1. Proofs

2. Math as a Language English : Math :: Words : Numbers
English : Math :: Sentences : Equations
English : Math :: Essays : Proofs
Math communicates ideas just like English does. Whereas rhetoric may blur facts in English, math is unique in that truth is its only focus.
A well-written proof indisputably shows something as true.

3. Prove: Base angles of an isosceles triangle are congruent.
Assume only the information given. In this case, we’re only given the triangle is isosceles. What’s the definition? Isosceles triangles have at least two congruent sides. TIP: Think about working backwards. The last thing in our proof would say the angles are congruent. What reasons do we have in our toolbox that result in congruent angles?

4. Prove: Base angles of an isosceles triangle are congruent.
Assume only the information given. In this case, we’re only given the triangle is isosceles. What’s the definition? Isosceles triangles have at least two congruent sides. TIP: Think about working backwards. The last thing in our proof would say the angles are congruent. What reasons do we have in our toolbox that result in congruent angles?

5. Prove: Base angles of an isosceles triangle are congruent.
Assume only the information given. In this case, we’re only given the triangle is isosceles. What’s the definition? Isosceles triangles have at least two congruent sides. TIP: Think about working backwards. The last thing in our proof would say the angles are congruent. What reasons do we have in our toolbox that result in congruent angles?

6. Routes Vertical angles are congruent.
Angles formed by two parallel lines and a transversal have congruent angle pairs.
Corresponding angles of two congruent triangles are congruent (CPCT).
Not all are necessarily going to be useful, but don’t be afraid to try a particular path.
Of these reasons, which one do we think will need to involve our known fact about congruent sides?

7. Routes Vertical angles are congruent.
Angles formed by two parallel lines and a transversal have congruent angle pairs.
Corresponding angles of two congruent triangles are congruent (CPCT).
Not all are necessarily going to be useful, but don’t be afraid to try a particular path.
Of these reasons, which one do we think will need to involve our known fact about congruent sides?

8. Routes Vertical angles are congruent.
Angles formed by two parallel lines and a transversal have congruent angle pairs.
Corresponding angles of two congruent triangles are congruent (CPCT).
Not all are necessarily going to be useful, but don’t be afraid to try a particular path.
Of these reasons, which one do we think will need to involve our known fact about congruent sides?

9. Prove: Base angles of an isosceles triangle are congruent.
If we’re going to use CPCT, then we need congruent triangles. Where are they?! Be creative!

10. Prove: Base angles of an isosceles triangle are congruent.
If we’re going to use CPCT, then we need congruent triangles. Where are they?! Be creative! Add a line going through A and perpendicular to BC.

11. Prove: Base angles of an isosceles triangle are congruent.
How do we know triangles AMB and AMC are congruent? HLR! Hypotenuse: 𝐴𝐵 ≅ 𝐴𝐶 , ΔABC is isosceles (given) Leg: 𝐴𝑀 ≅ 𝐴𝑀 , reflexive identity Right: ∠𝐴𝑀𝐵≅∠𝐴𝑀𝐶, right angles Thus, Δ𝐴𝑀𝐵≅Δ𝐴𝑀𝐶 via HLR Then ∠𝐴𝐵𝑀≅∠𝐴𝐶𝑀 via CPCT

12. Prove: Base angles of an isosceles triangle are congruent.
How do we know triangles AMB and AMC are congruent? HLR! Hypotenuse: 𝐴𝐵 ≅ 𝐴𝐶 , ΔABC is isosceles (given) Leg: 𝐴𝑀 ≅ 𝐴𝑀 , reflexive identity Right: ∠𝐴𝑀𝐵≅∠𝐴𝑀𝐶, right angles Thus, Δ𝐴𝑀𝐵≅Δ𝐴𝑀𝐶 via HLR Then ∠𝐴𝐵𝑀≅∠𝐴𝐶𝑀 via CPCT

13. Prove: Base angles of an isosceles triangle are congruent.
How do we know triangles AMB and AMC are congruent? HLR! Hypotenuse: 𝐴𝐵 ≅ 𝐴𝐶 , ΔABC is isosceles (given) Leg: 𝐴𝑀 ≅ 𝐴𝑀 , reflexive identity Right: ∠𝐴𝑀𝐵≅∠𝐴𝑀𝐶, right angles Thus, Δ𝐴𝑀𝐵≅Δ𝐴𝑀𝐶 via HLR Then ∠𝐴𝐵𝑀≅∠𝐴𝐶𝑀 via CPCT

14. Prove: Base angles of an isosceles triangle are congruent.
How do we know triangles AMB and AMC are congruent? HLR! Hypotenuse: 𝐴𝐵 ≅ 𝐴𝐶 , ΔABC is isosceles (given) Leg: 𝐴𝑀 ≅ 𝐴𝑀 , reflexive identity Right: ∠𝐴𝑀𝐵≅∠𝐴𝑀𝐶, right angles Thus, Δ𝐴𝑀𝐵≅Δ𝐴𝑀𝐶 via HLR Then ∠𝐴𝐵𝑀≅∠𝐴𝐶𝑀 via CPCT

15. Prove: Base angles of an isosceles triangle are congruent.
How do we know triangles AMB and AMC are congruent? HLR! Hypotenuse: 𝐴𝐵 ≅ 𝐴𝐶 , ΔABC is isosceles (given) Leg: 𝐴𝑀 ≅ 𝐴𝑀 , reflexive identity Right: ∠𝐴𝑀𝐵≅∠𝐴𝑀𝐶, right angles Thus, Δ𝐴𝑀𝐵≅Δ𝐴𝑀𝐶 via HLR Then ∠𝐴𝐵𝑀≅∠𝐴𝐶𝑀 via CPCT

16. Prove: Base angles of an isosceles triangle are congruent.
How do we know triangles AMB and AMC are congruent? HLR! Hypotenuse: 𝐴𝐵 ≅ 𝐴𝐶 , ΔABC is isosceles (given) Leg: 𝐴𝑀 ≅ 𝐴𝑀 , reflexive identity Right: ∠𝐴𝑀𝐵≅∠𝐴𝑀𝐶, right angles Thus, Δ𝐴𝑀𝐵≅Δ𝐴𝑀𝐶 via HLR Then ∠𝐴𝐵𝑀≅∠𝐴𝐶𝑀 via CPCT

17. Prove: Base angles of an isosceles triangle are congruent.
How do we know triangles AMB and AMC are congruent? HLR! Hypotenuse: 𝐴𝐵 ≅ 𝐴𝐶 , ΔABC is isosceles (given) Leg: 𝐴𝑀 ≅ 𝐴𝑀 , reflexive identity Right: ∠𝐴𝑀𝐵≅∠𝐴𝑀𝐶, right angles Thus, Δ𝐴𝑀𝐵≅Δ𝐴𝑀𝐶 via HLR Then ∠𝐴𝐵𝑀≅∠𝐴𝐶𝑀 via CPCT

18. Guidelines for Proof Writing
Think backwards
Don’t be afraid to try things out
Use your known information
Make no unfounded assumptions and state no unexplained facts
Be creative
There exists another, different, simple and ingenious proof for proving base angles of an isosceles triangle congruent. Can you think of it?

19. Proofs Answer Interesting Questions
What’s the minimum number of colors it would take to color in a map so no states/countries of the same color will be touching?

20. Proofs Answer Interesting Questions
What’s the minimum number of colors it would take to color in a map so no states/countries of the same color will be touching? Four.

21. Proofs Answer Interesting Questions
One of these is possible to draw without picking up your pencil from the paper (and not “backtracking” on your lines), the other one is impossible to do so.

22. Proofs Answer Interesting Questions
One of these is possible to draw without picking up your pencil from the paper (and not “backtracking” on your lines), the other one is impossible to do so.
Possible
(if you start from one of the bottom corners)
Impossible

23. Proofs Answer Interesting Questions
While math can prove a lot of interesting things, its power in doing so is not limitless.
We know this because the limits of math’s power has actually been proven by math itself.
Gödel’s Incompleteness Theorem
Kurt Gödel