### Video Transcript

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.

Now in these questions, the first
thing I always say is to mark on diagram what you know. Well, first of all, Iโm gonna mark
on 29 degrees for angle ๐ด๐ต๐. 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. And I could do this because the
angles on a straight line equal 180 degrees.

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. This kind of questions that come at
the exams always expect you to give reasons for your answers. So hereโs my reasoning is that the
angles on a straight line equal 180 degrees.

Okay, with this in mind, weโll move
on and Iโll actually add on what else we know from our diagram. And next, we can mark on that angle
๐๐ด๐ต is gonna be equal to 90 degrees. And thatโs because itโs a right
angle. And thatโs because thereโs always a
right angle between the radius of a circle and a tangent to a circle at point. To actually help you understand
where this has come from, I actually gonna prove it using contradiction.

So Iโve drawn a little sketch
here. So letโs start by actually
thinking- okay, ๐๐ถ, letโs suppose this isnโt perpendicular to ๐ท๐ธ, which is our
tangent. And from that, we then say โokay,
therefore ๐๐ต is perpendicular.โ 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.

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. But this doesnโt sound right
because we know that ๐๐ถ is equal to ๐๐ด because theyโre both radius of a
circle. So therefore, this would say that
๐๐ต being greater than ๐๐ถ is actually impossible. And therefore, this cannot be the
case and therefore ๐๐ต cannot be perpendicular.

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. Okay, so now weโve done that. Letโs get back on and solve the
problem. So therefore, we can actually say
that angle ๐ด๐๐ต is gonna be equal to 180 degrees minus 90 degrees minus 29
degrees. And therefore, this is gonna be
equal to 61 degrees.

But not forgetting, we need to give
our reasoning. And our reasoning for this
calculation is that the angles in a triangle are equal to 180 degrees. So therefore, weโve solved the
problem and we found the measure of angle ๐ด๐๐ต and it is 61 degrees.