# 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.