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# Top Drawer Teachers: Two proofs of the angle sum of a triangle

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Top Drawer Teachers
The Australian Association of Mathematics Teachers (AAMT) Inc.
http://topdrawer.aamt.edu.au

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### Top Drawer Teachers: Two proofs of the angle sum of a triangle

1. 1. The Angle Sum of a Triangle
2. 2. Prove that x + y + z = 180° A x B y z C
3. 3. Construct PQ through A so that PQ||BC P A x B Q y z C
4. 4. ∠PAB = y° (alternate angles, PQ||BC) P A y x B Q y z C
5. 5. ∠QAC = z° (alternate angles, PQ||BC) P A y x B z Q y z C
6. 6. Now PAQ is a straight line P A y x B z Q y z C
7. 7. x + y + z = 180° (PAQ is a straight line) P A y x B z Q y z C
8. 8. x + y + z = 180° A x B y z C
9. 9. The angle sum of a triangle is 180° A x B y z C
10. 10. A different proof for the same result
11. 11. Prove that x + y + z = 180° A x B y z C
12. 12. Produce BC to P A x B y z C P
13. 13. At C construct CQ||BA A x B Q y z C P
14. 14. ∠ACQ = x° (alternate angles, PQ||BC) A x B Q y z x C P
15. 15. ∠PCQ = y° (corresponding angles, PQ||BC) A x B Q y z x C y P
16. 16. Now BCP is a straight line A x B Q y z x C y P
17. 17. x + y + z = 180° (BCP is a straight line) A x B Q y z x C y P
18. 18. x + y + z = 180° A x B y z C
19. 19. The angle sum of a triangle is 180° A x B y z C