Introduction to Geometry Lesson 2 Angles - University of California ...

Oleg Gleizer

Introduction to Geometry

LAMC

Lesson 2

Angles

A ray is a half of a straight line, finite in one direction, but infinite in the other. A ray contains its boundary point.

A (plane) angle is a plane figure formed by two rays with a common vertex and by all the points in between.

Looking at the above picture, you have to imagine that each of the two rays goes to infinity and so does the green coloring. This way, the rays split the plane into two parts, the green and white. The green-colored part, together with the boundary rays, is an angle. The white-colored region, bounded by the same rays, is also an angle.

If the rays forming an angle coincide, the smaller one of the two is called a zero angle. The angle complementing it to a full

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plane is called a full angle.

If the rays forming an angle lie on a straight line, the angle is called a straight angle.

You have to imagine that the entire upper half-plane on the above picture is colored green.

The word congruent means coincide when superimposed. If two geometric figures are congruent, then all of their features, lengths of the corresponding sides, sizes of the corresponding angles, etc. are exactly the same. Problem 1 Define a straight angle in a way different from the above. Hint: consider the angle complementing it to a full angle.

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Problem 2 Draw a straight line in the space below. How can you make it into a straight angle?

Two angles are called supplementary, if they add up to a straight angle.

Problem 3 Using a ruler, draw an angle supplementary to the green-colored angle below. How many ways are there to solve the problem?

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A standard way to measure an angle is to divide a full angle into 360 equal parts. One part is called one (angular) degree and is denoted as 1. A circumference bearing the 1 marks, or a part of it, is called a protractor.

105 90 120

135

75 60

45

150

30

165

15

180

0

195

345

210

330

225

240 255

270

315

300 285

Question 1 What is the size, in degrees, of a straight angle?

It is customary to use lower-case Greek letters to denote angles and their sizes. You will find the table with letters of the Greek alphabet on page 25 of this handout.

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Problem 4 Use degrees to write an algebraic statement showing that angles and are supplementary.

Problem 5 Use a protractor to measure the following angle. Round to the nearest degree.

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Problem 6 Use a ruler and a protractor to draw a 75 angle in the space below.

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