Corresponding angles and corresponding sides are in the same position in polygons with an equal number of sides.
Two polygons are congruent polygons if and only if their corresponding sides are congruent. Thus triangles that are the same size and shape are congruent.

15. Example 1
If polygon LMNP  polygon EFGH, identify all pairs of corresponding congruent parts.
Angles: L  E, M  F, N  G, P  H
Sides: LM  EF, MN  FG, NP  GH, LP  EH

16. Example 2a Given: ∆ABC  ∆DEF a.) Find the value of x. b.) Find mF.
AB  DE
Corr. sides of  ∆s are .
AB = DE
Def. of  parts.
Substitute values for AB and DE.
2x – 2 = 6
2x = 8
Add 2 to both sides.
x = 4
Divide both sides by 2.