Welcome to a lesson on similar triangles
by angle angle
also known as AA.
The goals of the video are to determine whether two triangles are similar
by using angle angle similarity.
But let's first talk about the third angle theorem.
This theorem tells that if two angles in a triangle are congruent to two angles in another triangle
than the third angles
must be congruent also.
So looking at this diagram here.
This theorem tells us that if this angle congruent
to this angle.
And this angle here is congruent to this angle here.
Because their equal in measure
then these third angles
must also
be congruent.
And that should make sense because we know that the sum of the interior angles of any triangle
is 180 degrees.
To determine the measure of this third angle
in both triangles.
We would take 180 degrees.
And subtract.
The sum of the two given angles.
79 degrees
plus
47 degrees.
Well
79 degrees plus 47 degrees would be
one hundred
twenty-six degrees.
Leaving the third angle
to measure 54 degrees.
And this would be true for both triangles.
And this is really the basis for determining whether two triangles are similar
using angle angle. Let's take a look at that postulate.
The angle angle similarity postulate tells us that if two angles in one triangle
are congruent to two angles in another triangle
then the triangles are similar.
So really from the previous theorem.
We know the third angle would also be congruent.
But this postulate saves us from having to determine the measure of that third angle.
And remember if two triangles are similar.
Corresponding angles are congruent
and corresponding size are proportional.
So just a review.
The ratio of a to d.
Would have to equal the ratio of b to e.
Which would have to equal the ratio of c to f.
Let's take a look at some examples to see if we have similar triangles.
Here were ask to determine if the triangles are similar.
Just because at first glance we have two angles in two different triangles.
And the angles are not congruent.
Doesn't mean these triangles are not similar.
We need to go ahead and find the measure of this third angle here in this triangle.
And if it does measure 34 degrees
than these two triangles would be similar
by angle angle.
So looking at this large triangle.
We'll take a 180 degrees.
And we'll subtract
the sum of
132 degrees
and 15 degrees.
This is 147 degrees.
Leaving the third angle to measure
33 degrees.
So now we can see that this angle is congruent to this angle.
But notice how neither of the two remaining angles
would be congruent to
this angle here in the small triangle
therefore these two triangles are not similar.
Let's take a look at another example.
Again upon first glance.
We may think that these two triangles are not similar.
But that may not be true until we determine the third angle of one of these triangles.
Let's go ahead and find this third angle of this small triangle this time.
Again because the sum of the interior angles would be 180 degrees.
We'll take 180 degrees and we'll subtract
the sum of 77 degrees and 23 degrees.
Well 77 plus 23 is 100.
Or 100 degrees.
So the third angle
is equal to 80 degrees.
So now if we compare the two triangles.
This angle is congruent to this angle.
And this given angle here is congruent to this given angle here.
So these two triangles are similar
by angle angle similarity.
Let's take a moment and label these vertices. Let's go ahead and call this
ABC.
And let's go ahead and call this DEF.
Members it's important that we list
the vertices of similar triangles
in corresponding order.
So in this case we would say that triangle ABC.
Is similar to triangle.
Well angle A would correspond to
angle F this would be 77 degrees.
So we must label this as triangle F.
And then
angle B corresponds to angle E.
And then angle C
corresponds to angle D.
So the proper way to identify these similar triangle would be
triangle ABC is similar to triangle FED.
Let's take a look at one more example.
Again we want to determine if these two triangles are similar.
Given that segment AB is parallel to segment ED.
We know from our study of parallel lines.
If we have parallel lines cut by a transversal.
Then alternate interior angles would be congruent.
So angle A is congruent to angle D.
And for the same reason angle B is congruent to angle E.
So these two triangles are similar
by angle angle.
We would say that triangle ABC.
Is similar to
angle A correspond to angle D and so we have triangle
D.
Angle B corresponds to angle E so we'll have DE.
And then of course C corresponds to itself.
And we'll go ahead and stop here.
I hope you found this helpful.