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Given that ๐ด๐ต is a tangent to the circle ๐ at ๐ด and the measure of angle ๐๐ต๐ธ is 151 degrees, find the measure of angle ๐ด๐๐ต.
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Now in these questions, the first thing I always say is to mark on diagram what you know.
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Well, first of all, Iโm gonna mark on 29 degrees for angle ๐ด๐ต๐.
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And I found this because what I did that I took 151 degrees away from 180 degrees which gave me an answer of 29 degrees.
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And I could do this because the angles on a straight line equal 180 degrees.
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So itโs now that Iโm actually gonna point out something thatโs really key and important: whenever youโre doing a question like this, make sure you give reasoning.
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This kind of questions that come at the exams always expect you to give reasons for your answers.
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So hereโs my reasoning is that the angles on a straight line equal 180 degrees.
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Okay, with this in mind, weโll move on and Iโll actually add on what else we know from our diagram.
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And next, we can mark on that angle ๐๐ด๐ต is gonna be equal to 90 degrees.
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And thatโs because itโs a right angle.
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And thatโs because thereโs always a right angle between the radius of a circle and a tangent to a circle at point.
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To actually help you understand where this has come from, I actually gonna prove it using contradiction.
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So Iโve drawn a little sketch here.
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So letโs start by actually thinking- okay, ๐๐ถ, letโs suppose this isnโt perpendicular to ๐ท๐ธ, which is our tangent.
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And from that, we then say โokay, therefore ๐๐ต is perpendicular.โ
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Well, therefore, we would say that angle ๐๐ต๐ถ would be equal to 90 degrees and thus angle ๐๐ถ๐ต must be acute because obviously we canโt have two angles greater than 90 degrees in a triangle.
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So because of this, we could say that as the greater angle is always opposite the longest side, then in this case, ๐๐ต must be greater than ๐๐ถ.
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But this doesnโt sound right because we know that ๐๐ถ is equal to ๐๐ด because theyโre both radius of a circle.
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So therefore, this would say that ๐๐ต being greater than ๐๐ถ is actually impossible.
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And therefore, this cannot be the case and therefore ๐๐ต cannot be perpendicular.
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So what weโve done here just gonna show you how the actual tangent to a circle against the radius is actually 90 degrees true contradiction.
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Okay, so now weโve done that.
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Letโs get back on and solve the problem.
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So therefore, we can actually say that angle ๐ด๐๐ต is gonna be equal to 180 degrees minus 90 degrees minus 29 degrees.
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And therefore, this is gonna be equal to 61 degrees.
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But not forgetting, we need to give our reasoning.
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And our reasoning for this calculation is that the angles in a triangle are equal to 180 degrees.
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So therefore, weโve solved the problem and we found the measure of angle ๐ด๐๐ต and it is 61 degrees.