ShmoopTube

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Proving triangles congruent, a la Shmoop.

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<<cheesy TV announcer, canned applause>>

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

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Today we’ve got some very cooperative cats…

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

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

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to us from a viewer in Boston.

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ARE…THESE…EARS…SYMMETRICAL?

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

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in shape.

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

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

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…this assertion is known as the Isosceles Triangle Theorem.

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

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

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congruent…an extension of Angle-Angle-Side.

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

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

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

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

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

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also equiangular…

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

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

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the same size.

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

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measure 5 and a half centimeters.

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

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we have … what’s that?

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

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

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

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based on Angle-Side-Angle…

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

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…and the Hypotenuse-Leg Theorem.

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

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

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

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

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

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time.

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

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

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morphed together.

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

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are congruent, as are AD and AE.

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

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congruent sides are also congruent.

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

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

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

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…if they aren’t symmetrical…

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…your cat is defective.

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Good night.

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No animals were harmed in the making of this show.

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