Name___________________________________Date________________
1080 Stomps and Angles in Circles
Watch the video about snowboarding:
https://www.youtube.com/watch?v=gfTXUWlJCxk
 Make a list of all of the angle measurements that you hear.
 How do these angle measurements relate to circle measurements?
 Sketch a semicircle. How many degrees are in this shape?
A central angle of a circle is an angle whose vertex is the center of
the circle.
major arc
minor arc
An arc is a connected section of the circumference of a circle. Arcs
are measured in two ways: using the measure of the central
angle or as a length.
central angle
A minor arc is part of a circle that measures _______ ______
180º. Thus, a minor arc is less than half of a circle.
A major arc is part of a circle that measures _______ _______ 180º. Thus, a major arc is more
than half of a circle. A major arc must be named with three letters.
A semicircle is part of a circle that measures _______. Thus, a semicircle is half of a circle. A
semicircle must be named with three letters.
Arc Addition Postulate:
The measures of Adjacent arcs can be added together, just like the lengths of
adjacent line segments can. If you know the measures of arc RS and arc ST,
you can add them together to find the measure of arc RT. This works if you
are measuring in degrees or in length.
Use the following circle for (ex 1) and (ex 2):
(ex 1) Name each of the following in the circle.
a) A minor arc
b) A major arc
c) A semicircle
(ex 2) Find each measure.
a)
mIK  ______________________
b)
mJIL  ______________________
(ex 3) Find each measure.
mQS
_____________
mRQT _____________
Congruent Arcs:
Arcs with the same central angle measure will only be congruent if they come from the same
circle or from congruent circles.
Do you see why this is true?
THEOREM: Two arcs (from the same circle or
congruent circles) are congruent if and only if their
central angle measures are the same.
State this theorem for this figure using proper notation.
THEOREM: In the same circle or in congruent circles, two minor arcs are congruent if and
only if their corresponding chords are congruent.
Let’s investigate why this is true:
Why would the major arcs have to be congruent as well?
Practice:
(ex 4)





UTV  XTW, find WX.
(ex 6) Find m MN
Pair-Share: With a partner, talk through and sketch out a proof to show that
when are given GD is perpendicular to FH, we know that FM  MH.
 If the statement above is true this means that GD is a ____________ __________!
 Do you think that a radius is perpendicular to a chord will always create this
relationship? Why or why not?
THEOREM: If a diameter (or a radius) of a circle is perpendicular to a chord, then the diameter
bisects the chord and its arc.
Short form: In a , diameter  chord → diameter bisects chord and corr arc
In the diagram to the right, if DG  FH , then DG bisects FH and FH
THEOREM: If one chord is a perpendicular bisector of another chord, then the first chord is a
diameter.
Short form: In a
, 1st chord is  bisector of 2nd chord → 1st chord is a diameter (or radius)
So, if DG is the  bisector of FH , then DG is a radius
(ex 6) Find ZY.
(ex 7) Find EG.
Pair-Share: Try the proof below!
THEOREM: If two chords are parallel, then their intercepted arcs are
congruent.
(ex 8) Find the measure of arc CA.
Activity
MQ and NR are diameters.
Find the indicated measures.
1.
2.
m MN
m NQ
6.
7
m MR
m QMR
3.
m QR
8.
m PQ
4.
m NQR
9.
m PRN
5.
m MRP
10. m MQN
11. Find m MN
12. Find m MN .
Find the length of each chord. (Hint: Use the Pythagorean Theorem to find half the
chord length, and then double that to get the answer.)
13 .
CE 
14.
LN  __________