Name: ___________________________

Geometry Period _______

Unit 13: Circle Theorems - Part 1

In this unit you must bring the following materials with you to class every day:

Calculator

A Pencil This Booklet

Colored pencils/pens/highlighters

A device

Headphones!

Please note:

You may have random material checks in class

Some days you will have additional handouts to support your understanding of

the learning goals in that lesson. Keep these in a folder and bring to class every

day.

All homework for part one of this unit is in this booklet.

Answer keys will be posted as usual for each daily lesson on our website

Let’s Prepare for today! Try to label and identify the following parts you see in this circle

Part of Circle Describe it!

Radius Segment from _________ to the ________ of

the circle. All radii are _______________.

Arc Piece/part of the ____________________ of a circle.

Center* “ Middle” point of a circle. All points on the circle are

the ___________ distance away. We name the circle

by it center.

Chord Segment whose ____________________ lie on the

circle.

Diameter Special chord that passes through the ____________.

They create semicircles in a circle. Semicircles are

arcs whose measure is 180°

Flashback! Important Vocabulary/Concepts that we remember…

Major Arc- Measures___________ than 180°. Minor Arc- Measures___________ than 180°.

Central Angle- angle with a ________ at the center.

Intercepted Arc- Arc formed when segments ______________ a portion of the circle.

Let’s watch the simulation – as you watch, are there any notable observations that you can make? *The measure of a central angle and the measure of its intercepted arc are ________________________.

For example, if m ABC is 65 in circle B seen above,, then arc AC measures _______ and is

___________.

Today’s Learning Goals: What is a central angle, and what is its relationship with its intercepted arc? What is an inscribed angle, and what is its relationship with its intercepted arc? What is Thale’s Theorem?

13-1 Notes

Remember from our ed- puzzle!!

Measure of an arc is NOT same

as the length of an arc. Arcs are

measured in _______!

All Arcs of the circle

must add up to 360°

You try! In circle O, GE is the diameter. If the m = 80°, Find the m GOF Find the m FOE

Find the m

Find the m

Turn and Talk Describe , how is this different from a central angle?

Inscribed Angle

Let’s watch the simulation – as you watch, are there any notable observations that you can make? https://www.geogebra.org/m/aFXfGSNH

An Inscribed Angle is ________________________________________________________________________________ __________________________________________________________________________________________________ Let's Try It! Remember to label!

If find

Keep in

mind,

what’s the

measure

arc of a

Semi-

circle?

What if… the inscribed angle subtends an arc created by the diameter of the circle?

Sketch:

Let’s watch the simulation – as you watch, are there any notable observations that you can make?

http://www.mathopenref.com/thalestheorem.html

Thale’s Theorem: When an inscribed angle subtends a _______________, the inscribed angle is a ____________ angle.

Talk it out! With your partner answer the following...

1) What new vocabulary did you learn today?

2) How would you describe any relationships between angles and intercepted arcs in circles?

PRACTICE

Let's Practice These New Ideas! 1) The circle below has center A. Identify which intercepted arc goes with which central angle using the given chart.

2) The circle below has center A. Identify which intercepted arc goes with which inscribed angle using the given chart.

Angle Intercepted Arc

1) 1)__________

2)__________ 2)

Angle Intercepted Arc

1) 1)__________

2)__________ 2)

A

3) Given circle with center indicated, find the measure of angle x.

4) Given circle with center indicated,

a) find the measure of angle y

b) find the measure of angle x

c) Explain how you found x and y.

5) Find the measure of angle x:

6) Vertices of the triangle ΔABC lies on circle and divided it into arcs in the ratio 10:2:8.

Determine the size of the angles of the triangle ΔABC. SKETCH YOUR PICTURE!

7) An inscribed angle in a semicircle is a ______________ angle.

8) Find the measure of each arc of Circle P, where RT is a diameter.

a)

b)

c)

What do we do

with ratios? Use

x’s! What MUST all

arcs add up to in a

circle?

HOMEWORK 13-1

Directions: Answer the following questions to the best of your ability. Show all work!

1. In the accompanying diagram of circle O, the measure of is 38°. What is the measure of ?

2. In the accompanying diagram of circle O, and are chords and . What is ?

1) 32 2) 48

3) 96 4) 192

3. In the accompanying diagram of circle O,. What is ?

1) 210 3) 95

2) 105 4) 75

4. Is a diameter a chord? Are all chords diameters? Draw a diagram to defend your answer. Describe the arc which is

made by the diameter of a circle.

5. An angle inscribed in a circle measures 80 . Draw the diagram. What is the number of degrees in the intercepted arc?

6. Chords AB and AC are drawn in circle O. Radii OB and OC are also drawn. Draw the diagram.

Given that , what is the ?

7. In the accompanying diagram of circle O, and are diameters. Which statement is not true?

8. a) How do we know that AB is a diameter based on the markings in the picture?

b) Determine the radius of a circle with an inscribed right triangle with legs 7 cm and 4 cm. Round to the nearest

hundredth

A recent survey asked teenagers whether they would rather meet a famous musician, athlete, actor, inventor, or other

person. The circle graph shows the results in degrees. Find the indicated arc measures:

a. b.

d.

**Who would you rather meet?

1) 3)

2) 4)

108° 83°

61°

79°

Today’s Goal: What are Cyclic Quadrilaterals? How can we solve for angles formed by chords?

Re-activate our knowledge!

Yesterday, we looked a Thales Theorem (see right)

Thales Theorem told us, when an inscribed angle subtends a semi-circle, the angle

measure is ________.

Consider Quadrilateral ABCD inscribed in circle M:

By Thales Theorem we know:

m<B = _____

m<D = _____

What special relationship do these opposite angles have?

Do you think that is true for ALL quadrilaterals inscribed in circles?

Let’s Discover!

With your partner, examine the cyclic quadrilateral that was assigned to your team. Fill out the table below and answer

the questions.

1. My team has cyclic quad #_______

2. In my quadrilateral, ______ is opposite _______ and ______ is opposite _______

Angle measures SUM OF FIRST SET OF OPPOSITE ANGLES

SUM OF SECOND SET OF OPPOSITE ANGLES

What do you notice?

A________

B________

C _______

D_______

13-2 Notes

Challenge! If you have extra time, can you EXPLAIN why this works? Use the following diagram to help you explain!

CLASS SHARE OUT!

Cyclic Quadrilaterals

Together! EXAMPLE 1: Quadrilateral WZYX is inscribed in circle O. What is the measure of angle WXY?

Try a Regents question!

Angles formed by Intersecting Chords

What do you notice about the location of the vertex of the angle marked ‘X’ in the circle?

So, how do we find the ?

1) Circle O is shown below. Try it! Find

2) Circle O is shown below. Chords AB and CD intersect at E. m = 20 and m<AEC = 60. Solve for m

Arc #1

Arc #2

PRACTICE

Complete each of the following problems. Show all work.

1. The quadrilateral is inscribed in the given circle. Find the measure of angles x and y.

2. Circle O is shown below. Chords AB and CD intersect at E. m = 75° and Find m .

3. Circle O is shown below. Chords AB and CD intersect at E. m and m . What is the measure of

and

6. One of the following measurements is incorrect. Identify the error and correct it so that the properties about bowtie angles holds true.

,

4.

5.

HOMEWORK 13-2

Directions: Answer the following questions to the best of your ability. Show all work to receive full credit.

1. In the diagram below of circle O, chords and intersect at E, , and .

What is the degree measure of ?

1) 87

2) 61

3) 43.5

4) 26

2. In a circle, chords and intersect at E. Draw the diagram.

If , and =55, find

3. FC is a diameter find

4) In circle O shown below. Chords AB and CD intersect at E. AC:CB:BD:DA is a 1:3:2:4. Find m BED.

5)

6) In the diagram, a quadrilateral is inscribed in circle E. If the measure of = 96, solve for the measure of

Try it!

Learning goal: What is the relationship between the measure of angles and how these angles are formed?

New Vocabulary alert!

Tangent line/segment A line that passes through a circle ____________________________.

Sketch a tangent

Secant Line/segment A line that passes through a circle ____________________________.

Sketch a secant

When these special segments interact with other parts of the circle, new special relationships occur!

Relationship #1

Turn and Talk

1. What type of angle is ABC? Hint: Look at the vertex location!

2. What type of segments form this angle?

Together! When angles are formed by a ___________________ and a chord we call them ________________________________

>See example above. What is the value of x?

13-3 Notes

Another example: what is m ?

Relationship #2

1. What type of segments form this angle?

2. What type of angles are and ?

3. What is the measure of both arcs intercepted by chord ED?

4. What are the measures and ?

and are both ________________________________! Or….ED ______BC

5. Fill in the blank! We discovered that the Diameter/Radius will always be ____________ to a tangent line at the point

of tangency.

Arc 1st

w/ratios!!

Relationship #3 Sneaky Angles

Let’s take a look at .

This angle is NOT considered a special inscribed angle. Why do you think that is? What segments form it?

Since it’s NOT a special inscribed angle, we can’t use the inscribed angle rule.

Let’s use a different angle relationship to help us find the angle!

Let’s think!

What‘s the measure of , if ?

What is the relationship between and ?

Using these relationships- What’s the measure of ?

To calculate the measure of “sneaky angles” we: 1. Calculate the measure of ____________________________

2. Subtract from 180 (linear pair)

Practice Time!

1) In the following diagram

a) Solve for the measures of

b) Is Special inscribed or a sneaky angle? How do you know?

c) Solve for

2) In the accompanying diagram of circle , is a diameter, is a tangent , is a secant, is a chord,

, and

a. Find the measure of .

b. Find the measure of

c. Find the measure of

d. Find the measure of

**Sneaky angle

**Tangent Diameter!

3) A small round plate falls off of the china closet and breaks. The diagram below shows one of the broken

pieces. is tangent to at B and . What is the

measure in degrees of the arc , the outside edge of the broken piece?

4)

5) Practice DRAWING it! In circle M, Chord AB and CB are drawn. Chord CB extends from point B to a point Z outside of

the circle. If , then how big is m<ZBA?

HOMEWORK 13-3

Directions: Answer the following questions to the best of your ability. Show all work to receive full credit.

1) AB is a tangent to the circle below.

a) Solve for

b) What type of angle is DAB?

c) Solve for DAB

2) Given the circle shown, with 2 diameters drawn in, solve for x.

3) The following diagram shows a quadrilateral inscribed in a circle. What do we call these quadrilaterals? Find the

value of each variable.

1)

4) Solve for the measure of DBA.

5) In circle O shown below, diameter is perpendicular to at point C, and chords , , , and are drawn.

Which statement is not always true?

1)

2)

3)

4)

CHECK YOUR

HOMEWORK

Angles Outside of the Circle

Today’s Learning Goal: How do you solve for an angle outside of a circle using its intercepted arcs?

Let’s reactivate our knowledge so we can tap into some new concecpts!

What parts of this question can we answer now? What parts do we not know how to solve for yet?

In the accompanying diagram of circle O, is a diameter, PC is a tangent, is a secant, is a

chord, AO = 8 and m : m : m : m = 3:2:1:4.

Find:

a) m

b) m

c) m *how is this angle different?

What type of angle is this?

Inscribed?

Central?

Bow tie?

13-4 Notes

Discuss: o What do you notice about the location of in each diagram below? _________________________

Angles Outside of the Circle

Angles outside of the circle can occur by…

Example 1: Secants SQP and TRP intersect at P. If and What is the m

How are we doing so far?

-What is the relationship we learned today? *Let’s revisit the problem that

-When do we know when to use it? we tried at the start of class!

Rank yourself!

I get this! I’m ready to practice without hints!

I kind of get it, but I would like hints just in case.

I need a little more help on this topic. I would like some hints and then I’ll be good!

Practice Makes Progress!

1) Tangents and intersect at P. If measure of is 200o, what is the m<P?

2) In circle O, secant and tangent intersect at P. If m = 30o and m = 90o, what is the m<P?

3) In the accompanying diagram, and are tangent to circle O. If m<PBA = 50o, find m<P.

4) In the accompanying diagram of circle O, chords and intersect at E and

. What is ?

5) In the accompanying figure of circle O, chords and intersect at E and is a diameter. If ,

find .

6) Secant and tangent intersect at P. If m = 150 and m < P = 25, what is the m ?

13-4 Homework

Directions: Answer the following questions to the best of your ability. Show all work!

1) Use the video to fill out your graphic organize to prepare for tomorrow’s quiz!

2) Name the far arc and the near arc from angle A in the diagram below:

Far Arc:

Near Arc:

What is the formula for finding the m <A?

3) The accompanying diagram shows two lengths of wire attached to a wheel, so that and are

tangent to the wheel. If the major arc has a measure of 220°, find the number of degrees in .

4) In the accompanying diagram, is tangent to circle O at Q and is a secant. If and

, find .

D

5) Point P lies outside circle O, which has a diameter of . The angle formed by tangent and secant

measures 30°. Sketch the conditions above.

Find the number of degrees in the measure of minor arc CB.

6) In a circle, diameter is extended through B to external point P, tangent is drawn to point C on the

circle, and . Find .

7) Given: quadrilateral ABCD is inscribed in a circle; arcs AB and CD are 80 and 50 degrees respectively. SHOW that the angle between sides AD and BC (when you extend these sides) is 15 degrees. Explain your steps.

Parallel and Congruent Arcs in Circles

Learning Goal: What relationship do parallel and congruent chords have on intercepted arcs in a circle?

Relationship #1: Parallel Chords in a Circle

1) Using the fact that we have parallel lines cut by a transversal, what other angle must measure 20 degrees?

2) What is the m ? How do you know?

3) What is the m ? How do you know?

Turn and Talk! What conclusion can you and your partner make about your responses to the questions above?

What's the theorem? Theorem: If AB is parallel to CD then, Let's Try It! Using the theorem we just learned, what is the value of x?

13-5 Notes

Non-Negotiable facts!

All Arcs add up to 360.

Diameters create Semi-circles.

Identify TYPE of angle formed.

If ratio given- use X’s, add up

to 180 or 360.

Relationship #2: Congruent Chords In Circles

Think-Pair-Share What do you think is true about the arcs formed on the outside of these congruent chords (arc AB and arc CD)? Why?

What's the theorem? Theorem: In a circle if two chords are congruent then,

Let's Try It! Example 1: Find the measure of arc PQ and arc RS. Example 2:

Remember

your non-

negotiable

ratio fact!

Practice Time

Let's Practice using our new theorems! Remember MARK YOUR DIAGRAMS AS YOU GO and NON-NEGOTIABLE CIRCLE FACTS.

1) CD is a diameter of Circle O. AB || CD, m . Find m

2) AB || CD, m . Find m

3) AB || CD, . Find m

4) Identify one pair of inscribed angles that are congruent to each other and explain why they are congruent to each other.

5) In the accompanying diagram of circle o, is a diameter, is a tangent, is a secant, chords and

are drawn, m

Find:

A. Since AOD is a diameter,

Arc AD =

B.

C.

D.

E. F. G.

H.

13-5 Homework

Directions: Answer the following questions to the best of your ability. Show all work to receive full credit. The challenge

problem is optional.

REMEMBER THE NON-NEGOTIABLE CIRCLE FACTS!

4. Solve for x

What rule do we need here?

5. Solve for x

What rule do we need here?

6. In the figure below, is tangent to circle O at Q. and are chords. If m = 70 and m = 80, find

.

7. In the accompanying diagram of circle O, m and m Find the value of x.

8. In the accompanying diagram and are secants from external point P to circle O. Chords , , ,

and are drawn. , is twice and is 60 more than .

Find:

A.

B. .

C.

D.

E.

Smile, you did it!!