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Guias de estudo > ALGEBRA / TRIG I

Using Properties of Angles to Solve Problems

Learning Outcomes

  • Find the supplement of an angle
  • Find the complement of an angle
Are you familiar with the phrase 'do a 180180?' It means to make a full turn so that you face the opposite direction. It comes from the fact that the measure of an angle that makes a straight line is 180180 degrees. See the image below. The image is a straight line with an arrow on each end. There is a dot in the center. There is an arrow pointing from one side of the dot to the other, and the angle is marked as 180 degrees. An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle and the common endpoint is called the vertex. An angle is named by its vertex. In the image below, ∠A\angle A is the angle with vertex at point AA. The measure of ∠A\angle A is written m∠Am\angle A.
∠A\angle A is the angle with vertex at point A\text{point }A. The image is an angle made up of two rays. The angle is labeled with letter A.
We measure angles in degrees, and use the symbol ∘^ \circ to represent degrees. We use the abbreviation mm to for the measure of an angle. So if ∠A\angle A is 27∘\text{27}^ \circ , we would write m∠A=27∘m\angle A=27^ \circ. If the sum of the measures of two angles is 180∘\text{180}^ \circ, then they are called supplementary angles. In the images below, each pair of angles is supplementary because their measures add to 180∘\text{180}^ \circ . Each angle is the supplement of the other. The sum of the measures of supplementary angles is 180∘\text{180}^ \circ . Part a shows a 120 degree angle next to a 60 degree angle. Together, the angles form a straight line. Below the image, it reads 120 degrees plus 60 degrees equals 180 degrees. Part b shows a 45 degree angle attached to a 135 degree angle. Together, the angles form a straight line. Below the image, it reads 45 degrees plus 135 degrees equals 180 degrees. If the sum of the measures of two angles is 90∘\text{90}^ \circ, then the angles are complementary angles. In the images below, each pair of angles is complementary, because their measures add to 90∘\text{90}^ \circ. Each angle is the complement of the other. The sum of the measures of complementary angles is 90∘\text{90}^ \circ. Part a shows a 50 degree angle next to a 40 degree angle. Together, the angles form a right angle. Below the image, it reads 50 degrees plus 40 degrees equals 90 degrees. Part b shows a 60 degree angle attached to a 30 degree angle. Together, the angles form a right angle. Below the image, it reads 60 degrees plus 30 degrees equals 90 degrees.

Supplementary and Complementary Angles

If the sum of the measures of two angles is 180∘\text{180}^\circ , then the angles are supplementary.

If angle AA and angle BB are supplementary, then m∠A+m∠B=180∘m\angle{A}+m\angle{B}=180^\circ.

If the sum of the measures of two angles is 90∘\text{90}^\circ, then the angles are complementary.

If angle AA and angle BB are complementary, then m∠A+m∠B=90∘m\angle{A}+m\angle{B}=90^\circ.

In this section and the next, you will be introduced to some common geometry formulas. We will adapt our Problem Solving Strategy for Geometry Applications. The geometry formula will name the variables and give us the equation to solve. In addition, since these applications will all involve geometric shapes, it will be helpful to draw a figure and then label it with the information from the problem. We will include this step in the Problem Solving Strategy for Geometry Applications.

Use a Problem Solving Strategy for Geometry Applications.

  1. Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.
  2. Identify what you are looking for.
  3. Name what you are looking for and choose a variable to represent it.
  4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
  5. Solve the equation using good algebra techniques.
  6. Check the answer in the problem and make sure it makes sense.
  7. Answer the question with a complete sentence.
The next example will show how you can use the Problem Solving Strategy for Geometry Applications to answer questions about supplementary and complementary angles.

example

An angle measures 40∘\text{40}^ \circ. 1. Find its supplement 2. Find its complement Solution
1.
Step 1. Read the problem. Draw the figure and label it with the given information. .
Step 2. Identify what you are looking for. The supplement of a 40°40°angle.
Step 3. Name. Choose a variable to represent it. Let s=s=the measure of the supplement.
Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information. m∠A+m∠B=180°m\angle A+m\angle B=180° s+40°=180°s+40°=180°
Step 5. Solve the equation. s=140°s=140°
Step 6. Check: 140°+40°=?180°140°+40°\stackrel{?}{=}180° 180°=180°✓180°=180°\checkmark
Step 7. Answer the question. The supplement of the 40°40°angle is 140°140°.
2.
Step 1. Read the problem. Draw the figure and label it with the given information. .
Step 2. Identify what you are looking for.
The complement of a 40°40°angle.
Step 3. Name. Choose a variable to represent it. Let c=c=the measure of the complement.
Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information. m∠A+m∠B=90°m\angle A+m\angle B=90°
Step 5. Solve the equation. c+40°=90°c+40°=90° c=50°c=50°
Step 6. Check: 50°+40°=?90°50°+40°\stackrel{?}{=}90° 90°=90°✓90°=90°\quad\checkmark
Step 7. Answer the question. The complement of the 40°40°angle is 50°50°.
 

try it

[ohm_question]146495[/ohm_question]
In the following video we show more examples of how to find the supplement and complement of an angle. https://youtu.be/ZQ_L3yJOfqM Did you notice that the words complementary and supplementary are in alphabetical order just like 9090 and 180180 are in numerical order?

Exercises

Two angles are supplementary. The larger angle is 30∘\text{30}^ \circ more than the smaller angle. Find the measure of both angles.

Answer: Solution:

Step 1. Read the problem. Draw the figure and label it with the given information. .
Step 2. Identify what you are looking for. The measures of both angles.
Step 3. Name. Choose a variable to represent it. The larger angle is 30° more than the smaller angle. Let a=a= measure of smaller angle a+30=a+30= measure of larger angle
Step 4. Translate. Write the appropriate formula and substitute. m∠A+m∠B=180m\angle A+m\angle B=180
Step 5. Solve the equation. (a+30)+a=180(a+30)+a=180 2a+30=1802a+30=180 2a=1502a=150 a=75=a=75= measure of smaller angle. a+30=a+30= measure of larger angle. 75+3075+30 105105
Step 6. Check: m∠A+m∠B=180°m\angle A+m\angle B=180° 75°+105°=?180°75°+105°\stackrel{?}{=}180° 180°=180°✓180°=180°\quad\checkmark
Step 7. Answer the question. The measures of the angle are 75°75°and 105°105°.
 

 

try it

[ohm_question]146496[/ohm_question] [ohm_question]146497[/ohm_question]

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