Cogito
Integrated Math 2 · Chapter 6 · Lesson 2
Angles in Circles
Where an angle sits changes what it measures.
12 problems · about 21 minutes · G-C.A.2
What this lesson teaches
The student applies central angle, inscribed angle and tangent relationships.
- A central angle equals its arc; an inscribed angle is half of it.
- All inscribed angles on the same arc are equal.
- A tangent is perpendicular to the radius at the point of contact.
Warm Up
Straightforward practice. Get the method working first.
5 problemsAn arc of 120°. What is the inscribed angle on it, in degrees?
Answer 60
Why 60°.
What decides whether an angle equals its arc or halves it?
Answer Whether the vertex is at the centre or on the circle.
Why The position of the vertex.
An arc of 100°. Inscribed angle on it, in degrees?
Answer 50
Why Half.
An arc of 100°. Central angle on it, in degrees?
Answer 100
Why Equal to the arc.
An inscribed angle of 35°. What is its arc, in degrees?
Answer 70
Why Double it.
Build It Up
The same ideas with more to keep track of.
3 problemsThe angle between a tangent and the radius at the point of contact, in degrees?
Answer 90
Why Perpendicular.
One tangent from a point is 12 long. How long is the other?
Answer 12
Why They are equal.
How many points does a tangent share with the circle?
Answer 1
Why It only touches.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsMatch each angle type with its relationship to its arc.
Answer Central angle → Equal to the arc; Inscribed angle → Half the arc; Angle in a semicircle → Always 90 degrees
Why The vertex position decides everything.
An arc of 140°. Inscribed angle, in degrees?
Answer 70
Why Half.
The Inscribed Angle: An arc measures 80°. What is the inscribed angle on it, in degrees?
Answer 40 degrees
Why 40°.
The Semicircle: An angle inscribed in a semicircle. What is its measure, in degrees?
Answer 90 degrees
Why 90°.