Welcome to a lesson on determining
if two triangles are congruent.
The goals are to review the ways to prove
two triangles are congruent and then identify
congruent triangles.
Two triangles are congruent if and only if corresponding
angles and sides are congruent, as we see marked here.
It's important that when we identify congruent triangles
the vertices are listed in corresponding order.
So if we identify this triangle as triangle ABC,
it would be congruent to triangle MNO
because vertex A corresponds to vertex M,
vertex B corresponds to vertex N,
and vertex C corresponds to vertex O.
When referring to corresponding parts of congruent
triangles, it is called Corresponding Parts of Congruent
Triangles are Congruent, or CPCTC.
Remember from the previous video, there are four postulates
that can be used to verify two triangles congruent.
We have the Side-Side-Side Postulate, Side-Angle-Side
Postulate, Angle-Side-Angle Postulate,
and Angle-Angle-Side Postulate.
Notice that Side-Side-Angle and Angle-Angle-Angle
cannot be used to determine congruent triangles.
Let's take a look at our examples.
Here we're given a sketch, and we'll determine
if the triangles are congruent, and if they are,
give the reason and then state a congruence statement.
Notice that we're given that segment AB is congruent
to segment DE, and they're also parallel.
If these two segments are parallel, notice that BD
and AE would be transversals.
Therefore, we know that the alternate interior angles
would have to be congruent.
So angle A is congruent to angle E, and angle B
is congruent to angle D.
Looking at the information that we have,
notice we have Angle-Side-Angle and Angle-Side-Angle
in both of these triangles.
Therefore, by Angle-Side-Angle,
triangle ABC
would be congruent to the triangle on the right,
and again, we need to make sure we list these vertices
in the corresponding order.
Notice that angle A corresponds to angle E,
so vertex A corresponds to vertex E.
Angle B corresponds to angle D,
so vertex B corresponds to vertex D,
and vertex C corresponds to itself.
Let's take a look at another one.
Same question, different information.
Here, we're given that AB is congruent to DE.
Let's go ahead and mark that, and we're also given that C
is the midpoint of segment AE and segment DB.
Here's segment AE, so if C is the midpoint,
AC is congruent to CE, and if C is the midpoint
of segment BD, then BC would be congruent to CD.
So by Side-Side-Side, these two triangles are congruent.
We have triangle ABC and again, let's make sure
that we list the other triangle so the vertices correspond.
So A would correspond to E, notice how angle E
is formed by the two sides with one hash mark
and two hash marks, the same with angle A.
For vertex B, we have angle B formed by the two sides
with one hash mark and three hash marks.
So vertex B corresponds to vertex D.
Then vertex C corresponds to vertex C.
Same question, different diagram.
Notice the only information we're given here
is segment AB is congruent to segment BC
and segment AE is congruent to segment CD.
These two intersecting segments do form vertical angles,
so we could conclude that angle ABE is congruent
to angle CBD,
but notice we have Side-Side-Angle.
Side-Side-Angle or Angle-Side-Side does not fit
one of the four postulates, and therefore, we cannot
conclude that these are congruent triangles.
By the given information, we cannot conclude
that these are congruent triangles.
Let's go and take a look at one more example.
Notice we're given two pair of congruent sides,
and then both triangles share side BD.
Of course, it's congruent to itself,
so by Side-Side-Side, these two triangles
would be congruent.
We'll say triangle ABD
be congruent to triangle, A corresponds to C,
B corresponds to itself, and D corresponds to itself.
We're going to stop here for this video.
I hope you found these examples helpful.