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# angles in circles

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In the diagram below, triangle ABC is inscribed in the circle and AC = AB. The measure of angle BAC is 42 degrees and segment ED is tangent to the circle at point C. What is the measure of angle ACD? Dec 28, 2018

#1
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Isn't this an angle formed by a tangent and chord. If so, we have angle B and C as 180-42=138 divided by 2 equals 69. Since arc AC has the twice angle measure of angle B, we get 138. Learn this, pretty useful. Thus, angle ACD is 69 degrees.

Dec 28, 2018
edited by Guest  Dec 28, 2018
edited by Guest  Dec 28, 2018
#2
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We can start off by using the finding angle B. We have $$\frac{180-42}{2}=\frac{138}{2}=69^\circ$$ . Now, by the tangent formula, the arc AC is twice angle B, so $$69*2=138^\circ$$ . Now, finally the chord and tangent, we have $$\frac{1}{2}*138^\circ=\boxed{69^\circ}.$$

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Dec 28, 2018
#3
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Since AB = AC   the angles opposite these sides are equal

So m ABC =   [ 180 - m BAC] /2   =  [ 180 - 42 ] / 2  =  138/2 = 69°

And since ABC is an inscribed angle, then it intercepts an arc that is twice its measure

So  minor arc AC =  138°

And a chord meeting a tangent forms an angle =  1/2 the arc intercepted by the chord

So  chord AC intercepts minor arc AC....so....

m angle ACD  =  138 / 2  = 69°   Dec 28, 2018