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Appendix B: Geometry
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 [latex]180[/latex]?’ 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 [latex]180[/latex] degrees. See the image below.

[latex]\angle A[/latex] is the angle with vertex at [latex]\text{point }A[/latex].

We measure angles in degrees, and use the symbol [latex]^ \circ[/latex] to represent degrees. We use the abbreviation [latex]m[/latex] to for the measure of an angle. So if [latex]\angle A[/latex] is [latex]\text{27}^ \circ [/latex], we would write [latex]m\angle A=27[/latex].
If the sum of the measures of two angles is [latex]\text{180}^ \circ[/latex], then they are called supplementary angles. In the images below, each pair of angles is supplementary because their measures add to [latex]\text{180}^ \circ [/latex]. Each angle is the supplement of the other.
The sum of the measures of supplementary angles is [latex]\text{180}^ \circ [/latex].

The sum of the measures of complementary angles is [latex]\text{90}^ \circ[/latex].

Supplementary and Complementary Angles
If the sum of the measures of two angles is [latex]\text{180}^\circ [/latex], then the angles are supplementary .
If angle [latex]A[/latex] and angle [latex]B[/latex] are supplementary, then [latex]m\angle{A}+m\angle{B}=180^\circ[/latex].
If the sum of the measures of two angles is [latex]\text{90}^\circ[/latex], then the angles are complementary .
If angle [latex]A[/latex] and angle [latex]B[/latex] are complementary, then [latex]m\angle{A}+m\angle{B}=90^\circ[/latex].
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.
- Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.
- Identify what you are looking for.
- Name what you are looking for and choose a variable to represent it.
- Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
- Solve the equation using good algebra techniques.
- Check the answer in the problem and make sure it makes sense.
- 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.
An angle measures [latex]\text{40}^ \circ[/latex].
1. Find its supplement
2. Find its complement
Write the appropriate formula for the situation and substitute in the given information. [latex]m\angle A+m\angle B=90[/latex] Step 5. Solve the equation. [latex]c+40=90[/latex]
[latex]c=50[/latex] Step 6. Check:
[latex]50+40\stackrel{?}{=}90[/latex]
In the following video we show more examples of how to find the supplement and complement of an angle.
Did you notice that the words complementary and supplementary are in alphabetical order just like [latex]90[/latex] and [latex]180[/latex] are in numerical order?
Two angles are supplementary. The larger angle is [latex]\text{30}^ \circ[/latex] more than the smaller angle. Find the measure of both angles.
- Question ID 146497, 146496, 146495. Authored by : Lumen Learning. License : CC BY: Attribution
- Determine the Complement and Supplement of a Given Angle. Authored by : James Sousa (mathispower4u.com). Located at : https://youtu.be/ZQ_L3yJOfqM . License : CC BY: Attribution
- Prealgebra. Provided by : OpenStax. License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]

On solving optimal control problems by indirect and path following methods
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Solve the following initial-value problems starting from y(t=0)=1 and y(t=0)=-1 . Draw both solutions on the same graph
Combining two or more parts to make a whole is called the aggregation structure of addition. One quantity is increased by some amount is called an augmentation structure
We will adapt our Problem Solving Strategy for Geometry Applications. The next example will show how you can use the Problem Solving Strategy for Geometry Applications to answer questions about supplementary and complementary angles
There is no solution that satisfies y(0) = 1; the initial value problem y0 = y x,y(0) = 1 does not have a solution. the questions of existence and uniqueness o
On solving optimal control problems by indirect and path following methods. Read the brief description. Summary of HamPath possibilities