Grade 10 | Lesson 1

Mathematics

Lesson Overview

Geometry: Basics

***You will need a protractor and ruler to complete your 10th grade geometry lessons. If you need help with where to obtain these, please contact Mrs. Warren (dean@theschools.com) or your online academic advisor (if applicable).***

• Lines, Line Segments, Rays and Angles

Practice

• Angle Measures
• Classifying Angles

Practice

Geometry: Basics

Geometry is one of the most beautiful fields in math and is a gateway to more grand explorations of our planet, physics, art, architecture and the cosmos, just to name a few. Like algebra, The study of geometry depends on the building of techniques over time in a logical way so that when complex topics are encountered you have a strong understanding and knowledge of its foundation. Simply put, geometry is the study of shape, size, the position of figures, and the properties of space. It has many practical applications with a rich array of uses.

In these first two lessons, you'll learn some important new vocabulary words and classifications which will help lay the foundation for your understanding of Geometry.

Lines, Line Segments, Rays and Angles
Geometry is the study of shapes in space and because of this, its important to understand how shapes are defined in space. While some of the follow concepts many seem overly simplistic, they can actually be quite complex.

Lines
A line does not have a beginning or ending point. Although it can be mind boggling, imagine it continuing indefinitely in both directions. Lines are usually illustrated with arrows on each end to denote this.

Line Segments
In geometry, you'll rarely encounter lines, but instead will work with line segments. Line segments are just what they sound like, finite segments of an infinite line that have a measurable length in space.

Rays
A ray is a mix between a line and a line segment. In one direction it has a finite end point while in the other, it doesn't and continues forever.

None of these three has a specific way it needs to be oriented in space. Lines, line segments and rays can cross each other, run parallels or end at the same point.

Angles
An angle is comprised of two rays with share the same endpoint. This collective point is known as a vertex and the two rays are known as the sides of the angle.

The degree to which an angle is "open" or closed" will be covered a little later in this lesson.

Angle Measures
Angles are fundamental to geometry and you will be working with them countless times throughout 10th grade. Angles are measured in degrees (°) and are related to circles very closely, which have 360° total.

Imagine a pizza with a point in the exact center. Now imagine a pizza cutter cutting anywhere on the circle edge to that point. If that is repeated from another point on the edge of the circle, not only would there be a great snack (in your imagination), but you'd also have created an angle. In essence, the two cuts were rays and the point in the center a vertex. If you separate that slice of pizza from the entire thing, it would be hard to even know it came from a circle in the first place. Never mind the measure of the pizza angle for a second, the most important thing to remember from this thought exercise, is that angles are based on circles and that circles have 360° total.

In the practice problems below, use your protractor to measure the angles given and draw the angles prompted for. To measure angles with your protractor, the zero line needs be lined up with one side of the angle, and you'll be able read the angle's measurement. To draw an angle that is asked for, again, place the zero line on the ray and then make a pencil mark at the desired degree measurement. Use the flat edge of your protractor or a ruler to draw the second ray, thus creating the angle.

Practice

Practice Problems - Download Here

Practice Problems Answer Key - Download Here

Classifying Angles
There are five types of angles which you will encounter. Here they are from smallest to largest measure.

Zero Angle
A zero angle is one where the two rays which comprise the angle lay directly over top of each other. For practical purposes, a zero angle is identical to a single ray. In the diagram below, one of the rays is dashed so that you can see they are top of each other.

Acute Angle
If one of the rays in the zero angle begins to move counterclockwise from 0° it will be an acute angle all the way through 90°, but not including a 90° angle. Here is an example.

In mathematical terms an acute angle (A) can be understood this way: 0°<A<90°.

Right Angle
At 90°, an angle is a right angle. If it is slightly less or slight more, it is not. You'll usually see a small box in the angle to denote a right angle.

Obtuse Angle
This is an angle similar to an acute angle, but for angles between 90° all the way up to 180° but not including 180°.

In mathematical terms an obtuse angle (O) can be understood this way: 90°<O<180°.

Straight Angle
The final of the five angle classifications, a straight angle occurs when two rays point directly opposite from each other. The measurement of such an angle is 180°.

Practice

Practice Problems - Download Here

Practice Problems Answer Key - Download Here