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Proving Triangles Congruent 2 994 Views


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Description:

As you'll see, we use cats' ears to explore the congruency of various triangles. Oh, calm down and stop dialing the the Humane Society. They're still attached, silly.

Language:
English Language

Transcript

00:04

Proving triangles congruent, a la Shmoop.

00:06

<<cheesy TV announcer, canned applause>>

00:08

And now it’s time for Are Your Cat’s Ears Symmetrical?...

00:11

Today we’ve got some very cooperative cats…

00:11

…who I’m sure aren’t at all resentful that we’ve put them on TV for the sole purpose

00:12

of examining their ears. First up we have this calico that was sent

00:15

to us from a viewer in Boston.

00:18

ARE…THESE…EARS…SYMMETRICAL?

00:19

To tell if a cat’s ears are symmetrical, we must first find out if they are congruent

00:26

in shape.

00:27

This cat’s ears are in the shape of isosceles triangles; two sides of each are equal.

00:33

Which means that the angles opposite the equal sides are also equal…

00:37

…this assertion is known as the Isosceles Triangle Theorem.

00:44

We can conclude that if two triangles are known to be isosceles…

00:48

…they are congruent if a corresponding side and a corresponding angle are known to be

00:53

congruent…an extension of Angle-Angle-Side.

00:57

And… YES! We have confirmed that this cat’s ears are indeed symmetrical!

01:01

Next we have this Persian specifically bred to be perfectly symmetrical.

01:05

Now, this cat’s ears are equilateral. But ARE…THEY … SYMMETRICAL?

01:11

To check if this cat’s ears are really equilateral…

01:13

…we’d have to see if they fulfill the theorem that every equilateral triangle is

01:18

also equiangular…

01:20

…each angle measuring 180 degrees divided by 3, equaling 60 degrees.

01:26

Seems in order. The only other thing we really need to know is if this cat’s ears are both

01:30

the same size.

01:32

It seems the left ear has sides that measure 5 centimeters, and the right has sides that

01:36

measure 5 and a half centimeters.

01:38

Awww, what a shame. Oh, well, next cat. Ah, will you look at this gorgeous Abyssinian

01:45

we have … what’s that?

01:47

Well, folks, it appears that we have a RIGHT ANGLE ALERT!

01:51

This cat’s ears form perfect right angles, a very rare occurrence on this show, indeed.

01:53

To check for congruence, there are several theorems we can use: the Leg-Acute Angle Theorem,

01:58

based on Angle-Side-Angle…

02:00

…the Hypotenuse-Acute Angle Theorem, based on Angle-Angle-Side…

02:04

…and the Hypotenuse-Leg Theorem.

02:07

We’ve got congruent hypotenuses … and we’ve got congruent corresponding legs!

02:12

The ears are officially congruent! And now, our last cat before we have to wrap

02:16

up this segment; a very special pair of Siamese twin… Siamese cats whose ears overlap!

02:22

No really… that’s literally the only thing between them that’s connected.

02:22

That doesn’t mean they can’t STILL BE SYM… okay, okay, we’re running out of

02:22

time.

02:23

This is the general shape of the cats’ ears as they overlap.

02:27

Their owner apparently wants to know if they’re congruent to each other even though they’re

02:31

morphed together.

02:32

She tells us that sides AB and AC as labeled in this carefully drawn, representative diagram

02:38

are congruent, as are AD and AE.

02:42

The triangles share a base, and by the Isosceles Triangle Theorem, the angles opposite those

02:47

congruent sides are also congruent.

02:50

So by Side-Angle-Side, these ears are congruent, making these cats Siamese IDENTICAL twins!

02:56

Well, that’s all the time we have this week.

03:00

Thanks for watching Are Your Cat’s Ears Symmetrical? and remember…

03:04

…if they aren’t symmetrical…

03:05

…your cat is defective.

03:06

Good night.

03:07

No animals were harmed in the making of this show.

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