1. Triangle Angle Sum Theorem Proof
Mr. Erlin
Geometry
Fall 2010

2. Given: ABC, with angles 1, 2 & 3 as shown.
Mission:
Given: ABC, with angles 1, 2 & 3 as shown.
Prove: m1 + m2 + m3 = 180 A 3 1 2 B C
DON’T TAKE NOTES.
Just watch, follow along and try to understand the flow.

3. m l Step One: There exists a line m that is parallel to the
bottom side (line l), that contains the top vertex A. Draw it. m A 3 l 1 2 B C

4. m l Step Two: The new line, m, forms two additional angles,
adjacent to 3 shown. Label those angles 4 & 5 . m A 4 5 3 l 1 2 B C

5. Step Three: Outline the proof:
Draw the two column table with given, prove.
b) what do you know about m4, m 5 and m 3?
c) Consider side AB a transversal to lines l and m. Classify 4 & 1.
d) Do I have enough to say 4  1? Not quite…transversal, AIA, & ____
e) So, now 4  1, and by similar logic, show that 5  2
f) Since we can’t mix  & =. We need to get our  angles into = measures format.
g) Last step…substitution.
Statements Reasons _
ABC, with angles 1, 2 & 3 as shown
Given
m4, m 5 and m 3 form a straight angle
By construction
m4 + m5+ m 3 = 180
Definition of a straight angle/Protractor Postulate
Line AB is a transversal to l and m.
Definition of transversal
4 & 1 form Alternate Interior Angles
Definition of Alternate Interior Angles
Lines l and m are parallel
By construction
4  1
If parallel, transveral, AIA, then congruent
Line AC is a transversal to l and m.
Definition of transversal
5 & 2 form Alternate Interior Angles
Definition of Alternate Interior Angles
5  2
If parallel, transveral, AIA, then congruent
m4 = m1 & m 5 = m2
Definition of Congruent Angles
m1+ m2 + m3 =180
Substitution property of equality
QED

6. Step Four: Refine the proof:
There were some steps that were identical, yet came at different times. We could consolidate those, now that we know the whole picture.
Statements Reasons _
1) ABC, with angles 1, 2 & 3 as shown
1) Given
2) m4, m 5 and m 3 form a straight angle
2) By construction
3) Definition of a straight angle/Protractor Postulate
3) m4 + m5+ m 3 = 180
4) Line AB & AC are transversals to l and m.
4) Definition of transversal
5) Definition of Alternate Interior Angles
5) 4 & 1 and 5 & 2 form Alt Int Angles
6) Lines l and m are parallel
6) By construction
7) If parallel, transveral, AIA, then congruent
7) 4  1 & 5  2
8) m4 = m1 & m 5 = m2
8) Definition of Congruent Angles
9) m1+ m2 + m3 =180
9) Substitution property of equality
QED

7. Taking Notes
You’ve got a scaffolded proof in front of you, that was given to you as part of today’s warm up on TRI 01.
See if you can complete that proof yourself, now, simply based upon the instruction we’ve just gone thru.
Try your best, don’t give up. But after 10 minutes, we’ll post the answers on the board so everyone has a good copy in their notes

8. Triangle Angle Sum Theorem NOTESx A y B C
Triangle Angle Sum Theorem NOTES
Given: m 4 5 2
Given: m & n parallel.
Prove: m 1 + m2 +m 3 = 180º
Statement Reason 1 3 n
__Given__
_ Definition _ of Straight Angle
If Straight Angle, then 180
Angle Addition Postulate
Substitution __ Property_ of Equality _
Definition of Transversal(s)
Definition of Alt Interior Angles.
Definition of Alt Interior Angles
If then
Definition of _congruent_ Angles
Substitution Property of =
Lines _m_ and n are _parallel_
ABC is a _ Straight ___ angle.
__m ABC __ =180°
m4 + m2 + m5 = mABC
m4 + m2 + m5 =180°
X is _transversal_ forming 1 &  Y is _ transversal _ forming 3 & 5
1 & 4 are _ alternate _ Int. s
3 & _5_ are Alternate Int. s
1  _4_ & 3  5
m1 = m4 & m3 = m5
m1 + m2 + m3 = 180º
parallel
transversal
Alt. Int. 
congruent
QED