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The Fundamentals of Geometry Study Guide (page 2)

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Updated on Oct 5, 2011

Angles

What Is an Angle?

An angle is formed when two lines meet at a point: The lines are called the sides of the angle, and the point is called the vertex of the angle.

The symbol used to indicate an angle is

There are three ways to name an angle:

  1. By the letter that labels the vertex:

    Introducing Geometry

  2. By the three letters that label the angle: , with the vertex letter in the middle
  3. By the number inside the vertex:

An angle's size is based on the opening between its sides. Size is measured in degrees (°). The smaller the angle, the fewer degrees it has. Angles are classified by size. Notice how the are shows which of two angles is indicated:

Acute angle: less than 90°

 

Introducing Geometry

Right angle: exactly 90°

Introducing Geometry

The little box indicates a right angle. A right angle is formed by two perpendicular lines. (Perpendicular lines are discussed at the end of the lesson.)

Straight angle: exactly 180°

Introducing Geometry

Obtuse angle: more than 90°and less than 180°

Introducing Geometry

Tip

Since bisect means to cut into two congruent pieces, an angle bisector cuts an angle into two equal angles.

Introducing Geometry

is angle bisector to

 

Introducing Geometry

Congruent Angles

When two angles have the same degree measure, they are said to be congruent.

Congruent angles are marked the same way.

The symbol is used to indicate that two angles are congruent:.

Complementary, Supplementary, and Vertical Angles

Names are given to three special angle pairs, based on their relationship to each other:

Introducing Geometry

  1. Complementary angles: Two angles whose sum is 90°.

    are complementary angles.

    is the complement of , and vice versa.

  2. Supplementary angles: Two angles whose sum is 180°.

    Introducing Geometry

    are supplementary angles.

    is the supplement of , and vice versa.

     

    Hook: To prevent confusing complementary and supplementary:

    C comes before S in the alphabet, and 90 comes before 180.

    Complementary = 90° Supplementary = 180°

    Tip

    Sometimes more than two angles are combined to form a straight angle. When this occurs, the sum of all the angles is 180°.

     

    Introducing Geometry

  3. Vertical angles: Two angles that are opposite each other when two lines cross. Two sets of vertical angles are formed:

    Introducing Geometry

    Vertical angles are congruent.

    When two lines cross, the adjacent angles are supplementary and the sum of all four angles is 360°.

Angle-Pair Problems

Angle-pair problems tend to ask for an angle's complement or supplement.

Example: If = 35°, what is the size of its complement?
  To find an angle's complement, subtract it from 90°:   90°– 35°= 55°
  Check: Add the angles to be sure their sum is 90°.
Example: If = 35°, what is the size of its supplement?
  To find an angle's supplement, subtract it from 180°:   180°– 35°= 145°
  Check: Add the angles to be sure their sum is 180°.
Example: If = 70°, what are the sizes of the other three angles?
  Solution:
1. because they're vertical angles. Introducing Geometry
  Therefore, = 70°.
2. and are adjacent angles and therefore supplementary.
  Thus, = 110°(180°– 70°= 110°).
3. because they're also vertical angles.
  Therefore, = 110°.
  Check: Add the angles to be sure their sum is 360°.

To solve geometry problems more easily, draw a picture if one is not provided. Try to draw the picture to scale. If the problem presents information about the size of an angle or line segment, label the corresponding part of your picture to reflect the given information. As you begin to find the missing information, label your picture accordingly.

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