Congruent Triangles and SSS

Geometry

Congruence postulates

Congruent Triangles and SSS
SSS to Show a Radius is Perpendicular to a Chord that it Bisects
Other Triangle Congruence Postulates
Two column proof showing segments are perpendicular
Finding Congruent Triangles
Congruency postulates
More on why SSA is not a postulate
Perpendicular Radius Bisects Chord
Congruent Triangle Proof Example
Congruent Triangle Example 2
Congruent triangles 1
Congruent triangles 2

Congruent Triangles and SSS What it means for triangles to be congruent

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Congruent Triangles and SSS

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So, this diagram over here, I have this big triangle
I have all these little triangles inside of this big triangle.
And what I wanna do is see if I could figure out the measure
of this angle, right here.
And we'll call that measure theta.
And they tell us a few other things,
you might have seen this symbol before
these means they are right angles
or that they have a measure of 90 degrees.
So, that's a 90 degree angle, that is a 90 degree angle
and that is a 90 degree angle over there.
And they also tell us that this angle over here is 32 degrees.
So, let's see what we can do,
and maybe we can solve this in multiple different ways.
That's what really fun about this,
there's multiple ways to solve these problems.
So, if this angle is theta, we have theta is adjacent to this angle,
it's adjacent to this green angle.
And if you add them together you're gonna get this right angle
you're gonna get this right angle.
So, this pink angle, theta, plus this green angle
must be equal to 90 degrees.
When you combine them, you get a right angle.
So, you could call this one,
it's measure is gonna be 90 minus theta.
And now we have, now we have 3 angles in the triangle
and we just have to solve for theta.
Because we know that this angle plus this angle plus this angle
are gonna be equal to 180 degrees.
So you have 90 minus theta plus 90 degrees, so plus 90 degrees
plus 90 degrees plus 32 degrees,
so we're gonna use a different color,
plus 32 degrees, plus 32 degrees is going to be equal to 180 degrees.
The sum of the of the measures of the angle
inside the triangle add up to 180 degrees.
That's all we're doing over here.
And so let's see if we can simplify this a little bit.
So if we subtract these 2 guys, 90 plus 90 is gonna be a 180
so you get 180 minus theta plus 32 is equal to 180 degrees.
And then we have, what else do we have?
We have a 180 on both sides,
we can subtract that from both sides.
So, that cancels out that goes to 0.
And you have negative theta plus 32 degree is equal to 0.
You could add theta to both sides
and you get 32 degrees is equal to theta
or theta is equal to 32 degrees.
So, it's gonna be actually the same measure as this angle over here.
That's one way to the do the problem.
There's other ways that we could have done the problem.
We could have said and actually there's a ton of ways
we could have done this.
We can look at this big triangle over here,
we can say look if this is 90 degrees over here
this is 32 degrees over here, this angle up here
is gonna be 180 minus 90 degrees minus 32 degrees.
Because they all have to add up to 180 degrees.
And I just kinda skipped a step over there,
obviously this, actually let me not skip a step.
Let me call this X, if we call that the measure of that angle X
we would have X plus 90,
I'm looking at the biggest triangle in this diagram right here.
X plus 90 plus 32 plus 32 is going to be equal 180 degrees.
And so, if you subtract 90 and 32 from both sides,
so if you subtract 90 from both sides
you get X plus 32 is equal to 90.
And if you subtract 32 from both sides you get X is equal to
What is this?
58 degrees, X is equal to 58 degrees.
X is equal to 58 degrees.
Fair enough.
What else could we figure out?
Well if this angle here, if this angle over here is a right angle.
And I'm just redoing the problem all over again,
just to show you there's multiple ways to get the answer.
We were given, if this is a right angle,
if that is 90 degrees, then this angle over here is supplementary to it
and it also has to be 90 degrees.
So we have this angle plus 90 degrees plus this angle,
plus this angle, plus this angle have to equal to 180,
maybe we can call that Y.
So Y plus 58 plus 90 is equal to 180.
You can subtract 90 from both sides, subtract 90 from both sides.
This will become 90, subtract 58 from both sides
you get Y is equal to 32 degrees.
Y is equal to 32 degrees.
Well if Y is equal to 32 degrees, it's is complimentary
it is complimentary to this angle right over here.
it is complimentary, I'll give it a new color.
Not supplementary, it's complimentary it adds up to 90 degrees.
It is complimentary to this angle over here, we can call it Z.
So, these two combined are gonna add up to 90 degrees
or Z is going to be equal to 58 degrees.
And now we're inside the triangle that we care about
to figure out theta, we already figured out earlier on this video.
Well this is 58 degrees, if this over here is 90
then this one over here is also going to be 90
because they are supplementary.
So you have 58 degrees, you have 58 degrees
I wanna do that orange color.
So, if you have 58 degrees plus this 90, plus 90 plus theta now
plus theta now is going to be equal to 180 degrees.
You can subtract 90 from both sides, that becomes 90
and you have 58 plus theta is equal to 90 subtract 58 from both sides
you get theta is equal to 32 degrees again.
And so we got the same answer.
I just wanna do that, to show you that there are multiple ways
to do this problem, and as long as you're doing things
that are logically consistent you're making assumptions
that you can make and then logically deducing step by step,
there's multiple ways to get that right answer.

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