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# Geometry Word Problems Study Guide (page 2)

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Updated on Aug 24, 2011

#### Example

If angles A and B are vertical angles, and the measure of angle A is 65°, what is the measure of angle B?

We are given the measure of angle A, and we are told that it and angle B are vertical angles.

We are looking for the measure of angle B.

Underline the keywords and words that indicate formulas.

The word vertical tells us that angles A and B are congruent—their measures are the same.

Cross out extra information and translate words into numbers.

There is no extra information in this problem.

List the possible operations.

We don't need to perform any operations or write any number sentences. Vertical angles are congruent, so since angle A is 65°, angle B is 65°.

We didn't perform any operations, so we don't have any work to check. We can check the definition of vertical angles, which states that two vertical angles are equal in measure.

#### Example

If angles A and B are complementary angles, and the measure of angle A is 12°, what is the measure of angle B?

We are given the measure of angle A, and we are told that it and angle B are complementary angles.

We are looking for the measure of angle B.

Underline the keywords and words that indicate formulas.

The word complementary tells us that the measures of angles A and B total 90°.

Cross out extra information and translate words into numbers.

There is no extra information in this problem.

List the possible operations.

Since complementary angles add to 90° and we have the measure of one angle, we must use subtraction to find the measure of the other angle.

Write number sentences for each operation.

90 – 12

Solve the number sentences and decide which answer is reasonable.

90 – 12 = 78°

Since complementary angles add to 90°, the sum of the measures of angles A and B should be 90°: 78 + 12 = 90°.

Not all geometry questions require computations. Sometimes, a word problem may test our knowledge of a rule or property. Look closely at the question being asked to see if it is asking you for a number, word, or phrase.

#### Example

If a triangle contains a 91° angle, what kind of triangle is it?

We are given the measure of one angle of a triangle.

We must identify what kind of triangle it is.

Underline the keywords and words that indicate formulas.

There are no keywords in this problem.

Cross out extra information and translate words into numbers.

There is no extra information in this problem.

List the possible operations.

A 91° angle is an obtuse angle, because it is greater than 90° in measure. This means that the triangle is an obtuse triangle. We don't need to write any number sentences.

The definition of an obtuse triangle is a triangle that contains an angle that is greater than 90°, so our answer is correct.

#### Caution

Even if a word problem does require addition, subtraction, multiplication, or division, sometimes our answer can still be a word or phrase and not a number. The first step of the word-problem-solving process is always the most important: read the entire word problem. Equally important is determining the question being asked so that we know how to go about solving the problem.

#### Example

If a triangle contains two 33° angles, in what two ways can this triangle be classified?

We are given the measure of two angles of a triangle.

We are looking for two ways to name the triangle.

Underline the keywords and words that indicate formulas.

There are no keywords in this problem.

Cross out extra information and translate words into numbers.

There is no extra information in this problem.

List the possible operations.

Triangles can be classified as acute, right, obtuse, equilateral, and isosceles. A triangle is equilateral if all three angles measure 60°. Since two of the angles in this triangle are 33°, the triangle is not equilateral. However, the triangle is isosceles, because an isosceles triangle has two congruent angles. We have one name for this triangle, but we need another. In order to tell if this triangle is acute, right, or obtuse, we must find the measure of the third angle. There are 180° in a triangle, so we must find the sum of the two 33° angles, and subtract it from 180°.

Write number sentences for each operation.

33 + 33

Solve the number sentences and decide which answer is reasonable.

33 + 33 = 66°

Write number sentences for each operation.

Now subtract that sum from 180°:

180 – 66

Solve the number sentences and decide which answer is reasonable.

180 – 66 = 114°

Since this angle is greater than 90°, the triangle is also an obtuse triangle.

There are 180° in a triangle, so we can check that the third angle is 114° by finding the sum of the measures of the three angles: 33 + 33 + 114 = 180°.

### Triangles: Subject Review

The formula for area of a triangle is A = bh, where A is the area, b is the base of the triangle, and h is the height of the triangle. If the base of a triangle is 4 feet and the height is 5 feet, then the area of the triangle is equal to (4)(5) = (20) = 10 feet2.

The perimeter of a triangle is equal to the sum of the lengths of its three sides. If a triangle is equilateral, then its perimeter can be found by the formula P = 3s, where s is the length of one side of the triangle. If a triangle is isosceles, then its perimeter is equal to twice the length of one of the sides that are equal plus the length of the remaining side. A triangle with sides that measure 4 inches, 8 inches, and 6 inches has a perimeter of 4 + 8 + 6 = 18 inches.

If two triangles are similar, we can use the ratio of their sides or the ratio of their areas to find any missing measurements. If we know that the ratio of triangle A to triangle B is 2:1 and that the area of triangle A is 24 square units, then the area of triangle B, x, can be found by solving this proportion: , 2x = 24, x = 12. The area of triangle B is 12 square units.

If we are given the lengths of two sides of a right triangle, we can find the length of the missing side using the Pythagorean theorem: a2 + b2 = c2, where a and b are the bases of the right triangle and c is the hypotenuse. If one base of a right triangle measures 3 feet and the other measures 4 feet, then (3)2 + (4)2 = c2, 9 + 16 = c2, 25 = c2, and c = 5. The hypotenuse of the triangle is 5 feet.

#### Fuel for Thought

Area is the amount of two-dimensional space that a surface covers. Area is always expressed in square units.

Perimeter is the distance around an area. Perimeter is always expressed in linear units.

Similar triangles have identical angles. All three corresponding angles are congruent, although the sides of the triangles may not be congruent.

The Pythagorean theorem states that the sum of the squares of the bases of a right triangle is equal to the square of the hypotenuse of the right triangle: a2 + b2 = c2.

Most area word problems will contain the word area. Sometimes these problems give the base and height of a shape, and sometimes, they give the area and one of the two measurements and ask you for the other measurement. Perimeter word problems often contain the word perimeter, but a perimeter word problem may also ask you to find the distance around a figure.

#### Example

What is the area of a triangle whose base is 3 meters and whose height is 22 meters?

We are given the base and height of a triangle.

We are looking for the area of the triangle.

Underline the keywords and words that indicate formulas.

The phrase area of a triangle tells us that we must use the area formula for triangles.

Cross out extra information and translate words into numbers.

There is no extra information in this problem.

List the possible operations.

We are given the base and the height. The area of a triangle is the product of , the base, and the height, so we must multiply.

Write number sentences for each operation.

(3)(22)

Solve the number sentences and decide which answer is reasonable.

(3)(22) = (66) = 33 square meters

Since we used multiplication to find our answer, we can use division to check it. Divide the area by the base, and that value should equal half the height: = 11, which is half of 22.

#### Inside Track

Remember, you can always rewrite a formula by moving terms from one side of the equation to the other. If you are given the area and height of a triangle and you are looking for the base, you can rewrite the formula A = bh by dividing both sides of h. The base, b, is equal to .

#### Example

If an equilateral triangle has a perimeter of 111 inches, what is the length of one side of the triangle?

We are given the perimeter of an equilateral triangle.

We are looking for the length of one side of the triangle.

Underline the keywords and words that indicate formulas.

The word perimeter tells us that we have the distance around a triangle, and the phrase equilateral triangle tells us that every side of the triangle is equal in measure. These words together mean that we can use the formula for perimeter of an equilateral triangle, P = 3s.

Cross out extra information and translate words into numbers.

There is no extra information in this problem.

List the possible operations.

Rewrite the formula for perimeter of an equilateral triangle to solve for s, the side of the triangle: s =

Write number sentences for each operation.

Divide the perimeter by 3:

Solve the number sentences and decide which answer is reasonable.

= 37 inches

Since we used division to find our answer, we can use multiplication to check it. The perimeter of the equilateral triangle should be equal to three times the length of one side: (3)(37) = 111 inches.

Almost all word problems that involve similar triangles will include the word similar, but remember that similar triangles are two triangles whose angles are identical. If a word problem describes two triangles as having identical angles, be prepared to set up a ratio to find a missing side or area of one of the triangles. If a word problem is about similar triangles, draw a picture to help you match up corresponding sides.

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