Angles

When you open your math book part way as in the picture, the bottom edge of the cover forms an acute angle with the bottom edge of the first page.

An angle is formed when 2 lines meet at a point.

The point at which the lines meet is called the vertex.

The lines which meet to form the angle are called arms.

With our book, the bottom edges of the cover and first page are the arms and
the point where they meet on the book's spine, is the vertex .

In the diagram, point B is the vertex.
The line segments AB and BC are the arms.

BC is called the initial arm and AB is called the terminal arm.

It's as if BA started out lying on top of BC, pinned down at B.
Then we turned BA upwards to create the angle ABC.

Once we've got the cover of our math book open, we TURN the page. Turning is also called rotating -- so angles are the result of a rotation or turn. Think about the windshield wipers on the school bus. They rotate around a fixed point -- (if they didn't, they'd fly off the bus) -- through an angle large enough so that together, they wipe enough of the windshield for the driver to see clearly.

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Naming Angles

We generally name angles like the one in the diagram above with 3 letters and we always put the vertex letter between the other two just the way it appears in the picture. The angle above can be named either ABC or CBA -- note that B, the vertex, is in the middle, between A and C. Sometimes, we use only 1 letter to name an angle, but that can be confusing when there's more than one angle at the same vertex.

In the diagram above, it's easy to see what angle we mean when we say angle B. But in the next picture, there are four different angles at B, namely angles ABD, ABE, DBC and CBE, so if I say angle B, which one do I mean? Therefore, it's best to use 3 letters to name an angle if there's more than one angle at a common vertex. In triangle XYZ, we know exactly which angle we mean when we say angle X, so in such a case, 1 letter will do.

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Measuring Angles

When we open our first geometry kit, we see that it includes a ruler, 2 right triangles, a set of compasses and a protractor (diagram). These tools allow us not only to measure but also to construct angles. Since we're learning to measure angles in this lesson, we'll study how to use the protractor.

Notice how the degree measures run in both directions and how we place point O on the protractor, directly over the vertex of the angle we need to measure. In the diagram, angle DOB measures 45° and angle AOD measures 135° because we use the outer numbers to measure counter-clockwise rotations of point B and the inner numbers to measure clockwise rotations of point A. So, angle AOC measures 25° (using the inner numbers) and angle BOC measures 155° (using the outer numbers). Notice also that DOB and AOD are supplementary angles -- they add to 180°. So do AOC and BOC. That's because a straight angle = 180°.

Drawing an Angle of Given Size

Say we have to draw an angle of 57°. Here's how we do it. First we draw a line XY, upon which we pick a point O to be the vertex. We then place our protractor so that its vertex point is directly over O on the line. Then we mark a point A -- don't make a potato -- all we need is a point -- at the 57° mark on the protractor. Then we join point A to point O.
Angle AOY will be 57°.

Constructing a 57° angle.

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Angle Definitions:

A right angle is exactly 90°. A straight angle is exactly 180°.
an acute angle: measures more than ( > ) 0° and less than ( < ) 90°.
an obtuse angle: measures more than ( > ) 90° but less than ( < ) 180°.
a reflex angle: measures
more than ( > ) 180° but less than ( < ) 360°.
complementary angles: 2 acute angles that sum to 90° like 52° and 38°.
suplementary angles: 2 angles that sum to 180° like 152° and 28°.
opposite angles: also called "vertically opposite angles" are equal (look for a V).

Let's prove the last statement or theorem.

We want to show that equal angles are formed opposite to each other when two lines meet at a point to form 2 pairs of opposite angles. Let's look at the diagram.

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Practice Exercises

1) Label each angle acute, obtuse, straight, right or reflex.
 a) 128º b) 64º c) 258º d) 90º e) 180º f) 86º g) 172º h) 307º

2) a) Find the complementary angle for the acute angles in question 1.
b) Find the supplementary angle for the obtuse angles in question 1.

3/ Find the measure of all the angles at E in the diagram.

4) a) Using your protractor, draw acute angles of 29°, 57° and 78°.
b) Now draw their complementary angles.
c) Using your protractor, draw obtuse angles of 153°, 112° and 170°.
d) Now draw their supplementary angles.

5) Describe how to measure a reflex angle.

6) On a piece of graph paper, up near the top right corner, mark a point
and label it A. Now drop straight down 12 squares, mark that point and label it B. And finally, move 16 squares left, mark the point and label it C. Now join A to B, B to C and C to A, to form the right triangle ABC.

a) with your protractor, measure angle C.
b) use your answer to calculate angle A. (hint: angles A and C are complements).
c) now, measure angle A with your protractor.

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Solutions

1)
 a) 128º is obtuse b) 64º is acute c) 258º is reflex d) 90º a right angle e) 180º straight f) 86º is acute g) 172º is obtuse h) 307º is reflex

2 a)
 complement of 64º = 26° complement of 86º = 4°

2 b)
 supplement of 128º = 52° supplement of 172º = 8°

3/ Find the measure of all the angles at E in the diagram.

4)
 b) 61° complements 29° 33° complements 57° 12° complements 78°

 d) 27° supplements 153° 68° supplements 112° 10° supplements 170°

5) To measure a reflex angle, measure the obtuse angle or rotation,
then subtract that value from 360°.

6) You have made a 3, 4, 5 right triangle.
You should get angle C = 37° (really 36.9°)
and therefore angle A = (90 - 37)° = 53° (really 53.1°)

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