An angle is two rays with the same endpoint that do not lie on the same line. The common endpoint is the vertex of the angle. The rays are the sides of the angles. These two rays have the same endpoint.
The endpoint B is the vertex of the angle. The sides of the angle are rays BA and BC.
Label an angle using three letters, one letter from one side, one letter from the other side, and the letter of the vertex. Remember that the letter of the vertex is in the middle. Label the above angle ∠ABC or ∠CBA.
An angle can also be labeled using the letter of the vertex. This angle could just be labeled ∠Y. You can only use one letter to label an angle if no other angle shares the same vertex.
Perhaps the easiest way to label an angle is to put a number inside its vertex. When you use a number label, make a small arc inside the angle to make sure there is no confusion.
You can label this angle four different ways.
An angle divides a plane into three distinct regions:
•interior of the angle
•angle itself
•exterior of the angle
Points X and Z lie in the interior of the angle.
Points C and D lie on the angle.
Point Q lies in the exterior of the angle.
Based on the drawing, determine whether each of the following statements is true or false.
1.Each of the angles can be labeled in more than one way.
2.∠ABD is the same as ∠DBA.
3.The measure of angle 2 is equal to the measure of ∠ABD.
4.There are two different angles that could be labeled ∠B.
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You measure the length of a line segment using a ruler. You measure the size of an angle using a protractor. The unit of measure for an angle is the degree. A protractor is a semicircle that is marked off in 180 increments. Each increment is one degree.
Protractor Postulate
For every angle there corresponds a number between 0 and 180. The number is called the measure of an angle.
When expressing the measure of an angle, write “the measure of angle ABC is 25 degrees.” This is abbreviated as m∠ABC = 25. The symbol for degrees is °, but it is generally omitted when measurements are written in the abbreviated format. However, if the measurement is given in the angle itself, the degree sign is used.
Notice four things about a protractor.
1.There are numbers from 0 to 180.
2.The numbers are marked from right to left and from left to right around the curved part of the semicircle.
3.If the protractor is made of clear plastic, there is a horizontal line along the base of the protractor. The clear plastic line intersects the protractor at 0 and 180 degrees.
4.There is a mark at the halfway point of the straight side of a protractor.
To use a protractor, follow these painless steps:
Step 1:Place the center marker over the vertex of the angle.
Step 2:Place the horizontal line along the base of the protractor on one of the sides of the angle.
Step 3:Read the number where the other side of the angle intersects the protractor. This is the measure of the angle.
Watch as these angles are measured.
1.What is the measure of this angle?
It measures 30 degrees.
2.Place the protractor on this angle.
Notice that the angle measures 45 degrees.
3.Look at this angle. It measures 120 degrees.
Measure the following angles.
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You can add the measures of two angles together by adding the number of degrees in each angle together. For example, if the m∠A = 30 degrees and the m∠B = 20 degrees, then the m∠A plus the m∠B = 50 degrees.
You can also determine the difference between the size of two angles by subtracting the measure of the smaller angle from the measure of the larger angle.
If the m∠D = 45 degrees and the m∠E = 60 degrees, then the m∠E is 15 degrees larger than the m∠D, since 60 – 45 = 15.
If two angles share the same vertex and a single side, their measures cannot be added if one angle is a subset of the other. For example, in the following illustration the m∠ABD plus the m∠DBC is equal to the m∠ABC.
However, you cannot add the m∠ABD to the m∠ABC, since angle ABD is a subset of angle ABC.
The m∠ABC = 45 degrees and the m∠CBD = 50 degrees.
1.What is the sum of the two angles?
2.What is the difference between the two angles?
3.What is the difference between the m∠ABC and the m∠ABD?
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Once you know how to measure angles you can classify them.
An acute angle measures less than 90 degrees and more than 0 degrees. These are all acute angles.
A right angle measures exactly 90 degrees. These are both right angles.
An obtuse angle measures more than 90 degrees but less than 180 degrees. These are all obtuse angles.
A straight angle measures exactly 180 degrees. This is a straight angle.
Decide whether each of these statements is true or false.
1.A 63-degree angle is an acute angle.
2.A 90-degree angle is a straight angle.
3.A 179-degree angle is an obtuse angle.
4.A 240-degree angle is an obtuse angle.
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If two angles have a total measure of 90 degrees, they are called complementary angles.
Angles ABD and DBC are complementary.
If the measure of angle ABD is 40 degrees, then the measure of angle DBC is 50 degrees.
40 + 50 = 90
To find the complement of an angle, use these two painless steps:
Step 1:Find the measure of the first angle.
Step 2:Subtract the measure of the first angle from 90 degrees.
EXAMPLE:
Find the complement of a 10-degree angle.
Step 1:Find the measure of the first angle. It is given as 10 degrees.
Step 2:Subtract the measure of the first angle from 90 degrees.
90 – 10 = 80
The complement of a 10-degree angle is an 80-degree angle.
Theorem: If two angles are complementary to the same angle, the measures of the angles are equal to each other.
EXAMPLE:
If m∠A = 25
and ∠B is complementary to ∠A
and ∠C is complementary to ∠A,
then m∠B = m∠C.
Because ∠B is complementary to ∠A, m∠B is (90 – 25) or 65.
Because ∠C is complementary to ∠A, m∠C is (90 – 25) or 65.
If two angles have a total measure of 180 degrees, they are called supplementary angles.
Angles ABC and CBD are supplementary.
If ∠ABC measures 125 degrees, ∠CBD must measure 55 degrees, since 125 + 55 = 180.
To find the supplement of an angle, use these two painless steps:
Step 1:Find the measure of the first angle.
Step 2:Subtract the measure of the first angle from 180 degrees.
EXAMPLE:
Find the supplement of a 10-degree angle.
Step 1:Find the measure of the first angle. It is given as 10 degrees.
Step 2:Subtract the measure of the first angle from 180 degrees.
180 – 10 = 170
The supplement of a 10-degree angle is a 170-degree angle.
Theorem: If two angles form a straight line, the angles are supplementary.
EXAMPLE:
Angle ABD and angle DBC form a straight angle. Angle ABD and angle DBC are supplementary to each other.
Theorem: If two angles are supplementary to the same angle, the measures of the angles are equal to each other.
EXAMPLE:
If m∠A = 70
and ∠B is supplementary to ∠A
and ∠C is supplementary to ∠A,
then m∠B = m∠C.
Because ∠B is supplementary to ∠A, m∠B is (180 – 70) or 110.
Because ∠C is supplementary to ∠A, m∠C is (180 – 70) or 110.
An angle bisector is a line or ray that divides an angle into two equal angles.
EXAMPLE:
Ray BD bisects angle ABC. If angle ABC measures 60 degrees, then angle ABD measures 30 degrees, and angle DBC also measures 30 degrees.
30 + 30 = 60
Four angles are formed when two lines intersect. The opposite angles always have the same measure, and they are called vertical angles.
Angles 1 and 3 are vertical angles. Angles 2 and 4 are also vertical angles. The measure of angle 1 is equal to the measure of angle 3. The measure of angle 2 is equal to the measure of angle 4.
If angle 1 measures 70 degrees, then angle 3 must also measure 70 degrees.
If angle 2 measures 110 degrees, then angle 4 must also measure 110 degrees.
Two angles are adjacent if they share a common side and a common vertex but do not share any interior points.
EXAMPLE:
Angle ABC is adjacent to angle CBD. They share side BC. Angle ABD is not adjacent to angle CBD. They share side BD, but they have common interior points.
1.If ∠A measures 59 degrees, what is the measure of its complement?
2.If ∠B measures 12 degrees, what is the measure of its supplement?
3.If ∠C measures 50 degrees and it is bisected, what is the measure of the two resulting angles?
Look at these angles.
4.If ∠1 measures 30 degrees, what is the measure of ∠2?
5.If ∠1 measures 30 degrees, what is the measure of ∠3?
6.If a straight angle is bisected, what is the measure of each of the resulting angles?
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Two figures with exactly the same size and shape are congruent. How do you determine if two angles are congruent? Two angles are congruent if any one of the following conditions are met:
•They have the same measure.
•You can put one on top of the other, and they are identical.
•They are both right angles since both measure 90 degrees.
•They are complements of the same angle. Complements of the same angle always have the same measure.
•They are supplements of the same angle. Supplements of the same angle always have the same measure.
•They are both congruent to the same angle. If two angles are congruent to the same angle, they have the same measure as that angle and they must be congruent to each other.
To show that two angles are congruent, write ∠A ≅ ∠B. The symbol for congruence is a wavy line over an equals sign.
Look at the following diagram. Which angles are congruent to each other?
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It’s possible to use your skills in algebra and what you are learning about geometry to solve word problems.
EXAMPLE:
If an angle is twice as large as its complement, what is the measure of the angle and its complement?
Change this problem to an equation.
The phrases in parentheses are changed to mathematical language.
x (an angle)
= (is)
2 (twice)
90 – x (its complement)
Now write an equation.
x = 2(90 – x)
Solve this equation.
x = 2(90 – x)
x = 180 – 2x
3x = 180
x = 60
The angle is 60 degrees; its complement is 30 degrees.
EXAMPLE:
An angle is 50 degrees less than its supplement. What is the measure of the angle and its supplement?
Change the problem to an equation. Watch as the phrases in parentheses are changed to mathematical language.
x (an angle)
= (is)
180 – x (its supplement)
– 50 (50 degrees less than)
Now put all these expressions together to form an equation.
x = 180 – x – 50
Now solve this equation.
First simplify both sides of the equation.
x = 180 – x – 50
Simplified this equation becomes x = 130 – x.
Simplify both sides of the equation further to get 2x = 130.
Divide both sides by 2, and the result is x = 65.
The angle is 65 degrees and its supplement is 115 degrees.
How well do you know your geometry terms? Fill in the blanks with angle language.
1.___________ angles measure exactly 90 degrees.
2.The sum of two __________ angles is exactly 180 degrees.
3.Two angles with exactly the same measure are ____________.
4.An angle with a measure of less than 90 degrees is a(n) _______ angle.
5.The sum of two ________ angles is exactly 90 degrees.
6.__________ angles are formed when two lines intersect.
7.__________ angles measure greater than 90 degrees and less than 180 degrees.
8.A ray that divides an angle into two congruent angles is called an angle _________.
9.An angle that measures exactly 180 degrees is a(n) ________ angle.
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