Download Feb 28 Day 6 Inscribed Angle Theorem

Geometry - Semester 2
Mrs. Day-Blattner
2/28/2017
Agenda 2/28/2017
1) Bulletin
2) Page 39 “Lesson 4” Experiments with Inscribed angles
(homework - share learning and corrections)
3) Lesson 5 - The Inscribed Angle Theorem
4) Homework
Thursday - CAASPP Interim assessment
Quiz - return - Construction second chance.
Our Lesson 6. Page 39
Lesson 4. Experiments with Inscribed Angles.
Draw this diagram at the top of the page.
B
C
A
F
E
Use this diagram and
the Lesson Summary
on page 41 and your
textbook to describe
and explain all the
terms on page 39.
Use the diagram to help you identify the following
terms:
Arc
Minor and major arc
inscribed angle
central angle
intercepted arc of an angle
Use the diagram to help you identify the following
terms:
Arc
An arc is a portion of the circumference
of a circle
BC is an arc, so is DE or BE
http://www.mathopenref.com/arcminormajor.html
Minor and major arc
BE is a minor arc
EDB is a major arc
What’s the difference? Minor arc is
shortest arc linking 2 points on a
circumference, major arc is longest arc
linking the 2 points.
http://www.mathopenref.com/circleinscribed.html
inscribed angle
Angle BDC is an inscribed angle
- An angle whose vertex is on the circle,
and each side of the angle intersects
the circle in another point.
http://www.mathopenref.com/circlecentral.html
Central angle
angle BAC is a central angle
An angle whose vertex is the center of
the circle.
Use the diagram to help you identify the following
terms:
intercepted arc of an angle
angle CDB and angle CAB both intercept
arc BC . What do you think it means for
an angle to intercept an arc?
http://www.mathopenref.com/arcintercepted.html
intercepted arc of an angle
An angle intercepts an arc if the
endpoints of the arc lie on the angle, all
other points of the arc are in the interior of
the angle, and each side of the angle
contains an endpoint of the arc. e.g.ED
http://www.mathopenref.com/arcangle.html
http://www.mathopenref.com/arcintercepted.html
Angle measure of an arc - is the same as the
central angle that intersects that arc.
ADD this to your notes page.
Homework - to correct and grade
Problem Set, starts on page 41
Complete 1, 2, 3, 4, and 5.
Explain all your thinking. Write neatly
and in an organized way.
Homework
1. Using a protractor, measure both the inscribed
angle and the central angle shown on the
C
circle below.
B
Inscribed angle
measure angle
BCD =
D
A
Central angle
measure angle
BAD =
Homework -
1. Using a protractor, measure both the inscribed
angle and the central angle shown on the
C
circle below.
90°
B
Inscribed angle
m∠ BCD = 90°
D
A
Central angle
Homework -
1. Using a protractor, measure both the inscribed
angle and the central angle shown on the
C
circle below.
B
Inscribed angle
m∠ BCD = 90°
D
A
180°
Central angle
m∠BAD = 180°
Homework -
2. Using a protractor, measure both the inscribed
angle and the central angle shown on the circle
below.
B
A
Inscribed angle
m∠ BDC = °
Central angle
m∠BAC = °
C
D
Homework -
2. Using a protractor, measure both the inscribed
angle and the central angle shown on the circle
below.
B
A
Inscribed angle
m∠ BDC = 30°
Central angle
m∠ BAC = 60°
C
D
Homework -
3. Using a protractor, measure both the inscribed angle and
the central angle shown on the circle below.
B
C
A
Central angle
m∠ BAC = °
Inscribed angle
m∠ BDC = °
D
Homework
3. Using a protractor, measure both the inscribed angle and
the central angle shown on the circle below.
B
C
A
Central angle
m∠ BAC = 100°
Inscribed angle
m∠ BDC = 50°
D
Homework
4. What relationship between the measure of the inscribed
angle and the measure of the central angle that intercept the
same arc is illustrated by these examples?
Inscribed angle
m∠ BDC = 50°
Central angle
m∠ BAC = 100°
Homework -
4. What relationship between the measure of the inscribed
angle and the measure of the central angle that intercept the
same arc is illustrated by these examples?
The measure of the inscribed angle
appears to be half the measure of the
central angle that intercepts the same
arc.
Homework -
5. Is
your conjecture at least true for
inscribed angles that measure 90
degrees?
What is a “conjecture”?
5. Is your conjecture at least true for inscribed angles that measure 90 degrees?
Yes, according to Thales’ theorem, if A,
B, and C are points on a circle where AC
is a diameter of the circle, then angle
ABC is a right angle. Since a diameter
represents an angle of 180 degrees, our
conjecture is always true for inscribed
angles that measure 90 degrees.
7. Red(R) and Blue (B) lighthouses are located on the
coast of the ocean. Ships traveling are in safe waters
as long as the angle from the ship (S) to the two
lighthouses (angle RSB) is always less than or equal
to some angle theta, called the “danger angle.” What
happens to theta as the ship gets closer to shore and
moves away from shore? Why do you think a larger
angle is dangerous?
Discuss with neighbor, + 2 - plan a helpful diagram prepare to come up to explain. (3 groups)
θ
°
θ
°
θ+°
Angle is now greater than the danger angle and the ship
is too close to the shore where there is shallower water
and the ship might run aground.
When ship is further from the
shore, the angle is smaller than
the danger angle. The ship is in
deeper water, which is safer for
sailing in.
Lesson 5. The Inscribed Angle Theorem (p. 44)
1. A and C are points on a circle with Center O.
a) Draw a point B on the circle so that AB is a
diameter then draw angle ABC.
Lesson 5. The Inscribed Angle Theorem (p. 44)
1. i)
B
inscribed
angle
O
C
central
angle
A
Lesson 5. The Inscribed Angle Theorem (p. 44)
ii.
∠ ABC is an
inscribed
angle
(intercepting
minor arc AC)
B
inscribed
angle
C
O
central
angle
A
iii. ∠AOC is
a central angle
(also
intercepting
minor arc AC)
Lesson 5. The Inscribed Angle Theorem (p. 44)
iv.
The
intercepted arc
of ∠ABC is
minor arc
B
inscribed
angle
C
O
AC.
central
angle
A
v. The
intercepted
arc of angle
AOC is also
minor arc AC.
Lesson 5. The Inscribed Angle Theorem (p. 44)
b. The measure
of the inscribed
angle ACD is x
and the measure
of the central
angle CAB is y.
B
C
central
angle
A
inscribed
angle
D
Lesson 5. The Inscribed Angle Theorem (p. 44)
2.
Find measure
of angle CAB
in terms of x.
B
C
x
y
A
x
D
AB = AC = AD
triangle CAD is an
isosceles triangle,
since measure of
angle CDB is given
as x, we now know
measure of angle
ACD is also x (base
angles isosceles
triangle congruent)
Lesson 5. The Inscribed Angle Theorem (p. 44)
Sum of all
angles of a
triangle is 180,
so m∠CAD =
180 - 2x
B
C
x
central
angle, y
A
x
D
Lesson5. The Inscribed Angle Theorem (p. 44)
Sum of all
angles of a
triangle is 180,
so m∠CAD =
180 - 2x
B
C
x
Angle CAD and angle CAB
are supplementary which
means
m∠CAB = 180 - (180 - 2x)
= 2x;
central
angle, y
A
therefore y = 2x
x
inscribed
angle
D
Lesson 5. The Inscribed Angle Theorem (p. 49)
Theorem: The measure of a central
angle is twice the measure of any
inscribed angle that intercepts the same
arc as the central angle.
Lesson 5. The Inscribed Angle Theorem (p. 49)
so far we have shown that the inscribed
angle theorem is true for
i) inscribed angles that measure 90°
(Thales’ theorem).
ii) inscribed angles where a side of the
angle passes through the center of the
circle, even when the angle isn’t 90°.
Lesson 5. The Inscribed Angle Theorem (p. 49)
Theorem: The measure of a central
angle is twice the measure of any
inscribed angle that intercepts the same
arc as the central angle.
Need to also prove for following cases (plus one):
Center is inside the inscribed angle
a
C
and
center is outside the
inscribed angle
B
Need to also prove for following cases (plus one):
Center is inside the inscribed angle
a
C
and
center is outside the
inscribed angle
B
center of the circle is in the
interior of the inscribed
angle
center of the circle is in
the exterior of the
inscribed angle
Need to also prove for following cases (plus one):
Center is inside the inscribed angle
a
C
and
center is outside the
inscribed angle
B
center of the circle is in the
interior of the inscribed
angle
center of the circle is in
the exterior of the
inscribed angle
Example 1: A and C are points on a circle with
center O.
a. What is the
intercepted
arc of angle
COA ? color
it red
C
O
.
A
Example 1: A and C are points on a circle with
center O.
b. Draw
triangle AOC.
What type of B
triangle is it?
Why?
C
.
O
D
A
b. Draw
triangle AOC.
What type of B
triangle is it?
Why?
C
.
O
A
Triangle AOC
is an
isosceles
triangle
because it
D
has 2
congruent
legs (radii of
the circle).
c. What can
you conclude
about the
measures of
angle OCA B
and OAC?
Why?
C
.
O
D
A
c. What can
you conclude
about the
measures of
angle OCA
and OAC?
Why?
C
.
O
A
Angles OCA
and OAC are
the congruent
base angles
of an
D
isosceles
triangle
d. Draw a
point B on the
circle so that
O is in the
interior of the B
inscribed
angle ABC.
C
.
O
D
A
e. What is the
intercepted
arc of angle
ABC? Color it
green.
B
f. What do
you notice
about arc
AC?
C
.
O
D
A
e. What is the
intercepted
arc of angle
ABC? Color it
green.
B
f. What do
you notice
about arc
AC?
C
.
O
A
Arc AC is the
same one that
the inscribed
angle
intercepts and
D
the central
angle
intercepts.
Watch:
C
B
x
.
O
y
D
A
Let the measure of angle
ABC be x and the
measure of angle AOC by
y.
Prove y = 2x.
OC = OA = OB = radius
Triangles BOA and BOC
are isosceles triangles
C
x1
B
x2
.
O
Let angle CBD= x1
y1
y2
D
and angle ABD= x2
angle COD= y1
A
angle AOD= y2
x = x1+ x2
From our opening exercise
we know
C
x1
B
x2
y= y1 + y2
y1 = 2 x1 and y2 = 2 x2
.
O
y1
y2
D
so by substitution
y = 2x1 + 2x2 = 2(x1 + x2)
A
y = 2x
Does our conclusion support the inscribed angle
theorem?
Yes, even when the center of the circle is
in the interior of the inscribed angle, the
measure of the inscribed angle is equal
to half the measure of the central angle
that intercepts the same arc.
Need to also prove for following cases (plus one):
Center is inside the inscribed angle
a
C
and
center is outside the
inscribed angle
B
center of the circle is in the
interior of the inscribed
angle
✔
center of the circle is in
the exterior of the
inscribed angle
C
Example 2. A and C are
points on a circle with
center O.
a. Draw a point B on the
circle so that O is in the
exterior of the inscribed
angle ABC
.
O
A
C
Example 2. A and C are
points on a circle with
center O.
a. Draw a point B on the
circle so that O is in the
exterior of the inscribed
angle ABC
.
O
A
B
C
Example 2. A and C are
points on a circle with
center O.
b. What is the intercepted
arc of angle ABC? Color it
yellow.
.
O
Minor arc AC
A
B
C
Let the measure of angle
ABC be x and the
measure of angle AOC be
y.
Prove that y = 2x.
.
y
O
x
B
A
Let BD be the diameter
containing B
D
C
y2
y1
.
y
O
x2
x = x2 - x1 and y = y2 - y1
x1
x
B
Let x1 y1 x2 and y2 be the
measures of angles CBD,
COD, ABD and AOD
respectively.
A
x = x2 - x1 and y = y2 - y1
D
C
y2
Since we know from our
opening exercise that
y1 = 2x1 and y2 = 2x2
y1
.
y
O
y = y2 - y1 = 2x2- 2x1
y =2(x2- x1)
x2
x1
x
B
A
y = 2x
by substitution.
d. Does your conclusion support the inscribed angle
theorem?
Yes, even when the center of the circle is
in the exterior of the inscribed angle, the
measure of the inscribed angle is equal
to half the measure of the central angle
that intercepts the same arc.
e. Have we finished proving the inscribed angle
theorem?
We have shown all the cases of the
inscribed angle theorem (central angle
version), except when the location of B is
on the minor arc between A and C.
Lesson Summary
Inscribed angle theorem: The measure of a
central angle is twice the measure of an inscribed
angle that intercepts the same arc as the central
angle.
Consequence of inscribed angle theorem:
Inscribed angles that intercept the same arc are
equal in measure.
Exercises (page 46)
1. a) m∠D=25°
x is central angle that
Intercepts same arc as
Inscribed angle D.
x is 2(D)
x = 50°
x
.A
25°
D
Exercises (page 46)
C
1. b) m∠D=15°
x is other base angle
Of isosceles triangle
ACD with two radii for
Congruent legs
x = 15°
x
.A
15°
D
Exercises (page 46) - relocate x to solve this one.
1. c) m∠BAC=90°
x
x is an inscribed angle
Intercepting the same
Arc as central angle
BAC
.A
90°
B
C
Exercises (page 46) - relocate x to solve this one.
1. c) m∠BAC=90°
x
x is an inscribed angle
Intercepting the same
Arc as central angle
BAC
x = 45°
.A
90°
B
C
3. Let’s look at relationships between inscribed
angles.
a)
Draw a quadrilateral inscribed in a circle.
3. Let’s look at relationships between inscribed
angles.
a)
Draw a quadrilateral inscribed in a circle.
Arc intercepted by angle
A plus
Arc intercepted by angle C
must add up to 360 degrees.
3. Let’s look at relationships between inscribed
angles.
a)
Draw a quadrilateral inscribed in a circle.
Inscribed angles A and C
Have to be half the central
Angles that intercept those
Same arcs.
So A + C = 180°
3. Let’s look at relationships between inscribed
angles.
a)
Draw a quadrilateral inscribed in a circle.
And B + D = 180°
Opposite angles
supplementary
Homework
Page 46 - 47
1.
2.
3.
4.
d, e, f
Write in complete sentences.
b)
a, b, c, d, e, f
Page 49 -50
1, 2, 3, 5, 7 and 8.