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# Geometry

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In the diagram, if  \(\triangle ABC\) and \(\triangle PQR\) are equilateral, then what is the measure of

\(\angle CXY\) in degrees?

https://latex.artofproblemsolving.com/c/5/a/c5a3159832bd8c1395bc179beaf94b1c58a2e9eb.png

Feb 20, 2018

#1
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See diagram below

Feb 20, 2018
#3
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Great diagram! Thanks so much, EP!

tertre  Feb 20, 2018
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Angle YBP  = 180  - 65 - 60  = 55°

Angle BYP  =  180  - 75 - 60 = 45°

So

Angle BYP  =  180  - 55 - 45  = 80°

And Angle CYX is a vertical  angle to BYP  ....so it = 80°

So.... CXY  =  180  - 60  - 80   =   40°

Feb 20, 2018
#4
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Amazing Explanation, CPhill! Great job!

tertre  Feb 20, 2018
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Thanks, tertre.....!!!

CPhill  Feb 20, 2018
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Glad it helped , Tertre......waiting to see what the answer is to your red/white ball question to see if I got it correct!

Feb 20, 2018
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Let's see if I can solve this:

Since \(\triangle ABC\) and \(\triangle PQR\) are equilateral, then \(\angle ABC=\angle ACB=\angle RPQ=60^\circ\).

Therefore, \(\angle YBP = 180^\circ-65^\circ-60^\circ=55^\circ\)  and \(\angle YPB = 180^\circ-75^\circ-60^\circ=45^\circ\) .

In \(\triangle BYP\), we have \(\angle BYP = 180^\circ - \angle YBP - \angle YPB = 180^\circ - 55^\circ-45^\circ=80^\circ\).

Since \(\angle XYC = \angle BYP\) , then \(\angle XYC=80^\circ\).

In \(\triangle CXY\), we have \(\angle CXY = 180^\circ - 60^\circ - 80^\circ = 40^\circ\).

So our final answer is \(\boxed{40}\) degrees.

Feb 20, 2018