inscribed angles common core geometry homework answers

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High school geometry

Course: high school geometry   >   unit 8, inscribed angles.

• Challenge problems: Inscribed angles
• Inscribed angle theorem proof
• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  

Experiments with Inscribed Angles

Related Topics: Lesson Plans and Worksheets for Geometry Lesson Plans and Worksheets for all Grades More Lessons for Geometry Common Core For Geometry

New York State Common Core Math Geometry, Module 5, Lesson 4

Worksheets for Geometry, Module 5, Lesson 4

Student Outcomes

• Explore the relationship between inscribed angles and central angles and their intercepted arcs.

Opening Exercise

ARC: MINOR AND MAJOR ARC: INSCRIBED ANGLE: CENTRAL ANGLE: INTERCEPTED ARC OF AN ANGLE:

Exploratory Challenge 1

Your teacher will provide you with a straightedge, a sheet of colored paper in the shape of a trapezoid, and a sheet of plain white paper.

• Draw two points no more than 3 inches apart in the middle of the plain white paper, and label them 𝐴 and 𝐵.
• Use the acute angle of your colored trapezoid to plot a point on the white sheet by placing the colored cutout so that the points 𝐴 and 𝐵 are on the edges of the acute angle and then plotting the position of the vertex of the angle. Label that vertex 𝐶.
• Repeat several times. Name the points 𝐷, 𝐸, ….

Exploratory Challenge 2

a. Draw several of the angles formed by connecting points 𝐴 and 𝐵 on your paper with any of the additional points you marked as the acute angle was pushed through the points (𝐶, 𝐷, 𝐸, …). What do you notice about the measures of these angles? b. Draw several of the angles formed by connecting points 𝐴 and 𝐵 on your paper with any of the additional points you marked as the obtuse angle was pushed through the points from above. What do you notice about the measures of these angles?

Exploratory Challenge 3

a. Draw a point on the circle, and label it 𝐷. Create angle ∠𝐵𝐷𝐶. b. ∠𝐵𝐷𝐶 is called an inscribed angle. Can you explain why? c. Arc 𝐵𝐶 is called the intercepted arc. Can you explain why? d. Carefully cut out the inscribed angle, and compare it to the angles of several of your neighbors. e. What appears to be true about each of the angles you drew? f. Draw another point on a second circle, and label it point 𝐸. Create ∠𝐵𝐸𝐶, and cut it out. Compare ∠𝐵𝐷𝐶 and ∠𝐵𝐸𝐶. What appears to be true about the two angles? g. What conclusion may be drawn from this? Will all angles inscribed in the circle from these two points have the same measure? h. Explain to your neighbor what you have just discovered.

Exploratory Challenge 4

a. In the circle below, draw the angle formed by connecting points 𝐵 and 𝐶 to the center of the circle. b. Is ∠𝐵𝐴𝐶 an inscribed angle? Explain. c. Is it appropriate to call this the central angle? Why or why not? d. What is the intercepted arc? e. Is the measure of ∠𝐵𝐴𝐶 the same as the measure of one of the inscribed angles in Exploratory Challenge 2?

Lesson Summary

All inscribed angles from the same intercepted arc have the same measure.

Relevant Vocabulary

• ARC: An arc is a portion of the circumference of a circle.
• MINOR AND MAJOR ARC: Let 𝐶 be a circle with center 𝑂, and let 𝐴 and 𝐵 be different points that lie on 𝐶 but are not the endpoints of the same diameter. The minor arc is the set containing 𝐴, 𝐵, and all points of 𝐶 that are in the interior of ∠𝐴𝑂𝐵. The major arc is the set containing 𝐴, 𝐵, and all points of 𝐶 that lie in the exterior of ∠𝐴𝑂𝐵.
• INSCRIBED ANGLE: An inscribed angle is an angle whose vertex is on a circle, and each side of the angle intersects the circle in another point.
• CENTRAL ANGLE: A central angle of a circle is an angle whose vertex is the center of a circle.
• 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.

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Chapter 10, Lesson 4: Inscribed Angles

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Circles: Inscribed Angles, Arcs, and Chords Worksheets

Circles are really unique geometric shapes because they really have a single measure (radius) and that determines everything about it. There are number of different vocabulary words and phrases that are unique to this shape. These measures are critical for mastering many aspects of geometry, specifically proofs. If a line segment passes from one point on a circle to another it is called a chord. If two of those chords meet at a point on the circle it forms and inscribed angle. If you were to slice out any portion of the circumference of a circle, this curve that you have is referred to as an arc. This series of worksheets and lessons will put these new vocabulary words to work for you as you work on finding all these various measures.

Aligned Standard: High School Geometry - HSG-C.A.2

• Chords Step-by-step Lesson - Find the length of the chord based on the concept of congruency.
• Guided Lesson - Find some angles, lengths, and arc measures in this one.
• Guided Lesson Explanation - These are really neat because they require you to think past yourself and then come back and do the required math.
• Practice Worksheet - For half of this worksheet, you are just looking for x.
• Angles Circle Tangent Chord Intersecting Chords 5 Pack - Sometimes the most difficult part is just identifying what the question is looking for.
• Angles Outside Circle 5 Pack - My father retired years ago for a natural gas company. This skill was 75% of his job.
• Angles and Arcs in Circles 5 Pack - In a few cases you can negate the circle entirely.
• Matching Worksheet - Match the angles and measures to diagrams your are presented with.
• Circles: Diameter, Chord, Center, and Radius 5 Pack - Find all those values in each circle. Probably would have been good to add that in the directions.
• Arcs of Circles Worksheet Five Pack - Find arc lengths, minor arcs, and circumference too.
• Chords and Circles Worksheet Five Pack - Some of the chords here are bit off center.
• Answer Keys - These are for all the unlocked materials above.

Homework Sheets

Find the lengths of chords and positions of center.

• Homework 1 - FI and GH are chords in a circle, and their corresponding arcs are congruent. So, FI is congruent to GH.
• Homework 2 - < ABC is an inscribed angle that intercepts the same arc as the central angle.
• Homework 3 - The center of the circle is J. GJI is a diameter of the circle.

Practice Worksheets

I have used the mechanics of arcs to actually analyze a basketball shot. They are really fun.

• Practice 1 - What is the length of a chord and a few unknown angles to figure out.
• Practice 2 - We start to find the length of arcs here. We then move on to other missing measures and fill it all in.
• Practice 3 - Two chords in a circle are congruent if their corresponding arcs are congruent.

Math Skill Quizzes

Some of these are tricky to determine which part they are looking for you to find.

• Quiz 1 - Find the value of x. It is presented to you in so many different ways.
• Quiz 2 - In the figure, O is the center of the circle. What is m< PQR, if m < OQR = 35?
• Quiz 3 - The inscribed angle theorem states that the measure of an inscribed angle is half the measure of the central angle that intercepts the same arc.

What are Inscribed Angles, Arcs and Chords in Circles?

A circle is simply a series of points that are all an equal distance away from the center. That distance to the center is called the radius. Each circle that you run into that is a different size differs by the length of its radius. We all are familiar with an 'O' like shape known as a circle. In fact, we observe and sometimes use it daily. A circle is a line forming a closed loop, every point on which is equidistant from a fixed point that is its center. But there is more to a circle than just a circular boundary. There are three very commonly used vocabulary terms and phrases that we use while working with circles. We explore them below:

Chord - Before we discuss the mathematical definition of a chord, let's visualize it with an example. Imagine you are standing on the edge of a perfectly round lake and gazing at picnic tables on the other side of the lake. The chord is the straight line extending from you to the picnic tables. Mathematically, a chord is a line that connects two points on a circle. The diameter of this line would be the distance of the chord.

Arcs - An arc is commonly defined as a small portion of the circumference of a circle, which is the distance around the edge of a circle. Arcs can be a small portion of some other curved shapes, too, like ellipses. To avoid the confusion, we generally use the term arc of a circle as a circular arc. Due to gravity, arcs as very commonly used to describe the plot trajectories of the motion of projectile objects.

Inscribed Angles - An inscribed angle is formed by connecting the points present on the circumference of a circle. Mathematically saying, an inscribed angle is formed by two chords that meet at the same endpoint. This endpoint is referred to as the vertex. This is where geometric shapes sometimes overlap and form complex new figures.

Why is Circle Geometry Important?

Our world is literally surrounded with this shape, pun intended. They are so ubiquitous we often forget that they are even present. The first major human invention, the wheel, is simply the proper use of this shape. In ancient cultures circles were thought to represent perfection and balance. In math this unparalleled shape can be used and modelled to understand a great deal about the world around us. Any thing that is tethered from a central point can be used to model a circle. In construction this phenomenon is often used to create all types of circular patterns. We live on a big sphere that is built off of the circle. To find out where you are located on that sphere, we use GPS which uses circle geometry to plot our location. Many scientific principles that are used to model and predict motion are based on circle geometry. Anything that does not travel in a straight line is modelled and understood this way.

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