List of Common Properties and Vocabulary Terms of Circles
Circles have diameters, radius, tangent, and chords.
Tangent is a line that touches a circle only once. It only touches the circle but
does not cut through the circle.
Chord is a line segment that cuts a circle into two uneven pieces. (Chords are not
diameters, because chords do not pass through the center of a circle.)
By definition of a circle:
All radii of a circle are congruent to each other.
All diameters of a circle are congruent to each other.
Euclid Book III, Proposition #16:
Tangents to a circle will always form 90o with a diameter of the circle at the point where
the tangent touches the circle.
Euclid Book VI, Proposition #33:
In a given circle, an arc length is always proportional to the central angle it subtends.
Example: arc BC = arc DC
mBAC
mDAC
Other interpretations / conclusions from this same theorem:
If the central angles subtending an arc are congruent, then the arcs themselves
will be congruent as well.
Each arc length subtends a unique central angle.
“Intermediate” Proof Practice
1. In this exercise, we are going to prove that the three angles of a triangle must sum
up to 180o. This is an incredibly important fact in Geometry and underlies most
of our analyses of diagrams, but can you prove that it is always true?
You start by drawing a scalene triangle ABC, and then adding a line segment
DE such that D, A, and E are collinear and DE || BC . Using this diagram and
basic line and angle relationships, prove that
mABC  mACB  mBAC  180o
Given Facts:
Goal:
Diagram:
Step-by-Step Reasoning or “Proof”:
2. In the diagram below, AB and CD are both diameters of the circle O. Write a formal
proof to show that the circular arc lengths AC and DB are congruent. To help you out,
some simple circular properties are listed at the start of this packet.
Given Facts:
Goal:
Diagram: (No need as it has been provided above.)
Step-by-Step Reasoning or “Proof”:
2. In the following diagram below, you are given that O is the center of the circle.
Prove that ∆ABO  ∆ACO.
Given Facts:
Goal:
Diagram: (No need as it has been provided above.)
Step-by-Step Reasoning or “Proof”:
4. In the following diagram below, you are given that CD is a diameter and AD is a
tangent of circle O. Prove that ∆DCA ~ ∆BCD. (Note: You may NOT assume any
triangle in this picture is isosceles!)
Given Facts:
Goal:
Diagram: (No need as it has been provided above.)
Step-by-Step Reasoning or “Proof”:
5. In the following diagram, angles ABC and AOC are subtended by the same circular
arc AC, except one of the angles starts in the middle of the circle and the other angle
starts on the far side of the circle. You are given that O is the center of the circle, and
you must prove that mAOC  2mABC .
Given Facts:
Goal:
Diagram: (No need as it has been provided above; you may modify the diagram as
you see fit.)
Step-by-Step Reasoning or “Proof”:
(Hint: My recommendation is to draw a new segment BO and to represent all
angles in the diagram using algebra, recycling variables whenever possible and
appropriate. In the end you should be able to draw the correct conclusion based on
your algebra. Remember that you cannot assume ABC to be bisected by BO !)
5. In this exercise, you are going to prove that two tangents from the same exterior
point to the same circle are always congruent.
Given Facts:
Goal:
Diagram:
Step-by-Step Reasoning or “Proof”:
(Hint: Starting by finding two triangles in your diagram and proving them
congruent to each other.)