# 9.3 Use properties of angles, triangles, and the pythagorean theorem

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By the end of this section, you will be able to:
• Use the properties of angles
• Use the properties of triangles
• Use the Pythagorean Theorem

Before you get started, take this readiness quiz.

1. Solve: $x+3+6=11.$
If you missed this problem, review Solve Equations using the Subtraction and Addition Properties of Equality .
2. Solve: $\frac{a}{45}=\frac{4}{3}.$
If you missed this problem, review Solve Proportions and Their Applications .
3. Simplify: $\sqrt{36+64}.$
If you missed this problem, review Solve Equations using the Subtraction and Addition Properties of Equality .

So far in this chapter, we have focused on solving word problems, which are similar to many real-world applications of algebra. In the next few sections, we will apply our problem-solving strategies to some common geometry problems.

## Use the properties of angles

Are you familiar with the phrase ‘do a $180\text{’?}$ 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 $180$ degrees. See [link] .

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 [link] , $\text{∠}A$ is the angle with vertex at point $A.$ The measure of $\text{∠}A$ is written $m\angle A.$ ∠ A is the angle with vertex at point A .

We measure angles in degrees, and use the symbol $°$ to represent degrees. We use the abbreviation $m$ to for the measure of an angle. So if $\text{∠}A$ is $\text{27°},$ we would write $m\angle A=27.$

If the sum of the measures of two angles is $\text{180°},$ then they are called supplementary angles    . In [link] , each pair of angles is supplementary because their measures add to $\text{180°}.$ Each angle is the supplement of the other. The sum of the measures of supplementary angles is 180° .

If the sum of the measures of two angles is $\text{90°},$ then the angles are complementary angles    . In [link] , each pair of angles is complementary, because their measures add to $\text{90°}.$ Each angle is the complement of the other. The sum of the measures of complementary angles is 90° .

## Supplementary and complementary angles

If the sum of the measures of two angles is $\text{180°},$ then the angles are supplementary.

If $\text{∠}A$ and $\text{∠}B$ are supplementary, then $m\text{∠}A+m\text{∠}B=\text{180°.}$
If the sum of the measures of two angles is $\text{90°},$ then the angles are complementary.
If $\text{∠}A$ and $\text{∠}B$ are complementary, then $m\text{∠}A+m\text{∠}B=\text{90°.}$

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.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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Good
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Jeannette has $5 and$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
What is the expressiin for seven less than four times the number of nickels
How do i figure this problem out.
how do you translate this in Algebraic Expressions
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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