Download triangle congruency proofs 4.0

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. This is a way to see all the pieces
come together to formulate the congruency theorems. By pieces, we mean definitions, basic
properties of triangles they’ve learned thus far, and theorems they’ve learned thus far. The
following proofs include: SAS, ASA, SSS, AAS. Please note the informal side notes will be in red
to signify key hints or important instructions for teachers when using teaching these proofs. Be
sure to read carefully though these proofs to make sure your students know the basic
components of geometry (definitions and basic properties of triangles) needed to comprehend
the following 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.
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.
We are given two triangles, ∆𝐴𝐵𝐶 and ∆𝐷𝐸𝐹. We also know that ̅̅̅̅
𝐴𝐵 = ̅̅̅̅
𝐷𝐸 , ̅̅̅̅
𝐴𝐶 = ̅̅̅̅
𝐷𝐹 , and
that ∠𝐴𝐵𝐶 = ∠𝐷𝐸𝐹. We want to show ∠𝐵𝐴𝐶 ≅ ∠𝐸𝐷𝐹.
To prove this is very simple. Simply realize that when you line up the angles ∠𝐴𝐵𝐶 with ∠𝐷𝐸𝐹
then the two sides that make up the angles will line up since the angles equal. Since the sides
on either side of the angle line up AND they are congruent, they will lay exactly on top of one
another. Specifically, point 𝐵 and point 𝐸 will be the same point as will points 𝐶 and 𝐹. Then if
we use Euclid’s first axiom (A unique line segment can be drawn between any two points), we
can then draw a unique line between the two points. Before introducing this proof, be sure to
go over what Euclid’s first axiom means. Perhaps have them explore it on their own to become
familiar with the first axiom before talking about it in this proof. But since the points are the
same, it will be the same line for the two triangles. Since both triangles will line up exactly, this
̅̅̅̅ , and ∠𝐵𝐴𝐶 ≅ ∠𝐸𝐷𝐹 .
proves ∠𝐵𝐴𝐶 ≅ ∠𝐸𝐷𝐹. Thus, we have ̅̅̅̅
𝐴𝐵 = ̅̅̅̅
𝐷𝐸 , ̅̅̅̅
𝐴𝐶 = 𝐷𝐹
We have proven the SAS congruency theorem; 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. From this,
∠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 (Angle Addition Postulate):
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) - Postulate 19- If three sides of one triangle are congruent to three
sides of a second triangle, then the triangles are congruent.
If
Side
Side
Side
Then ∆ABC ≡ ∆DEF
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.
Prove: ∆ABC ≡ ∆DEF
Given:
∎
Boswell, Laurie, Timothy D. Kanold, Ron Larson, and Lee Stiff. Geometry. Evanston, IL:
McDougal Littell, 2007. Print.
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.
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.
Given:
∠𝐶𝐴𝐵 ≅ ∠𝐹𝐷𝐸
∠𝐵𝐶𝐴 ≅ ∠𝐸𝐹𝐷
𝐴𝐵 ≅ 𝐷𝐸
Prove: ∠𝐴𝐵𝐶 ≅ ∠𝐷𝐸𝐹
The sum of angles equal 180,
𝑚∠𝐶𝐴𝐵 + 𝑚∠𝐴𝐵𝐶 + 𝑚∠𝐵𝐶𝐴 = 180
(1)
𝑚∠𝐹𝐷𝐸 + 𝑚∠𝐷𝐸𝐹 + 𝑚∠𝐸𝐹𝐷 = 180
(2)
Notice we can substitute 𝑚∠𝐶𝐴𝐵 + 𝑚∠𝐴𝐵𝐶 + 𝑚∠𝐵𝐶𝐴 from equation (1) for 180 in equation
(2) to obtain
𝑚∠𝐹𝐷𝐸 + 𝑚∠𝐷𝐸𝐹 + 𝑚∠𝐸𝐹𝐷 = 𝑚∠𝐶𝐴𝐵 + 𝑚∠𝐴𝐵𝐶 + 𝑚∠𝐵𝐶𝐴
However, we are given ∠𝐶𝐴𝐵 ≅ ∠𝐹𝐷𝐸, and ∠𝐵𝐶𝐴 ≅ ∠𝐸𝐹𝐷. After cancelation, we are left
with ∠𝐴𝐵𝐶 ≅ ∠𝐷𝐸𝐹.
By the Angle-Angle similarity, we have similar triangles,
𝐴𝐵 𝐵𝐶
=
𝐷𝐸 𝐸𝐹
We know, 𝐴𝐵 ≅ 𝐷𝐸 so we have
1=
𝐵𝐶
𝐸𝐹
𝐵𝐶 ≅ 𝐸𝐹
and
𝐶𝐴 𝐴𝐵
=
𝐹𝐷 𝐷𝐸
We know, 𝐶𝐴 ≅ 𝐹𝐷
1=
𝐴𝐵
𝐷𝐸
𝐴𝐵 ≅ 𝐷𝐸
This shows that ∆ABC is congruent to ∆DEF. Thus, proving Angle-Angle-Side congruency; 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.
∎
Alternative proof for AAS only if your students have explored and understand properties of
parallel lines this will be useful and may be easier to understand than the previous proof:
Credit: http://www.mathwarehouse.com/geometry/congruent_triangles/angle-angle-sidepostulate.php