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 l 1 2 B C

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

Statements Reasons _ 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. ABC, with angles 1, 2 & 3 as shown Given By construction m4, m 5 and m 3 form a straight angle Definition of a straight angle/Protractor Postulate m4 + m5+ m 3 = 180 Line AB is a transversal to l and m. Definition of transversal Definition of Alternate Interior Angles 4 & 1 form Alternate Interior Angles Lines l and m are parallel By construction If parallel, transveral, AIA, then congruent 4  1 Line AC is a transversal to l and m. Definition of transversal Definition of Alternate Interior Angles 5 & 2 form Alternate Interior Angles If parallel, transveral, AIA, then congruent 5  2 Definition of Congruent Angles m4 = m1 & m 5 = m2 m1+ m2 + m3 =180 Substitution property of equality QED

Statements Reasons _ 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. 1) ABC, with angles 1, 2 & 3 as shown 1) Given 2) By construction 2) m4, m 5 and m 3 form a straight angle 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) By construction 6) Lines l and m are parallel 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

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

y x parallel transversal Alt. Int.  congruent C A B Triangle Angle Sum Theorem NOTES Given: m Statement Reason 4 5 2 Given: m & n parallel. Prove: m 1 + m2 +m 3 = 180º 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° • Xis _transversal_ forming 1 & 4 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 QED

Triangle Angle Sum Theorem Proof