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Master Angles in a Circle: Formulas, Rules, and Examples

Derive proofs involving properties of angles and triangles

Classifying triangles

Isosceles and equilateral triangles

Pairs of lines and angles

Parallel lines and transversals

Parallel line proofs

Perpendicular line proofs

Solve problems involving properties of angles and triangles

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Unit circle

Angles in a circle

Determining non-permissible values for trig expressions

Solve problems involving cosine law and sine law, including ambiguous case

Law of sines

Ambiguous case: SSA triangles

Law of cosines

Applications of the sine law and cosine law

Analyze and prove conjectures using inductive and deductive reasoning to solve problems

Null hypothesis and alternative hypothesis

Proving claims

Confidence levels, significance levels and critical values

Test statistics

Analyze puzzles and games involving spatial reasoning using problem-solving strategies

Line symmetry

Rotational symmetry and transformations

Solve problems involving application of rates

Direct variation

Identifying direct variations

Rate of change

Applications of linear relations

Solve problems involving scale diagrams using proportional reasoning

Enlargements and reductions with scale factors

Scale diagrams

Similar triangles

Similar polygons

Understand relationships among scale factors, areas, surface areas, and volumes of similar 2-D shapes and 3-D objects

Surface area and volume of prisms

Surface area and volume of pyramids

Surface area and volume of cylinders

Surface area and volume of cones

Surface area and volume of spheres

Model and solve problems involving systems of linear inequalities in two variables

Graphing linear inequalities in two variables

Graphing quadratic inequalities in two variables

Graphing systems of quadratic inequalities

Inequalities of combined functions

Inequalities of Combined Functions

Graphing systems of linear inequalities

Applications of inequalities

Demonstrate understanding of characteristics of quadratic functions

Characteristics of quadratic functions

Transformations of quadratic functions

Quadratic function in general form: y = ax^2 + bx + c

Converting from general to vertex form by completing the square

Shortcut: Vertex formula

Converting general form into vertex form by applying the vertex formula

Vertex Formula Application

Graphing quadratic functions: General form VS. Vertex form

Finding the quadratic functions for given parabolas

Applications of quadratic functions

Adding functions

Subtracting functions

Multiplying functions

Dividing functions

Operations of Functions – In a Nutshell

Inverse functions

Quadratic function in vertex form: y = a(x-p)^2 + q

Demonstrate understanding of normal distribution, including standard deviation and z-scores

Introduction to normal distribution

The Normal Distribution and the 68-95-99.7 Rule

Using "invNorm" to Solve Normal Distribution Problems

Normal distribution and continuous random variable

Z-scores and random continuous variables

Interpret statistical data using confidence intervals, confidence levels, and margin of error

Point estimates

Confidence levels and critical values

Margin of error

Margin of Error

Making a confidence interval

Confidence intervals to estimate population mean

Critically analyze society's use of inferential statistics

Traditional hypothesis testing

Traditional Mean Hypothesis Testing

P-value hypothesis testing

Proportion Hypothesis Testing

Mean hypothesis testing with t-distribution

Hypothesis Testing Mean Claims without Knowing <katex> \sigma </katex>

Mathematics 11

Logical Reasoning

Measurement

Relations and Functions

math

Intro

- Hi, welcome to this question right here. So, in the following diagram which is here, the radius of the triangle is 24 centimeters, right? And we know the angle B, D, and C. That's this angle right here, is 75 degrees. So, we're trying to figure out the angle BAC, which is right over here. So, that's the missing angle that you're trying to figure out, okay? 
Now, there's a relationship between this angle and this angle. Now, the angle 75 is called the central angle. The central angle is called the C, for example, okay? 
And the inscribed angle of that central angle is only just half of that, okay? So, it's equal to the inscribed angle but you need to put a two times right there, okay? 
So, basically the central angle is double, double of the inscribed angle. Now, how do you know that one is really an inscribed while the other is the center? Well, the central angle is always from a point and then the other point, and they form with the center line, okay? The center point, okay? So, this is your central angle right here. Now, if you start and finish with the same points, which is B and C, and you form any triangle around your circle right here, those are called your inscribed angles, okay? Or they have the same arc lengths. You can actually look at the arc lengths right here. This is the arc length of the triangles, right? 
So, as long as they share the same arc length. Doesn't have to start from here to here. It could be somewhere else. As long as the arc length, the A-R-C, is the same. Therefore, this equation right here will exist, okay? The central angle is the double of the inscribed angle. So, if that's 75 over here, then, basically it's just two times the missing angle. Which is, let me put down. . . 
75 degrees and there's your two theta that's missing, okay? So, just grab your calculator and take the 75 divided by two, you get yourself the angle, which is this one right here, the BAC will just be 37.5 degrees. And that's your final answer for this question. Thanks for watching.

In a circle, chords, angles, inscribed angles and arc length all have special relationships with each other. This lesson focuses on exploring the relationships among inscribed angles in a circle as well as those of inscribed angle and central angle with the same arc. We will make use of the relationships to solve related questions in this lesson.  

Using formulas to find angles in a circle

Central angles and proofs

Inscribed angles and proofs