Welcome to a video on congruent and similar triangles
the goals in this video were to determine congruent triangles,
determine similar triangles
and also to use Similar triangles to solve problems.
Congruent triangles have the same size and the same shape.
Therefore, the corresponding sides had the same length
and the corresponding angles are congruent.
So, the side is the same length as the side
this side is same length as a side and side is the same length as this side
and the corresponding angles are also congruent.
Which means this angle equal to this angle
this angle equals to this angle
and this angle is equal in measure to this angle.
Similar triangles have the exact same shape,
but not necessarily the same size.
So, the conditions for similarity are number one
corresponding angles must have the same measure
so again this angles is Congruent to this angle
those are congruent
and those are congruent.
But, corresponding sides are proportional.
Which means, the ratios at the corresponding sides are
equal.
So, what that's telling us is, let's say this is eight inches long
and this is four inches long
the ratio between these two would be to two to one
Therefore, if this was ten inches long
this would have to be five inches and this was twelve inches long
this would have to be six inches
the ratio stays consistent with corresponding sides.
Here's a nice visual three Similar triangles
as I animate one vertex the large triangle
the other two triangles remain similar. Meaning, they are the same shape but not necessarily
the same size.
again these three triangles
are Similar triangles
So hopefully that gives you feel of what similarity means
Throughout various other conditions that can be used to determine if
triangles are similar
Number one, two triangles are similar
if they have three pairs of congruent
angles
sometimes called Angle-Angle-Angle or AAA
but actually Angle-Angle is enough.
So, for example here we have two triangles. Where,
two pairs of corresponding angles are congruent
However, since we know that the sum of these angles
must equal 180 degrees
we could also just conclude that these
two angles with equal seventy degrees
and if we have three pairs of congruent angles
this guarantees that these two triangles are Similar.
Number two,
Two triangles are Similar
if all three pairs of corresponding sides are proportional.
So, what that means is if we take one pair
corresponding sides
For example, this side nine corresponds to this side six
nine to six simplifies to the ratio of three to two
and if all the other corresponding sides
maintain this ratio they would also be proportional.
and therefore, we would have two Similar triangles
we can quickly see that fifteen to ten simplifies to three to two
and eighteen to twelve
simplifies to three to two
so we have three pairs of corresponding sides are proportional.
Therefore, these two triangles are Similar
and lastly,
two triangles are Similar,
If two pare corresponding sides are proportional
and included angles are congruent.
So, what that means is, if we know this side was fifteen, and this side was ten
this side was eighteen, this side was twelve.
We know, these corresponding sides are proportional
and because the angle formed by these two sides are Congruent
this guarantees similarity. And, this is called Side-Angle-Side or SAS.
So, take a look at couple of problems that demonstrate those ideas
The two triangles are similar. Determine the measure of angle A and angle B
what's first? Write the given information. We know, angle D is thirty-five.
degrees
and we know angle F is forty-three degrees
and we know since they are Similar,
the corresponding angles are congruent so
angle C is also forty-three degrees
an angle A is thirty-five degrees
and the question they want us to determine is angle A
which we just found
but they also want angle B the angle here
we know the sum of these angle
so we can do is take the total sum 180
subtract thirty-five degrees
give us 145 degrees
and then subtract forty-three degrees
we have 102 degrees
so the measure of angle B
equals 102 degree
okay. On this problem we have similar triangles again, we want to determine the length of
sides DE
We will call this X
and EF
which, we'll call Y.
and now lets go ahead and mark the given information
CB equals fifteen
AC equals nine
AB equals ten
and DF equals six.
Okay, since we know, these triangles are Similar,
these sides are proportional
so we can setup proportions to solve for the unknowns
we know the ratio
of these two sides would be nine to six
in that must be the same as a ratio of
ten to x
and the same thing for side Y.
we can setup the proportion that would say nine to six
must be equal to fifteen to y
and now I can solve this apportioned
to find
the length of the missing sides.
So, performing cross products here we would have nine x
equals sixty.
dividing both sides by nine
X equals twenty-third
which is equal to six and two third.
To solve for y, we would have nine y
equal six times fifty which equals ninety
dividing by nine on both sides we have Y equals
ten
so we were able to solve for the missing sides because we knew the
triangles were Similar.
Which means, their corresponding sides were proportional.
Now lets take a look at another type of application
so for example here we can see that
this building is casting a shadow
onto the grass
and so is this person standing over here
but the idea here is that the shadow
forms a triangle and these two triangles are Similar
even as time passes the length of that shadow is going to increase
but the result is to similar triangles
so what this means is that the sides are these two triangle will be proportional.
So, we could have proportioned that will compare
corresponding sides have these two similar triangles
let's take a look at the problem like this
We have a tree that casts a shadow forty-five feet long,
and at the same time the shadow cast by a vertical three-foot stick
is five feet long.
so here we have a three foot stick
which cast a shadow five feet long
and again the shadows forty-five feet
so we have our too Similar triangles
that will allow us to find
the height of this tree
without ever having to climb it
because we know these size of proportion
so X is to three
as
forty-five is to five
so now I can solve for the unknown by performing cross-product
five x
must equal
135
dividing by five
X equals
twenty-seven feet
so without ever having to climb this tree we were able to use Similar triangles
to determine
that the high of this tree is twenty-seven feet
I hope you found this video helpful.
Thank you for watching