Reasoning and Proof An angle is formed by two rays with the same endpoint. The endpoint is called the vertex. The symbol Ð is used to represent an angle.
Naming Angles
Name the angle by its vertex alone:
Name the angle by its vertex and two points, with the vertex as the middle point:
Name the angle by its vertex and two point, but switch the order of the two points:
Using a Protractor: A protractor is a tool you can use to draw and measure angles. Angles are measured in units called degrees ( ° ).
Classifying Angles
If you take a look around you, you can probably see many types of angles. Angles are classified by their measures.
Classifying Angles |
A right angle is an angle whose measure is exactly 90 ° .
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An acute angle is an angle whose measure is less than 90 ° .
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An obtuse angle is an angle whose measure is between 90 ° and 180 ° .
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A straight angle is an angle whose measure is exactly 180 ° .
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The following angles are all acute angles
The following angles are all obtuse.
The following angles are both right angles
Example: Classify the angles in the figure as acute , right , or obtuse .
Ð A is marked as a right angle.
Ð B is an acute angle because m Ð B is less than 90 ° .
Ð C and Ð are obtuse angles because m Ð C and m Ð D are between 90 ° and 180 ° .
Vertical Angles When two lines intersect, the angles opposite each other are called vertical angles. In the diagram. Ð 1 and Ð 3 are vertical angles, and Ð 2 and Ð 4 are vertical angles. Vertical angles have equal measures.
Key Concept
Complementary and Supplementary Angles
Complementary angles Two angles are complementary if the sum of their measures is 90 ° .
m Ð 1 + m Ð 2 = 90 °
Supplementary angles Two angles are supplementary if the sum of their measures is 180 ° .
m Ð 3 + m Ð 4 = 180 °
Example: Decide whether the angles are complementary , supplementary , or neither .
a. |
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b. |
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Solution: a. The angles are supplementary because 48 ° + 132 ° = 180 °
b. The angles are complementary because 40 ° + 50 ° = 90 °
Alternate Interior Angles
For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle D are called alternate interior angles. Alternate interior angles have the same degree measurement. Angle B and angle C are also alternate interior angles.
Alternate Exterior Angles
For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle D are called alternate exterior angles. Alternate exterior angles have the same degree measurement. Angle B and angle C are also alternate exterior angles.
Corresponding Angles
For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle C are called corresponding angles. Corresponding angles have the same degree measurement. Angle B and angle D are also corresponding angles.
Angle Bisector
An angle bisector is a ray that divides an angle into two equal angles.
Example:
The blue ray on the right is the angle bisector of the angle on the left.
The red ray on the right is the angle bisector of the angle on the left.
Perpendicular Lines Two lines that meet at a right angle are perpendicular.
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