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## High school geometry

### Course: High school geometry > Unit 8

Lesson 10: Properties of tangents- Proof: Radius is perpendicular to tangent line
- Determining tangent lines: angles
- Determining tangent lines: lengths
- Proof: Segments tangent to circle from outside point are congruent
- Tangents of circles problem (example 1)
- Tangents of circles problem (example 2)
- Tangents of circles problem (example 3)
- Tangents of circles problems
- Challenge problems: radius & tangent
- Challenge problems: circumscribing shapes

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# Tangents of circles problem (example 1)

CCSS.Math:

Sal finds missing angles using the property that tangents are perpendicular to the radius.

## Want to join the conversation?

- So the opposite angles of a quadrilateral are supplementary.

Would that be proof of the first answer? Or is the first answer proof of that fact? Because I'll be honest, I try not to just jump to the easy answer when I'm learning a new concept - but the answer to the first question was practically reflex.(3 votes)- The opposite angles of a quadrilateral are
*not*necessarily supplementary. The only condition on the angles is that all four of them must sum to 360˚.

If the opposite angles*do*happen to be supplementary, however, some interesting properties arise. One such property is that such a quadrilateral must be*cyclic*(that is, it must be inscribable in a circle).(4 votes)

- How did you get that 50 on that sec problem(2 votes)
- Because since the degree measure of a minor arc is the same as its central angle, and Sal's central angle was 100, Arc BC was = 100. And since an inscribed angle's degree measure is 1/2 of the minor arc it intercepts, and D intercepts Arc BC, D = 50.(5 votes)

- so at4:25what's basically happening is that an angle at the center of the circle is double the angle made anywhere else on the circle, and conversely the angle anywhere else on the circle would be double the angle at the angle at the center.(2 votes)
- Yes you are correct. It is a theorem, the inscribed angle is half of the central angle if and only if they are on the same arc.(3 votes)

- 3:00instead of making it long a little long can I use the statement : "the opposite angles in a quadrilateral are equal to 180" so I can directly subtract 80 from 180 which will give me 100, which is angle COB.(2 votes)
- not all quadrilaterals will have opposite angles equal to 180. but you can use the angle sum property of quadrilaterals is 360.

here, the radii make 90 degree angles with the tangents thus, the opposite angles make 180.(2 votes)

- What does it mean when a line is "tangent to a circle?"(2 votes)
- That it touches the circle at one and only one point. With this, it can be shown that a radius drawn to the point of tangency will be perpendicular to the line of tangency.(1 vote)

- Is this stuff on the S.A.T.?

Cuz like I'm taking it before 7th grade and I have no clue whether to study this stuff cuz i never took high school :)(1 vote)- There is a bit geometry on the SAT, yeah. Most of it is algebra and "data analysis"/charts, but a few geometry questions will be there.(2 votes)

- How do you prove that a line is tangent to a circle(1 vote)
- You can prove that the line is a tangent to a circle by just seeing if it intersects the circle 1 time. If the line intersects the circle more than once then it would be a chord.(2 votes)

- what does tangent mean?(1 vote)
- For a circle or curve, the tangent is a line or line segment that touches the circle at one point only. For a circle, the tangent will be perpendicular to a radius drawn to the tangent point.(1 vote)

- At4:05, can you just skip the step in which Sal finds the measure of the arc and go straight to finding the solution?(1 vote)
- Yes, you can. The important fact in the problem is to know that for any arc the inscribed angle is half the central angle.(1 vote)

- What does circumscribed mean(1 vote)

## Video transcript

- [Voiceover] So we're
told that angle A is circumscribed about circle O. So this is angle A right over here, we're talking about this
angle right over there. And when they say it's
circumscribed about circle O that means that the two sides of the angle they're segments that would
be part of tangent lines, so if we were to continue, so for example that right over there, that
line is tangent to the circle and (mumbles) and this line is also
tangent to the circle. So you see that the sides of angle A are parts of those tangents,
and points B and point C are where those tangents
actually sit on the circle. So given all of that they're asking us what is the measure of angle A, so we're trying to figure
this out right here and I encourage you to pause the video and figure it out on your own. So the key insight here
and there's multiple ways that you could approach
this is to realize, that a radius is going to be
perpendicular to a tangent or a radius that intersects a tangent is going to be perpendicular to it. So let me label this, so this is going to be a right angle and this is going to be a right angle. In any quadrilateral,
the sum of the angles are going to add up to 360 degrees, and if you wonder where that comes from well you can divide a
quadrilateral into two triangles, where the sum of all the interior angles of a triangle are 180 and
since you have two of them, it's 360 degrees. So this is 92 + 90 + 90 + question mark is going to be equal to 360 degrees. So let me write that down. 92 plus 90 plus 90 so plus 180. Plus, let's just call this X. X degrees. Plus X, they all have to
add up to be 360 degrees so let's see, we could
subtract 180 from both sides and so, if we do that we would have 92 plus X is equals to 180 and if we subtract 92 from both sides we get X is equals to let's
see 180 minus 90 would be 90 and then we're just going
to subtract two more so it's going to be X is equals to 88. So the measure of angle A is 88 degrees. Now let's do one more of these. These are surprisingly fun. All right. So it says, angle A is
circumscribed about circle O, we have seen that before
in the last question and they said what is
the measure of angle D. So we want to find, let me make
sure I'm on the right layer. We want to find the measure of that angle and let's call that,
let's call that x again. So what can we figure out? Well just like in the last question we have a quadrilateral here,
quadrilateral A, B, O, C and we know two of the angles, we know this is going to be a right angle we have a radius
intersecting with a tangent, or part of a tangent
I guess you could say. And then this would be a right angle and so by the same logic as
we saw in the last question, this angle, plus this
angle, plus this angle, plus the central angle are
going to add up to 360 degrees. So let's call the measure
of the central angle let's call that Y over there. So we have Y plus 80 degrees or we'll just assume
everything is in degrees, so Y plus 80 plus 90
plus 90, so I could say plus another 180 is going
to be equal to 360 degrees sum of the interior
angles of a quadrilateral. And so let's see you have, we could have Y plus 80 is equal to 180, if I just subtract 180 from both sides if I subtract 80 from both
sides, we get Y is equal to 100 or the measure of this is 100, the measure of this interior angle right over here is 100 degrees, which also tells us that
the measure of this arc cause that interior angle intercepts... intercepts arc C, B, right over here, that tells us that the measure of arc C, B, is also 100 degrees. And so if we're trying to find this angle, the measure of angle D,
that's an inscribed angle that intercepts the same arc. And we've seen in previous
videos that an inscribed angle that intercepts that arc is going to have half the arc's measure. So if this is a 100 degree measured arc, then the measure of this angle right over here is going to be 50 degrees. So the measure of angle D is 50 degrees. And we are done.