Download The basic idea of utilizing these proofs is for high school teachers to

The basic idea of utilizing these proofs is for high school teachers to use these simple, basic
proof examples so get students started on thinking about the mechanics of a derivation.
“Where does it come from?” or “why?” is heard all too often. The following proofs include: SAS,
ASA, SSS, AAS. Please note the informal side notes will be keyed in red to signify key hints or
important instructions for teachers when using teaching these proofs.
G-CO.10. Prove theorems about triangles.
1. (Side-Angle-Side) - If two sides and included angle of one triangle are congruent
to two sides and the included angle of another triangle, then the triangles are
congruent.
2. (Angle-Side-Angle)- If, in a triangle, 2 angles are congruent, then the sides that
subtend these two angles will also be congruent.
We are given ∆𝐴𝐵𝐶 and ∆𝐷𝐸𝐹 with ∠𝐴𝐵𝐶 ≅ ∠𝐷𝐸𝐹, ∠𝐴𝐶𝐵 ≅ ∠𝐷𝐹𝐸, and 𝐵𝐶 = 𝐸𝐹. We are
going to assume to the contrary, 𝐴𝐵 ≠ 𝐷𝐸. We will then construct a point on 𝐴𝐵 called 𝐺 such
that 𝐵𝐺 = 𝐸𝐷. We will arrive at a contradiction rendering our original statement true. Here is a
depiction:
Here, it’s good to draw two triangles that AREN’T congruent to the naked eye.
Otherwise, your students will think these are congruent when we haven’t proven it
yet.
By SAS, ∆𝐺𝐵𝐶 ≅ ∆𝐷𝐸𝐹 this means
∠𝐺𝐶𝐵 ≅ ∠𝐷𝐹𝐸.
(1)
However, we know that any two angles that make up a bigger angle their measures
must add up to the bigger angle:
𝑚∠𝐺𝐶𝐵 + 𝑚∠𝐴𝐶𝐺 = 𝑚∠𝐷𝐹𝐸.
(2)
Notice we can substitute ∠𝐷𝐹𝐸 from relation (1) for 𝑚∠𝐺𝐶𝐵 in equation (2) to
obtain
𝑚∠𝐷𝐹𝐸 + 𝑚∠𝐴𝐶𝐺 = 𝑚∠𝐷𝐹𝐸.
The measure of an angle cannot be equal to itself when it is added to another angle
measure no matter what measure that second angle measure is. Thus, we have
reached a contradiction rendering our original statement true. If, in a triangle, 2
angles are congruent, then the sides that subtend these two angles will also be
congruent.
3.
SSS (Side-Side-Side) - If three sides of one triangle are congruent to
three sides of a second triangle, then the two triangles are congruent.
Show ∆ABC ≡ ∆DEF
Theorem:
N7 tells us that since the segments AB ≡DE and BC≡EF, there is no other point for B and E. Since
B must be at the same point as E, then we know that ∆ABC ≡ ∆DEF, we have shown through SSS
that the two triangles are congruent.
4. AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are
congruent to two angles and the corresponding non-included side of a second
triangle, then the two triangles are congruent.
Prove: <B = <E
Given: <A and <C
<A ≡ <F
AB ≡ DE
*Because the sum of angles needs to be 180..
<A + <B + <C = 180
<D + <E + <F = 180
<A + <B + <C = <D + <E + <F
*Because we are given, <A ≡ <D, and <C ≡ <F, so we are left with <B = <E
By AA we have similar ∆’s
AB/ED = BC/EF
1=BC/EF
BC ≡ EF
And..
AC/DF = AB/DE
1 = AB/DE
AB ≡ DE
This shows that ∆ABC is congruent to ∆DEF.