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How to Use CASTC after Proving Triangles Similar

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2016-03-26 20:31:59
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Geometry Essentials For Dummies
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CASTC is simply an acronym that stands for ‘Corresponding angles of similar triangles are congruent.’ You often use CASTC in a proof immediately after proving triangles similar (in precisely the same way that you use CPCTC after proving triangles congruent).

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The following proof shows you how CASTC works:

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Here’s how your game plan might go: When you see the two triangles in this proof diagram and you’re asked to prove that the lines are parallel, you should be thinking about proving the triangles similar. Then, using CASTC, you’ve got congruent angles that you can use with the parallel-line theorems to finish.

So here’s the solution. You’ve got the pair of congruent vertical angles, angle 3 and angle 4, so if you could show that the sides that make up those angles are proportional, the triangles would be similar by SAS~ (Side-Angle-Side). So check that the sides are proportional:

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(Note that the similarity is written so that corresponding vertices pair up.)

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Mark Ryan has more than three decades’ experience as a calculus teacher and tutor. He has a gift for mathematics and a gift for explaining it in plain English. He tutors students in all junior high and high school math courses as well as math test prep, and he’s the founder of The Math Center on Chicago’s North Shore. Ryan is the author of Calculus For Dummies, Calculus Essentials For Dummies, Geometry For Dummies, and several other math books.