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Proofs
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.
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?
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?
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?
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?
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?
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?
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!
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.
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
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
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
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
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
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
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
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?
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?
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.
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.
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.
Impossible
Possible
(if you start from one of the bottom corners)
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