Geometry Circles: Angles, Chords and Tangents Lesson (Free PDF)
This circle angles lesson for Geometry covers circles: angles, chords and tangents (Common Core G-C.A.1, G-C.A.2, G-C.A.3, G-C.A.4), with a free PDF to print.
A central angle equals its intercepted arc, and an inscribed angle is half of its intercepted arc. An angle inscribed in a semicircle is a right angle, and opposite angles of an inscribed quadrilateral add to 180 degrees. A tangent is perpendicular to the radius at the point of tangency, and two tangent segments from one point are equal.
Geometry · Common Core State Standards: G-C.A.1, G-C.A.2, G-C.A.3, G-C.A.4
Key facts in this circle angles lesson
- Central angle = intercepted arc; inscribed angle = half the arc.
- An angle inscribed in a semicircle measures 90 degrees.
- Opposite angles of an inscribed quadrilateral are supplementary.
- A tangent is perpendicular to the radius at the point of tangency.
- Tangent segments from one outside point are congruent.
What are central and inscribed angles?
A central angle has its vertex at the center of the circle. Its measure equals the measure of its intercepted arc. An inscribed angle has its vertex on the circle, and its measure is half of its intercepted arc.
- Inscribed angles that intercept the same arc are congruent.
- An angle inscribed in a semicircle is a right angle.
What is true about an inscribed quadrilateral?
When all four vertices of a quadrilateral lie on a circle, opposite angles are supplementary. Each angle is half of the arc opposite it, and the two opposite arcs together make 360°.
What are the properties of tangents?
A tangent touches the circle at exactly one point. It is perpendicular to the radius drawn to that point. Two tangent segments from the same outside point have equal length.
What about chords, circumscribed circles and similarity?
A radius perpendicular to a chord bisects the chord. The center of the circle that passes through all three vertices of a triangle (the circumcenter) is where the perpendicular bisectors meet. The center of the inscribed circle (the incenter) is where the angle bisectors meet. All circles are similar: a dilation maps any circle to any other, so the ratio of circumference to diameter is always \(\pi\).
How do you solve a circle angle problem?
- Mark central angles, inscribed angles and arcs on a sketch.
- Pick the relationship: central = arc, inscribed = half the arc.
- Write an equation and solve.
- Check that the arcs of a circle add to 360°.
Example: An inscribed angle is \((x + 20)^{\circ}\) and the central angle on the same arc is \((4x – 30)^{\circ}\). Then \(4x – 30 = 2(x + 20)\), so \(x = 35\) and the inscribed angle is 55°.
Circle angles lesson: questions and answers
What is the difference between a central angle and an inscribed angle?
A central angle has its vertex at the center and equals its arc. An inscribed angle has its vertex on the circle and is half of its intercepted arc.
Why is an angle in a semicircle a right angle?
The intercepted arc is a semicircle of 180 degrees, so the inscribed angle is half of that, 90 degrees.
Why are opposite angles of an inscribed quadrilateral supplementary?
Each angle is half of the arc opposite it, and the two opposite arcs add to the full circle, 360 degrees, so the angles add to 180 degrees.
How do you know a line is tangent to a circle?
A line is tangent if it is perpendicular to a radius at the point where it meets the circle. For example, a segment of 12 from a point 13 units from the center of a circle of radius 5 is tangent.
Why are all circles similar?
A translation moves one center onto the other, and a dilation with the ratio of the radii maps one circle onto the other.
How to use this circle angles lesson
Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.
Download the circle angles lesson as a PDF with the button below and keep it with your notes.
Download the circle angles lesson as a PDF with the button below and keep it with your notes.
All Geometry chapters: Geometry math.
More Geometry math
Lessons
See all
Worksheets
See all
Tests
See all
Translations, Reflections and Rotations Test with Point Values- Parallel Lines and Transversals Test with Point Values
- Symmetry of Polygons and Rigid Motions Test with Point Values
- Arc Length and Sector Area Test with Point Values
- Geometric Modeling Applications Test with Point Values
- Equations of Circles and Parabolas Test with Point Values
