Name: ____________________________
Bell Ringer:
A.
Lesson 74
NYS Geometry
In circle
O , inscribed angle ABC has a measure of 54 . What is the
measure of AC ?
______________
B. In circle
O , mPQ  72 . What is the measure of inscribed angle
PRQ ?
______________
Circle Proofs I
Some tools we’ll need:
A ___________________ can be drawn from the ____________________ to any ______________
on the ________________________ .
Any 2 ______________________ of the same ____________________ are ____________________
Recall that the definition of _________________________ is that two figures have the same
____________________ and ____________________ . It allows us to go from two figures
having the same ____________________________ to the same figures being
_____________________________ or visa versa.
Circle proofs can be more complex than triangle and quadrilateral proofs so, for now,
you will be given a “bank” of statements and reasons that could be used to produce a
valid proof. It is advised that you challenge yourself to cover these in your homework
problems and try the proof without the help first. If you come up with a proof that does
not use the same statements and reasons, but is still a valid proof, don’t feel that you
need to change yours.
Geometry
Lesson 74
1
Proof 1:
Congruent chords intersect congruent arcs.
AB  CD in circle O
Prove: AB  CD
Given:
Definition of congruent
mBA  mCD
Given
Substitution
AB  CD
A radius can be drawn from the
center to any point on the circle.
Any two radii in the same circle are
.
OAB  ODC
BOA  COD
Definition of congruent.
mBA  mBOA; mCD  mCOD
Draw radii OA, OB, OC , OD
The measure of a central angle and
the arc it intercepts are equal.
OA  OD, OB  OC
BA  CD
CPCTC
Statements
Geometry
Lesson 74
SSS
mBOA  mCOD
Reasons
2
Proof 2:
Parallel chords intercept congruent arcs.
Given: AB CD in circle
Prove:
O
AC  BD
Definition of congruent.
mAC  mBD
When parallel lines are cut by a
transversal, alternate interior
angles are congruent.
AC  BD
AB CD
Between any two points a line can
be drawn.
Given
Division
Statements
Geometry
Lesson 74
The measure of an inscribed angle
is half the measure of the
intercepted arc.
Substitution
B  C
Draw BD .
Definition of congruent.
1
1
m AC  mBD
2
2
mB  mC
1
1
mB  mAC ; mC  mBD
2
2
Reasons
3
Geometry
Lesson 74
4
Name: _____________________________
Lesson 74
NYS Geometry
On Your Own:
1. *Prove that a trapezoid inscribed in a circle is isosceles.
Given:
ABCD is a trapezoid with bases AB and CD
ABCD is inscribed in circle O
Prove: ABCD is isosceles.
chords intercept congruent arcs
AC  BD
Definition of trapezoid.
Given
Given
ABCD is a trapezoid with bases AB
and CD
ABCD is isosceles
Congruent arcs intercept congruent
chords.
ABCD is inscribed in circle O .
AC  BD
AB CD
Definition of isosceles.
Statements
Geometry
Lesson 74
Reasons
5
2. * PQRS is a trapezoid inscribed in circle
O . If leg PQ has a measure of 15,
what is the measure of leg QR ?
3. In circle
of
O , inscribed angle ABC
______________
has a measure of 102 . What is the measure
AC ?
______________
4. In circle
O , mXZ  56 . If XYZ is inscribed in circle O , what is its measure?
______________
5. Construct an equilateral triangle with sides congruent to AB .
A
B
6. Construct the angle bisector of
ABC .
A
B
C
Geometry
Lesson 74
6
7. Find the equation of a circle with a diameter that has endpoints at
 3, 5
and
 1, 7  .
_____________________
8. Graph the equation found in #8.
9. If altitude AD is drawn in
nearest hundredth.
ABC , BD  10 and CD  26 , find AD to the
______________
10. If sin A 
2
, what is the degree measure of  A ?
2
______________
Geometry
Lesson 74
7
11. Triangle
ABC is circumscribed about a circle. If AB is tangent to the circle at
R , AC is tangent to the circle at S , BR  3 , and CS  8 , find BC .
______________
12. If the length of a diagonal of a square is 12, the length of a side of the square is
(1)
6
(2)
(3)
6 2
(4)
3 2
12 2
______________
13. Which point lies on the locus of points equidistant from
(1)
 0, 4 
(2)
 2,5
(3)
A  2,0 and B  6,0  ?
 5, 4 
(4)
 4,5
______________
Math Musings:
14. What property of circle geometry was most helpful in proving the hypotheses in this
lesson? Explain.
_______________________________________________________________________________
_______________________________________________________________________________
_______________________________________________________________________________
SAT Prep Question:
In the xy -plane, line m is the reflection of line
is 
l about the x -axis. If the slope of line m
4
, what is the slope of line l ?
5
(A)
5
4
(B)
4
5
(C)
1
5
(D) 
4
5
(E) 
5
4
______________
Geometry
Lesson 74
8