DISCOVERING CONGRUENT TRIANGLES THROUGH TRANSFORMATIONS
OBJECTIVE
Review over Rigid Transformations
Go over the definition of Congruence
See some examples of Congruent Triangles
Go over CPCTC
Homework.
Review Over Rigid Transformations
So, we know there are a few different ways to manipulate objects on a graph.
We can:
Translate it – or move it either up, down, left, or right.
Rotate it – Either around a point outside of the object, or a point on the object
Reflect it – Pick a line of reflection and reflect the object
So then, what is Congruence?
Basically, in order to understand what congruent triangles are, we need to understand what it means to be congruent.
So:
According to Wikipedia – “In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.”
In other words, two triangles are congruent if the have the same shape as well as the same size.
Now, there are a few important things about congruent triangles, and honestly, there is a whole bunch of topics about proving them, but for right now, just know they are important.
So what do congruent triangles look like?
So again, two triangles are congruent only if they share the same shape and size.
This means each of their corresponding sides are going to be the same length.
It also means each of their corresponding angles are going to be the same.
So, some congruent triangles are:
So again, if you can either rotate, reflect, or translate (or any sequence of those three) one of the triangles to and can map it on perfectly to the other triangle, then the two triangles are congruent.
So how do we know that they are congruent?
We look at their corresponding parts.
If their corresponding parts are congruent, then the triangle is congruent.
So what is a corresponding part?
Well let’s look.
Let’s say we have a triangle:
C
A B
Now, let’s say we translate that triangle down a few units:
F
D E
We haven’t rotated, or reflected, we just picked up our first triangle
And moved it.
So, let’s look at the corresponding parts.
And corresponding side (or like side) DF
We have side AC
We have side CB
And corresponding side (or like side) FE
We have side AB
And corresponding side (or like side) DE
So, since we didn’t rotate, reflect, or make anything bigger, are these sides all the same length as their corresponding counterparts?
Or, in other words, are their corresponding parts the same?
Well, we can see that AC = DF
Well, we can also see that CB = FE
And finally, we can also see that AB = DE
But that’s not all!
We can see that the angles are also the same!
For example, we can see that:
C
A B
F
D E
So, what does this all mean?
Well, it would seem that:
In congruent triangles,
all corresponding parts of each triangle,
are congruent.
CPCTC
And that’s what CPCTC stands for:
Corresponding
Parts of
Congruent
Triangles are
Congruent
Now, it’s important to note that this works both ways.
In other words, yes it’s true that if two triangles are congruent, then their parts are congruent as well.
But, it’s also true that if the corresponding parts of a triangle are congruent, then the triangles are congruent.
So, let’s look at an example to show what that means:
EXAMPLE 1:
Are these two triangles congruent?
Well, we know that Corresponding Parts of Congruent Triangles are Congruent.
Which means, we need to know some congruent parts.
So, let’s take a look at the lengths of each side:
Now let’s record what we know:
We know that:
AB = 3
AC = 3.2
CB = 3.6
And
DE = 3
DF = 3.2
FE = 3.6
So:
AB = DE�AC = DF
CB = FE
Since we can see that the corresponding sides of both of these triangles are congruent, then we know that
These two triangles are congruent.
EXAMPLE 2:
Are these two triangles congruent?
Well, we know that Corresponding Parts of Congruent Triangles are Congruent.
Which means, we need to know some congruent parts.
So, let’s take a look at the measurements of each angle:
Now let’s record what we know:
We know that:
Since we can see that the corresponding angles of both of these triangles are congruent, then we know that
These two triangles are congruent.
(It’s important to note that just because the
Angles are the same, doesn’t mean the triangle are the same. However, since we can see that AC = FD, then with the added bonus of having a congruent side, we can know for sure that the two triangles are the same.)
EXAMPLE 3:
Are these two triangles congruent?
Well, we know that Corresponding Parts of Congruent Triangles are Congruent.
Which means, we need to know some congruent parts.
So, let’s take a look at the lengths of each side:
Now let’s record what we know:
We know that:
AB = 3
AC = 3.2
CB = 3.6
And
DE = 3
DF = 3.2
FE = 3.6
So:
AB = DE�AC = DF
CB = FE
Since we can see that the corresponding sides of both of these triangles are congruent, then we know that
These two triangles are congruent.
EXAMPLE 4
Are these two triangles congruent?
Well, we know that Corresponding Parts of Congruent Triangles are Congruent.
Which means, we need to know some congruent parts.
So, let’s take a look at the measurements of each angle:
Now let’s record what we know:
We know that:
Since we can see that the corresponding angles of both of these triangles are congruent, then we know that
These two triangles are congruent.
(It’s important to note that just because the
Angles are the same, doesn’t mean the triangle are the same. However, since we can see that AC = FD, then with the added bonus of having a congruent side, we can know for sure that the two triangles are the same.)