Geometry
Name_____________________
Circles (1): Chords/Tangents & Angles/Arcs
Date: 1/3 or 1/4
Basic Theorems/Definitions/Postulates
Before embarking on our study of the circle, we define the following:
Circle: the set of all points in a plane equidistant from a common point (the center of the circle)
Radius: a line segment connecting the center point to a point on the circle
Chord: a line segment connecting two points on the circle (what is the longest chord of a circle called?)
We also present a postulate regarding radii of a circle:
Postulate: All radii of a circle are congruent.
Using these definitions and postulate, present a brief, but convincing argument to prove the following
theorem:
Theorem: If a radius is perpendicular to a chord of a circle, then it bisects
the chord.
The converse of this theorem is also true! Take a moment to think about how you would prove this.
Here is an additional definition regarding circles:
Tangent: a line that intersects a circle at only one point.
A postulate that follows from this definition:
Postulate: A tangent is perpendicular to a radius drawn to the point of tangency.
Using this postulate and the definition of a tangent, present a brief, but convincing
argument to prove the following theorem:
Theorem: If two tangent segments share a common endpoint outside the circle, then they are congruent.
A Central Angle & Its Intercepted Arc
We define the following:
Central Angle: an angle whose vertex is the center of a circle.
Using this definition, we can now work with the arc that a central angle
intercepts on a circle:
P
The measure of an arc ( m AB ) is equal in measure to that of the central
angle (units: same as the angle, in degrees).
The length of an arc (length of AB ) is simply a fraction of the circle’s
circumference (units: same as the radius, in cm/ft/inches, etc).

length of AB = 


 2 r

Exercises:
1. Find the arc length.
6
82º
82º
2. This circle has radius 10 and AB has length 15.
Find the m AB .
A
AB=15
B
A
r
x°
x°
B
An Inscribed Angle & Its Intercepted Arc
C
We define the following:
(2x)°
Inscribed Angle: an angle whose vertex is on the circle.
Using this definition, we can now present the following
theorem, which we will prove in our next lesson.
Theorem: If an angle is an incribed angle, then its measure is
one-half the measure of its intercepted arc.
D
x°
T
Exercise:
3. Find the arcs and angles labeled (hint: the center is
irrelevant in this exercise).
100º
w
58º
y
z
v
x
120º
Extensions to the Theorem (involving inscribed angles)
Below are some corollaries, which you should be able to justify. Take a
moment to reflect why each of these statements are true.
Corollary: If a triangle is inscribed with one side a diameter, then it is a right
triangle.
x°
Corollary: If a quad is inscribed in a circle, then its opposite angles are
supplementary.
(180-y)°
y°
Corollary: If two chords intersect and both sets of endpoints are connected
(creating a “butterfly” shape), then the created triangles are similar.
(180-x)°
Exercises:
4. Label all missing angles and arcs.
68º
65º
72º
5. Find the measures of angles X and Y.
Then find A.
80º
3
2
Y
5
A
36º
X
6. Find angle X (hint: draw in AB).
X
47º
A
149º
B