High School Math : Geometry

Study concepts, example questions & explanations for High School Math

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Example Questions

Example Question #2 : How To Find The Length Of The Side Of A Pentagon

Find the length of the side of the following pentagon.

Angle_length_of_side_pentagon

The perimeter of the pentagon is .

Possible Answers:

 

 

 

Correct answer:

 

 

 

Explanation:

The formula for the perimeter of a regular pentagon is

,

where  represents the length of the side.

Plugging in our values, we get:

Example Question #3 : How To Find The Length Of The Side Of A Pentagon

Find the length of the side of the following pentagon.

Angle_length_of_side_pentagon

The perimeter of the pentagon is .

Possible Answers:

 

 

 

Correct answer:

Explanation:

The formula for the perimeter of a regular pentagon is

,

where  represents the length of the side.

Plugging in our values, we get:

 

 

Example Question #501 : Plane Geometry

What is the area of a regular pentagon with side length ?

Possible Answers:

Correct answer:

Explanation:

The area of a regular polygon can be solved using the following equation: .

To find the perimeterof the pentagon, we multiply the side length by five.

The apothem of a regular polygon ( = number of sides;  = side length) is given by the equation below.

Pluggin in our numbers, we can solve for the apothem.

Finally, we can solve for the area.

Example Question #502 : Plane Geometry

Find the area of the following pentagon:

Area_pentagon

Possible Answers:

Correct answer:

Explanation:

The formula for the area of of a pentagon is

.

Plugging in our values, we get:

Example Question #1 : How To Find The Area Of A Pentagon

Find the area of the following pentagon:

Screen_shot_2014-03-11_at_8.34.46_pm

Possible Answers:

Correct answer:

Explanation:

The formula for the area of of a pentagon is:

Plugging in our values, we get:

Example Question #1 : How To Find An Angle Of A Line

Lines  and  are parallel. Which angle is congruent to angle ?

Possible Answers:

Correct answer:

Explanation:

When two lines are parallel, corresponding angles are congruent. Because we don't know if lines and  are parallel, we can't make any conclusions about angles 2 and 3.

With the given information, we can ignore line , as it has no relation to angle . This leaves angles 1 and 4 as possible answers. Angle 4 will be congruent to angle , while angle 1 will supplement angle .

Example Question #1 : Geometry

Angles

Figure not drawn to scale.

In the figure above, APB forms a straight line. If the measure of angle APC is eighty-one degrees larger than the measure of angle DPB, and the measures of angles CPD and DPB are equal, then what is the measure, in degrees, of angle CPB?

Possible Answers:

33

40

50

114

66

Correct answer:

66

Explanation:

Let x equal the measure of angle DPB. Because the measure of angle APC is eighty-one degrees larger than the measure of DPB, we can represent this angle's measure as x + 81. Also, because the measure of angle CPD is equal to the measure of angle DPB, we can represent the measure of CPD as x.

Since APB is a straight line, the sum of the measures of angles DPB, APC, and CPD must all equal 180; therefore, we can write the following equation to find x:

x + (x + 81) + x = 180

Simplify by collecting the x terms.

3x + 81 = 180

Subtract 81 from both sides.

3x = 99

Divide by 3.

x = 33.

This means that the measures of angles DPB and CPD are both equal to 33 degrees. The original question asks us to find the measure of angle CPB, which is equal to the sum of the measures of angles DPB and CPD.

measure of CPB = 33 + 33 = 66.

The answer is 66.

Example Question #1 : How To Find An Angle Of A Line

One-half of the measure of the supplement of angle ABC is equal to the twice the measure of angle ABC. What is the measure, in degrees, of the complement of angle ABC?

Possible Answers:

36

18

72

54

90

Correct answer:

54

Explanation:

Let x equal the measure of angle ABC, let y equal the measure of the supplement of angle ABC, and let z equal the measure of the complement of angle ABC.

Because x and y are supplements, the sum of their measures must equal 180. In other words, x + y = 180. 

We are told that one-half of the measure of the supplement is equal to twice the measure of ABC. We could write this equation as follows:

(1/2)y = 2x.

Because x + y = 180, we can solve for y in terms of x by subtracting x from both sides. In other words, y = 180 – x. Next, we can substitute this value into the equation (1/2)y = 2x and then solve for x.

(1/2)(180-x) = 2x.

Multiply both sides by 2 to get rid of the fraction.

(180 – x) = 4x.

Add x to both sides.

180 = 5x.

Divide both sides by 5.

x = 36.

The measure of angle ABC is 36 degrees. However, the original question asks us to find the measure of the complement of ABC, which we denoted previously as z. Because the sum of the measure of an angle and the measure of its complement equals 90, we can write the following equation:

x + z = 90.

Now, we can substitute 36 as the value of x and then solve for z.

36 + z = 90.

Subtract 36 from both sides.

z = 54.

The answer is 54. 

Example Question #1 : Geometry

Parallellines

 

 

In the diagram, AB || CD. What is the value of a+b?

Possible Answers:

None of the other answers.

160°

80°

60°

140°

Correct answer:

160°

Explanation:

Refer to the following diagram while reading the explanation:

Parallellines-answer

We know that angle b has to be equal to its vertical angle (the angle directly "across" the intersection).  Therefore, it is 20°. 

Furthermore, given the properties of parallel lines, we know that the supplementary angle to a must be 40°.  Based on the rule for supplements, we know that a + 40° = 180°.  Solving for a, we get a = 140°.

Therefore, a + b = 140° + 20° = 160°

Example Question #1552 : Plane Geometry

In rectangle ABCD, both diagonals are drawn and intersect at point E.  

Let the measure of angle AEB equal x degrees.

Let the measure of angle BEC equal y degrees.

Let the measure of angle CED equal z degrees.

Find the measure of angle AED in terms of x, y, and/or z.

Possible Answers:

360 – x + y + z

180 – 2(x + z)

180 – (x + y + z)

180 – 1/2(x + z)

180 – y

Correct answer:

180 – 1/2(x + z)

Explanation:

Intersecting lines create two pairs of vertical angles which are congruent. Therefore, we can deduce that y = measure of angle AED.

Furthermore, intersecting lines create adjacent angles that are supplementary (sum to 180 degrees). Therefore, we can deduce that x + y + z + (measure of angle AED) = 360.

Substituting the first equation into the second equation, we get

x + (measure of angle AED) + z + (measure of angle AED) = 360

2(measure of angle AED) + x + z = 360

2(measure of angle AED) = 360 – (x + z)

Divide by two and get:

measure of angle AED = 180 – 1/2(x + z)

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