# 9.3 Use properties of angles, triangles, and the pythagorean theorem  (Page 5/15)

 Page 5 / 15

Use the Pythagorean Theorem to find the length of the leg.

8

Use the Pythagorean Theorem to find the length of the leg.

12

Kelvin is building a gazebo and wants to brace each corner by placing a $\text{10-inch}$ wooden bracket diagonally as shown. How far below the corner should he fasten the bracket if he wants the distances from the corner to each end of the bracket to be equal? Approximate to the nearest tenth of an inch.

## Solution

 Step 1. Read the problem. Step 2. Identify what you are looking for. the distance from the corner that the bracket should be attached Step 3. Name. Choose a variable to represent it. Let x = the distance from the corner Step 4. Translate. Write the appropriate formula. Substitute. Step 5. Solve the equation. Isolate the variable. Use the definition of the square root. Simplify. Approximate to the nearest tenth. Step 6. Check: Yes. Step 7. Answer the question. Kelvin should fasten each piece of wood approximately 7.1" from the corner.

John puts the base of a $\text{13-ft}$ ladder $5$ feet from the wall of his house. How far up the wall does the ladder reach?

12 feet

Randy wants to attach a $\text{17-ft}$ string of lights to the top of the $\text{15-ft}$ mast of his sailboat. How far from the base of the mast should he attach the end of the light string?

8 feet

## Key concepts

• Supplementary and Complementary Angles
• If the sum of the measures of two angles is 180°, then the angles are supplementary.
• If $\text{∠}A$ and $\text{∠}B$ are supplementary, then $m\text{∠}A+m\text{∠}B=180$ .
• If the sum of the measures of two angles is 90°, then the angles are complementary.
• If $\text{∠}A$ and $\text{∠}B$ are complementary, then $m\text{∠}A+m\text{∠}B=90$ .
• Solve 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.
• Sum of the Measures of the Angles of a Triangle

• For any $\text{Δ}ABC,$ the sum of the measures is 180°
• $m\text{∠}A+m\text{∠}B=180$
• Right Triangle

• A right triangle is a triangle that has one 90° angle, which is often marked with a $\text{⦜}$ symbol.
• Properties of Similar Triangles
• If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths have the same ratio.

## Practice makes perfect

Use the Properties of Angles

In the following exercises, find the supplement and the complement of the given angle.

$\text{53°}$

1. 127°
2. 37°

$\text{16°}$

$\text{29°}$

1. 151°
2. 61°

$\text{72°}$

In the following exercises, use the properties of angles to solve.

Find the supplement of a $\text{135°}$ angle.

45°

Find the complement of a $\text{38°}$ angle.

Find the complement of a $27.5°$ angle.

62.5°

Find the supplement of a $109.5°$ angle.

Two angles are supplementary. The larger angle is $\text{56°}$ more than the smaller angle. Find the measures of both angles.

62°, 118°

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
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it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
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how did you get the value of 2000N.What calculations are needed to arrive at it
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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.
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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.
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why surface tension is zero at critical temperature
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