Geometry Review Assignment Find the Measure of Angle a Answers

Math Models

28 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem

Learning Objectives

By the end of this section, you will be able to:

Solve applications using properties of triangles
Apply the Pythagorean Theorem
Solve applications using rectangle properties

Solve Applications Using Properties of Triangles

In this section we will use some mutual geometry formulas. Nosotros will adapt our problem-solving strategy so that we can solve geometry applications. The geometry formula will proper name the variables and requite us the equation to solve. In add-on, since these applications will all involve shapes of some sort, almost people find it helpful to describe a figure and label it with the given data. We will include this in the first step of the problem solving strategy for geometry applications.

Solve Geometry Applications.

Read the problem and brand sure all the words and ideas are understood. Draw the figure and label it with the given data.
Place what nosotros are looking for.
Label what nosotros are looking for by choosing a variable to represent it.
Translate into an equation past writing the appropriate formula or model for the situation. Substitute in the given information.
Solve the equation using good algebra techniques.
Check the answer past substituting information technology back into the equation solved in step v and by making sure it makes sense in the context of the trouble.
Answer the question with a complete sentence.

We volition offset geometry applications by looking at the properties of triangles. Let'southward review some basic facts virtually triangles. Triangles have three sides and three interior angles. Usually each side is labeled with a lowercase letter of the alphabet to match the capital alphabetic character of the opposite vertex.

The plural of the word vertex is vertices. All triangles have three vertices. Triangles are named past their vertices: The triangle in (Effigy) is chosen $\text{△}ABC.$

Triangle ABC has vertices A, B, and C. The lengths of the sides are a, b, and c.

A triangle with vertices A, B, and C. The sides opposite these vertices are marked a, b, and c, respectively.

The three angles of a triangle are related in a special way. The sum of their measures is $180\text{°}.$ Notation that we read $m\text{∠}A$ as "the measure out of bending A." So in $\text{△}ABC$ in (Figure),

$m\text{∠}A+m\text{∠}B+m\text{∠}C=180\text{°}$

Because the perimeter of a figure is the length of its boundary, the perimeter of $\text{△}ABC$ is the sum of the lengths of its iii sides.

$P=a+b+c$

To find the area of a triangle, we need to know its base and height. The height is a line that connects the base to the reverse vertex and makes a $90\text{°}$ angle with the base. We volition draw $\text{△}ABC$ once again, and at present show the height, h. See (Effigy).

The formula for the surface area of $\text{△}ABC$ is $A=\frac{1}{2}bh,$ where b is the base and h is the meridian.

A triangle with vertices A, B, and C. The sides opposite these vertices are marked a, b, and c, respectively. The side b is parallel to the bottom of the page, and it has a dashed line drawn from vertex B to it. This line is marked h and makes a right angle with side b.

Triangle Backdrop

For $\text{△}ABC$

Angle measures:

$m\text{∠}A+m\text{∠}B+m\text{∠}C=180$

The sum of the measures of the angles of a triangle is $180\text{°}.$

Perimeter:

$P=a+b+c$

The perimeter is the sum of the lengths of the sides of the triangle.

Area:

$A=\frac{1}{2}bh,b=\text{base},h=\text{height}$

The area of a triangle is half the base times the summit.

The measures of 2 angles of a triangle are 55 and 82 degrees. Find the mensurate of the third bending.

The measures of ii angles of a triangle are 31 and 128 degrees. Find the mensurate of the tertiary bending.

21 degrees

The measures of two angles of a triangle are 49 and 75 degrees. Find the measure of the 3rd angle.

56 degrees

The perimeter of a triangular garden is 24 feet. The lengths of ii sides are four feet and nine feet. How long is the 3rd side?

The perimeter of a triangular garden is 48 feet. The lengths of two sides are 18 feet and 22 anxiety. How long is the third side?

8 feet

The lengths of two sides of a triangular window are seven feet and five feet. The perimeter is eighteen anxiety. How long is the tertiary side?

6 feet

The area of a triangular church window is 90 square meters. The base of the window is 15 meters. What is the window'southward height?

The expanse of a triangular painting is 126 square inches. The base is 18 inches. What is the height?

fourteen inches

A triangular tent door has surface area 15 square anxiety. The height is five feet. What is the base?

half dozen feet

The triangle properties we used so far apply to all triangles. Now nosotros will expect at one specific type of triangle—a right triangle. A right triangle has 1 $90\text{°}$ angle, which we normally mark with a small square in the corner.

A right triangle with the largest angle marked 90 degrees.

Right Triangle

A right triangle has one $90\text{°}$ angle, which is ofttimes marked with a square at the vertex.

One bending of a right triangle measures $28\text{°}.$ What is the measure out of the third angle?

1 angle of a right triangle measures $56\text{°}.$ What is the mensurate of the other small angle?

$34\text{°}$

One angle of a right triangle measures $45\text{°}.$ What is the mensurate of the other small angle?

$45\text{°}$

In the examples nosotros take seen so far, we could depict a figure and label it directly after reading the problem. In the adjacent case, we will take to define i bending in terms of another. Nosotros will wait to depict the figure until we write expressions for all the angles nosotros are looking for.

The mensurate of one angle of a right triangle is 20 degrees more the measure of the smallest angle. Detect the measures of all 3 angles.

The measure out of one angle of a correct triangle is l° more than the measure of the smallest angle. Detect the measures of all three angles.

$20\text{°},70\text{°},90\text{°}$

The measure of one angle of a right triangle is 30° more than the mensurate of the smallest angle. Find the measures of all three angles.

$30\text{°},60\text{°},90\text{°}$

Use the Pythagorean Theorem

We have learned how the measures of the angles of a triangle chronicle to each other. Now, we volition learn how the lengths of the sides relate to each other. An important holding that describes the relationship among the lengths of the three sides of a right triangle is chosen the Pythagorean Theorem. This theorem has been used around the globe since ancient times. It is named subsequently the Greek philosopher and mathematician, Pythagoras, who lived around 500 BC.

Before we country the Pythagorean Theorem, we need to introduce some terms for the sides of a triangle. Remember that a right triangle has a $90\text{°}$ angle, marked with a pocket-size square in the corner. The side of the triangle opposite the $90\text{°}$ bending is called the hypotenuse and each of the other sides are called legs.

Three right triangles with different orientations. The right angles are marked with two small lines that make a small square with the angle. Opposite these angles, hypotenuse is written. The other sides are marked

The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. Information technology states that in any right triangle, the sum of the squares of the lengths of the ii legs equals the foursquare of the length of the hypotenuse. In symbols nosotros say: in any right triangle, ${a}^{2}+{b}^{2}={c}^{2},$ where $a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b$ are the lengths of the legs and $c$ is the length of the hypotenuse.

Writing the formula in every exercise and saying it aloud as you write it, may help you call back the Pythagorean Theorem.

The Pythagorean Theorem

In whatever correct triangle, ${a}^{2}+{b}^{2}={c}^{2}.$

A right triangle with sides marked a, b, and c. The side marked c is the hypotenuse.

where a and b are the lengths of the legs, c is the length of the hypotenuse.

To solve exercises that use the Pythagorean Theorem, nosotros will need to find square roots. We take used the annotation $\sqrt{m}$ and the definition:

If $m={n}^{2},$ and so $\sqrt{m}=n,$ for $n\ge 0.$

For example, nosotros found that $\sqrt{25}$ is 5 because $25={5}^{2}.$

Considering the Pythagorean Theorem contains variables that are squared, to solve for the length of a side in a right triangle, we volition have to utilize foursquare roots.

Use the Pythagorean Theorem to notice the length of the hypotenuse shown below.

A right triangle with legs marked 3 and 4.

Use the Pythagorean Theorem to find the length of the hypotenuse in the triangle shown beneath.

A right triangle with legs marked 6 and 8. The hypotenuse is marked c.

$c=10$

Use the Pythagorean Theorem to find the length of the hypotenuse in the triangle shown beneath.

No Alt Text

$c=13$

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

A right angle with one leg marked 5. The hypotenuse is labeled 13.

Use the Pythagorean Theorem to observe the length of the leg in the triangle shown below.

A right triangle with legs marked b and 15. The hypotenuse is marked 17.

Use the Pythagorean Theorem to detect the length of the leg in the triangle shown beneath.

A right triangle with legs marked b and 9. The hypotenuse is marked 15.

A gazebo is shown. In one of its corners, a triangle is made with the wood. The hypotenuse is marked 10 inches, and one of the legs is marked x

Kelvin is building a gazebo and wants to brace each corner by placing a $10\text{″}$ slice of forest diagonally as shown above.

If he fastens the wood and so that the ends of the caryatid are the same altitude from the corner, what is the length of the legs of the correct triangle formed? Gauge to the nearest tenth of an inch.

John puts the base of operations of a 13-pes ladder five anxiety from the wall of his house as shown below. How far upwardly the wall does the ladder reach?

A house is shown with a ladder leaning against it. The ladder is marked 13', and the distance from the house to the base of the ladder is marked 5'.

12 anxiety

Randy wants to attach a 17 foot string of lights to the top of the 15 pes mast of his sailboat, as shown below. How far from the base of the mast should he attach the stop of the calorie-free string?

A sailboat is shown with a 15' mast (the straight tall part). From the top of the mast, a series of colored dots stretches down to the back of the boat and is marked 17'.

8 feet

Solve Applications Using Rectangle Properties

You may already be familiar with the properties of rectangles. Rectangles have four sides and four right $\left(90\text{°}\right)$ angles. The opposite sides of a rectangle are the same length. We refer to 1 side of the rectangle as the length, Fifty, and its adjacent side every bit the width, West.

A rectangle with sides marked W and L.

The distance around this rectangle is $L+W+L+W,$ or $2L+2W.$ This is the perimeter, P, of the rectangle.

$P=2L+2W$

What about the area of a rectangle? Imagine a rectangular rug that is two-feet long by iii-feet wide. Its surface area is 6 square feet. There are vi squares in the figure.

A rectangles composed of 6 squares that is three high and two wide. The height is marked 3 and the width is marked 2.

$\begin{array}{c}A=6\hfill \\ A=2·3\hfill \\ A=L·W\hfill \end{array}$

The area is the length times the width.

The formula for the expanse of a rectangle is $A=LW.$

Backdrop of Rectangles

Rectangles have four sides and four right $\left(90\text{°}\right)$ angles.

The lengths of opposite sides are equal.

The perimeter of a rectangle is the sum of twice the length and twice the width.

$P=2L+2W$

The area of a rectangle is the product of the length and the width.

$A=L·W$

The length of a rectangle is 32 meters and the width is twenty meters. What is the perimeter?

The length of a rectangle is 120 yards and the width is l yards. What is the perimeter?

340 yards

The length of a rectangle is 62 feet and the width is 48 feet. What is the perimeter?

220 anxiety

The surface area of a rectangular room is 168 foursquare feet. The length is fourteen feet. What is the width?

The area of a rectangle is 598 foursquare feet. The length is 23 anxiety. What is the width?

26 feet

The width of a rectangle is 21 meters. The area is 609 square meters. What is the length?

29 meters

Find the length of a rectangle with perimeter 50 inches and width 10 inches.

Detect the length of a rectangle with: perimeter fourscore and width 25.

fifteen

Find the length of a rectangle with: perimeter xxx and width half dozen.

nine

We have solved problems where either the length or width was given, along with the perimeter or area; now we will learn how to solve problems in which the width is defined in terms of the length. We will expect to depict the figure until we write an expression for the width so that we can label ane side with that expression.

The width of a rectangle is two feet less than the length. The perimeter is 52 feet. Find the length and width.

The width of a rectangle is seven meters less than the length. The perimeter is 58 meters. Find the length and width.

xviii meters, 11 meters

The length of a rectangle is eight feet more than than the width. The perimeter is 60 feet. Observe the length and width.

xix anxiety, xi feet

The length of a rectangle is four centimeters more than twice the width. The perimeter is 32 centimeters. Detect the length and width.

The length of a rectangle is viii more than twice the width. The perimeter is 64. Find the length and width.

24, eight

The width of a rectangle is six less than twice the length. The perimeter is 18. Notice the length and width.

5, four

The perimeter of a rectangular swimming pool is 150 feet. The length is fifteen feet more the width. Find the length and width.

The perimeter of a rectangular pond puddle is 200 feet. The length is 40 feet more than the width. Find the length and width.

lxx feet, 30 anxiety

The length of a rectangular garden is 30 yards more the width. The perimeter is 300 yards. Find the length and width.

90 yards, lx yards

Central Concepts

Problem-Solving Strategy for Geometry Applications
1. Read the trouble and make all the words and ideas are understood. Depict the effigy and label it with the given data.
2. Identify what we are looking for.
3. Name what nosotros are looking for past choosing a variable to represent information technology.
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.
Triangle Properties For $\text{△}ABC$
Bending measures:
- $m\text{∠}A+m\text{∠}B+m\text{∠}C=180$
Perimeter:
- $P=a+b+c$
Area:
- $A=\frac{1}{2}bh,\phantom{\rule{0.2em}{0ex}}\text{b}=\text{base},\text{h}=\text{height}$
A correct triangle has 1 $90\text{°}$ angle.
The Pythagorean Theorem In whatsoever right triangle, ${a}^{2}+{b}^{2}={c}^{2}$ where c is the length of the hypotenuse and a and b are the lengths of the legs.
Properties of Rectangles

Do Makes Perfect

Solving Applications Using Triangle Properties

In the post-obit exercises, solve using triangle backdrop.

The measures of two angles of a triangle are 26 and 98 degrees. Discover the measure of the third angle.

56 degrees

The measures of two angles of a triangle are 61 and 84 degrees. Find the measure of the third angle.

The measures of ii angles of a triangle are 105 and 31 degrees. Notice the measure of the third angle.

44 degrees

The measures of two angles of a triangle are 47 and 72 degrees. Find the mensurate of the third angle.

The perimeter of a triangular pool is 36 yards. The lengths of two sides are x yards and 15 yards. How long is the third side?

eleven feet

A triangular courtyard has perimeter 120 meters. The lengths of 2 sides are xxx meters and 50 meters. How long is the tertiary side?

If a triangle has sides half dozen feet and 9 feet and the perimeter is 23 feet, how long is the 3rd side?

eight anxiety

If a triangle has sides 14 centimeters and eighteen centimeters and the perimeter is 49 centimeters, how long is the third side?

A triangular flag has base one foot and height i.five foot. What is its surface area?

0.75 sq. ft.

A triangular window has base eight feet and tiptop six feet. What is its area?

What is the base of a triangle with area 207 foursquare inches and height 18 inches?

23 inches

What is the top of a triangle with expanse 893 square inches and base 38 inches?

One angle of a right triangle measures 33 degrees. What is the measure of the other small angle?

One angle of a right triangle measures 51 degrees. What is the measure of the other small bending?

I angle of a right triangle measures 22.v degrees. What is the mensurate of the other small angle?

67.five

One angle of a correct triangle measures 36.5 degrees. What is the mensurate of the other small bending?

The perimeter of a triangle is 39 anxiety. One side of the triangle is i foot longer than the 2d side. The third side is 2 feet longer than the second side. Observe the length of each side.

13 ft., 12 ft., 14 ft.

The perimeter of a triangle is 35 feet. One side of the triangle is 5 anxiety longer than the second side. The third side is three feet longer than the second side. Observe the length of each side.

1 side of a triangle is twice the shortest side. The 3rd side is five feet more than the shortest side. The perimeter is 17 feet. Find the lengths of all three sides.

three ft., six ft., viii ft.

Ane side of a triangle is three times the shortest side. The third side is three feet more than the shortest side. The perimeter is 13 feet. Discover the lengths of all 3 sides.

The two smaller angles of a right triangle have equal measures. Notice the measures of all 3 angles.

$45\text{°},45\text{°},90\text{°}$

The measure of the smallest bending of a right triangle is 20° less than the measure of the next larger angle. Detect the measures of all iii angles.

The angles in a triangle are such that i angle is twice the smallest angle, while the third angle is three times equally big as the smallest angle. Observe the measures of all three angles.

$30\text{°},60\text{°},90\text{°}$

The angles in a triangle are such that one bending is twenty° more than the smallest angle, while the third angle is 3 times as big equally the smallest angle. Find the measures of all iii angles.

Use the Pythagorean Theorem

In the post-obit exercises, use the Pythagorean Theorem to find the length of the hypotenuse.

A right triangle with legs marked 9 and 12.

A right triangle with legs marked 16 and 12.

A right triangle with legs marked 15 and 20.

A right triangle with legs marked 5 and 12.

In the following exercises, use the Pythagorean Theorem to detect the length of the leg. Round to the nearest tenth, if necessary.

A right triangle with one leg marked 6 and hypotenuse marked 10.

A right triangle with one leg marked 8 and hypotenuse marked 17.

A right triangle with one leg marked 5 and hypotenuse marked 13.

A right triangle with one leg marked 16 and hypotenuse marked 20.

A right triangle with one leg marked 8 and hypotenuse marked 13.

10.2

A right triangle with both legs marked 6.

No Alt Text

9.8

A right triangle with legs marked 5 and 7.

In the following exercises, solve using the Pythagorean Theorem. Estimate to the nearest tenth, if necessary.

A xiii-foot string of lights will be attached to the tiptop of a 12-foot pole for a holiday display, as shown beneath. How far from the base of the pole should the end of the string of lights be anchored?

A right triangle with one leg marked 12 and hypotenuse marked 13.

5 feet

Pam wants to put a banner across her garage door, as shown below, to congratulate her son for his higher graduation. The garage door is 12 feet high and 16 feet broad. How long should the banner be to fit the garage door?

A house is shown with a banner over the garage door. The garage door is marked 16 ft wide and 12 ft high.

Chi is planning to put a path of paving stones through her flower garden, every bit shown below. The flower garden is a square with side 10 feet. What will the length of the path be?

A square garden is shown that is marked 10' on the side. There is a path of stones along the diagonal of the square.

fourteen.1 feet

Brian borrowed a 20 pes extension ladder to use when he paints his firm. If he sets the base of the ladder 6 feet from the house, as shown below, how far upwardly will the acme of the ladder achieve?

A house is shown with a ladder leaning against it. The ladder is marked 20', and the distance from the house to the base of the ladder is marked 6'.

Solve Applications Using Rectangle Properties

In the following exercises, solve using rectangle properties.

The length of a rectangle is 85 feet and the width is 45 anxiety. What is the perimeter?

260 feet

The length of a rectangle is 26 inches and the width is 58 inches. What is the perimeter?

A rectangular room is 15 anxiety wide by fourteen anxiety long. What is its perimeter?

58 feet

A driveway is in the shape of a rectangle xx feet wide by 35 feet long. What is its perimeter?

The area of a rectangle is 414 foursquare meters. The length is eighteen meters. What is the width?

23 meters

The expanse of a rectangle is 782 square centimeters. The width is 17 centimeters. What is the length?

The width of a rectangular window is 24 inches. The expanse is 624 square inches. What is the length?

26 inches

The length of a rectangular poster is 28 inches. The area is 1316 square inches. What is the width?

Notice the length of a rectangle with perimeter 124 and width 38.

Find the width of a rectangle with perimeter 92 and length 19.

Observe the width of a rectangle with perimeter xvi.ii and length iii.two.

4.nine

Find the length of a rectangle with perimeter twenty.two and width vii.8.

The length of a rectangle is nine inches more than than the width. The perimeter is 46 inches. Notice the length and the width.

16 in., vii in.

The width of a rectangle is eight inches more than the length. The perimeter is 52 inches. Discover the length and the width.

The perimeter of a rectangle is 58 meters. The width of the rectangle is 5 meters less than the length. Find the length and the width of the rectangle.

17 m, 12 one thousand

The perimeter of a rectangle is 62 feet. The width is seven feet less than the length. Find the length and the width.

The width of the rectangle is 0.7 meters less than the length. The perimeter of a rectangle is 52.6 meters. Discover the dimensions of the rectangle.

13.5 m length, 12.8 m width

The length of the rectangle is 1.1 meters less than the width. The perimeter of a rectangle is 49.4 meters. Detect the dimensions of the rectangle.

The perimeter of a rectangle is 150 feet. The length of the rectangle is twice the width. Find the length and width of the rectangle.

50 ft., 25 ft.

The length of a rectangle is three times the width. The perimeter of the rectangle is 72 anxiety. Detect the length and width of the rectangle.

The length of a rectangle is three meters less than twice the width. The perimeter of the rectangle is 36 meters. Find the dimensions of the rectangle.

7 m width, 11 g length

The length of a rectangle is 5 inches more than twice the width. The perimeter is 34 inches. Find the length and width.

The perimeter of a rectangular field is 560 yards. The length is 40 yards more than than the width. Notice the length and width of the field.

160 yd., 120 yd.

The perimeter of a rectangular atrium is 160 feet. The length is 16 feet more than the width. Find the length and width of the atrium.

A rectangular parking lot has perimeter 250 feet. The length is five anxiety more than twice the width. Find the length and width of the parking lot.

85 ft., xl ft.

A rectangular rug has perimeter 240 inches. The length is 12 inches more than twice the width. Detect the length and width of the rug.

Everyday Math

Christa wants to put a contend effectually her triangular flowerbed. The sides of the flowerbed are six feet, eight feet and x feet. How many feet of fencing will she need to enclose her flowerbed?

24 feet

Jose just removed the children'due south playset from his dorsum yard to make room for a rectangular garden. He wants to put a fence around the garden to keep out the domestic dog. He has a 50 foot curlicue of fence in his garage that he plans to utilise. To fit in the backyard, the width of the garden must be 10 feet. How long tin he make the other length?

Writing Exercises

If you need to put tile on your kitchen floor, do yous demand to know the perimeter or the area of the kitchen? Explain your reasoning.

area; answers will vary

If you need to put a fence around your backyard, do you need to know the perimeter or the area of the lawn? Explain your reasoning.

Look at the ii figures below.

On the left, we have a rectangle with height 2 and width 8. On the right, we have a square with height 4 and width 4.

ⓐ Which figure looks like it has the larger area?
ⓑ Which looks like information technology has the larger perimeter?
ⓒ Now summate the area and perimeter of each figure.
ⓓ Which has the larger area?
ⓔ Which has the larger perimeter?

ⓐ Answers volition vary.
ⓑ Answers will vary.
ⓒ Answers will vary.
ⓓ The areas are the aforementioned.
ⓔ The 2×8 rectangle has a larger perimeter than the four×4 square.

Write a geometry give-and-take problem that relates to your life experience, then solve it and explain all your steps.

Self Check

ⓐ Later completing the exercises, employ this checklist to evaluate your mastery of the objectives of this section.

This is a table that has four rows and four columns. In the first row, which is a header row, the cells read from left to right

ⓑ What does this checklist tell you most your mastery of this section? What steps will you have to meliorate?

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Source: https://opentextbc.ca/elementaryalgebraopenstax/chapter/solve-geometry-applications-triangles-rectangles-and-the-pythagorean-theorem/

Geometry Review Assignment Find the Measure of Angle a Answers

28 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem

Learning Objectives

Solve Applications Using Properties of Triangles

Use the Pythagorean Theorem

Solve Applications Using Rectangle Properties

Central Concepts

Do Makes Perfect

Everyday Math

Writing Exercises

Self Check

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