Transcript Triangle Angle Sum Theorem Proof

Triangle Angle Sum Theorem
Proof
Mr. Erlin
Geometry
Fall 2010
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.
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
1
B
l
2
C
Step Two: The new line, m, forms two additional angles,
adjacent to 3 shown. Label those angles 4 & 5 .
m
A
4
1
B
3
5
l
2
C
Step Three: Outline the proof:
a) Draw the two column table with given, prove.
b)
c)
d)
e)
f)
g)
what do you know about m4, m 5 and m 3?
Consider side AB a transversal to lines l and m. Classify 4 & 1.
Do I have enough to say 4  1? Not quite…transversal, AIA, & ____
So, now 4  1, and by similar logic, show that 5  2
Since we can’t mix  & =. We need to get our  angles into = measures format.
Last step…substitution.
Statements
Reasons
ABC, with angles 1, 2 & 3 as shown
m4, m 5 and m 3 form a straight angle
m4 + m5+ m 3 = 180
Line AB is a transversal to l and m.
4 & 1 form Alternate Interior Angles
Lines l and m are parallel
_
Given
By construction
Definition of a straight angle/Protractor Postulate
Definition of transversal
Definition of Alternate Interior Angles
By construction
If parallel, transveral, AIA, then congruent
4  1
Line AC is a transversal to l and m.
5 & 2 form Alternate Interior Angles
Definition of transversal
Definition of Alternate Interior Angles
5  2
m4 = m1 & m 5 = m2
If parallel, transveral, AIA, then congruent
Definition of Congruent Angles
m1+ m2 + m3 =180
Substitution property of equality
QED
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) m4 + m5+ m 3 = 180
3) Definition of a straight angle/Protractor Postulate
4) Line AB & AC are transversals to l and m.
4) Definition of transversal
5) 4 & 1 and 5 & 2 form Alt Int Angles
5) Definition of Alternate Interior Angles
6) Lines l and m are parallel
6) By construction
7) 4  1 & 5  2
7) If parallel, transveral, AIA, then congruent
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
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
Given: m
x
n
C
A
B
4 2
1
Triangle Angle Sum Theorem NOTES
5
3
Given: m & n parallel.
y
Prove: m 1 + m2 +m 3 = 180º
Statement
Reason
1) Lines _m_ and n are _parallel_
1) __Given__
2) ABC is a _ Straight ___ angle.
2) _ Definition _ of Straight Angle
3) __m ABC __ =180°
3) If Straight Angle, then 180
4) m4 + m2 + m5 = mABC
5)
m4 + m2 + m5 =180°
6)
X is _transversal_ forming 1 & 4
Y is _ transversal _ forming 3 & 5
4) Angle Addition Postulate
5) Substitution __ Property_ of Equality _
6) Definition of Transversal(s)
7) 1 & 4 are _ alternate _ Int. s
7) Definition of Alt Interior Angles.
8) 3 & _5_ are Alternate Int. s
8)
9) 1  _4_ & 3  5
9) If
10) m1 = m4 & m3 = m5
10) Definition of _congruent_ Angles
11) m1 + m2 + m3 = 180º
11) Substitution Property of =
QED
Definition of Alt Interior Angles
parallel
transversal
Alt. Int. 
then
congruent