Complementary and Supplementary Angles

Summary

This video script delves into the concepts of complementary and supplementary angles, explaining that complementary angles sum to 90 degrees while supplementary angles total 180 degrees. Using a right triangle as an example, the script illustrates these concepts and guides viewers through various practice problems, including finding supplements and complements of given angles, and solving for unknown angles based on their relationships. The script also covers a challenge problem involving the relationship between an angle's supplement and complement, providing a step-by-step solution to engage and educate viewers on these fundamental geometric principles.

Takeaways

• 📐 Complementary angles are two angles that add up to 90 degrees.
• 🔍 In a right triangle, the two non-right angles are always complementary to each other.
• 🌐 Supplementary angles are pairs of angles that add up to 180 degrees.
• 📉 To find the supplement of an angle, subtract the angle from 180 degrees.
• 📈 To find the complement of an angle in degrees, minutes, and seconds (DMS), convert 90 degrees to DMS and subtract the given angle.
• 🤔 If one angle is larger than the other by a certain degree, you can set up an equation to find the measure of both angles.
• 📝 When two angles are in a ratio, you can express each angle in terms of a variable and solve for the ratio to find their measures.
• 🔢 For angles in a ratio, dividing the sum of the angles by the sum of the ratio parts will give you the value of one part.
• 🧩 If one supplementary angle is a certain amount more than a multiple of the other, set up an equation to find the measure of the smaller angle.
• 🔄 The supplement of an angle is four times its complement can be solved by assigning variables and creating equations based on the relationships between the angles.
• 🎯 The video provides a systematic approach to solving geometry problems involving complementary and supplementary angles.

Q & A

What are complementary angles?

Complementary angles are two angles that add up to 90 degrees. For example, if one angle measures 30 degrees, its complement would measure 60 degrees to make the sum 90 degrees.

Can you explain the concept of supplementary angles with an example?

Supplementary angles are two angles that add up to 180 degrees. For instance, if one angle measures 120 degrees, its supplement would be 60 degrees because 120 + 60 equals 180.

What is the relationship between supplementary angles and a straight line?

Supplementary angles form a linear pair when they are positioned along a straight line. The sum of the angles in a linear pair is always 180 degrees, which represents a straight line.

How do you find the supplement of a given angle?

To find the supplement of an angle, subtract the given angle from 180 degrees. For example, the supplement of a 55-degree angle would be 180 - 55, which equals 125 degrees.

What is the process of finding the complement of an angle given in degrees, minutes, and seconds (DMS)?

To find the complement of an angle in DMS, subtract the angle from 90 degrees. You may need to convert 90 degrees into DMS and then perform the subtraction, borrowing from minutes and seconds as necessary.

How can you determine the measures of two complementary angles if one is 20 degrees larger than the other?

Let the smaller angle be 'b' and the larger angle be 'a'. If 'a' is 20 degrees larger than 'b', you can set up the equation a = b + 20. Since they are complementary, a + b = 90. Solving these equations will give you the measures of both angles.

What is the measure of the larger angle if two complementary angles exist in a 2 to 3 ratio?

If the angles are in a 2 to 3 ratio, let the smaller angle be 2x and the larger angle be 3x. Since they are complementary and add up to 90 degrees, (2x + 3x) = 90. Solving for x gives you the measure of the larger angle as 3x.

How do you calculate the measure of the smaller angle if one of two supplementary angles is 12 more than three times the value of the other?

Let x be the measure of the smaller angle and y be the measure of the larger angle. If y is 12 more than three times x, you can express y as 3x + 12. Since they are supplementary, x + y = 180. Solving this equation will give you the measure of the smaller angle.

What is the measure of an angle if its supplement is four times the complement of the angle?

Let a be the measure of the angle, c be the complement, and s be the supplement. If s is four times c, and knowing that a + c = 90 and s + a = 180, you can set up an equation to solve for a, which will be 60 degrees in this case, making the supplement 120 degrees.

How can you verify the solution to the challenge problem where the supplement of an angle is four times the complement?

After finding the angle to be 60 degrees, the complement would be 30 degrees (90 - 60), and the supplement would be 120 degrees (180 - 60). You can verify the solution by checking if the supplement (120) is indeed four times the complement (30 * 4 = 120).

Outlines

00:00

📚 Introduction to Complementary and Supplementary Angles

This paragraph introduces the concepts of complementary and supplementary angles. Complementary angles are two angles that sum up to 90 degrees, exemplified by angles in a right triangle where one angle is 30 degrees, making its complement 60 degrees. Supplementary angles, on the other hand, add up to 180 degrees, as illustrated with a linear pair forming a straight line. The paragraph also presents practice problems involving finding the supplement of a 55-degree angle and the complement of an angle given in degrees, minutes, and seconds.

05:01

🔍 Solving Problems with Complementary and Supplementary Angles

The second paragraph delves into solving more complex problems involving complementary and supplementary angles. It explains how to find the measure of two angles if one is 20 degrees larger than the other, using the elimination method to solve a system of equations. The paragraph also addresses a problem where two angles exist in a 2:3 ratio and how to find the larger angle by setting up variables and solving for them. Additionally, it introduces a problem where one supplementary angle is 12 more than three times the other, guiding viewers through the substitution method to find the measure of the smaller angle.

10:01

🧩 Advanced Angle Problems and Challenge

The final paragraph presents advanced problems involving the relationships between angles. It discusses a scenario where the supplement of an angle is four times its complement, guiding viewers to assign variables and create equations to solve for the measure of the angle. The solution process involves algebraic manipulation to isolate the variable and find the angle's measure. The paragraph concludes with a summary of the findings, confirming that the supplement is indeed four times the complement, thus reinforcing the understanding of complementary and supplementary angles.

Keywords

💡Complementary angles

Complementary angles are two angles whose measures add up to 90 degrees. They are fundamental in geometry, particularly in the context of right triangles where one angle and its complement together form a right angle. In the script, complementary angles are exemplified by angles in a right triangle, such as angle A being 30 degrees and angle C being 60 degrees, thereby adding up to 90 degrees.

💡Supplementary angles

Supplementary angles are two angles that add up to 180 degrees. They are often found in geometric configurations where two lines form a straight line, such as a linear pair. The script explains supplementary angles by showing how the sum of angle ABD (120 degrees) and angle CBD (60 degrees) equals 180 degrees, forming a straight line.

💡Right triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. It is a key shape in geometry and trigonometry, with properties that are used to define and understand complementary and supplementary angles. In the script, the right triangle ABC is used to illustrate how angles A and C are complementary, each measuring 30 and 60 degrees respectively.

💡Degrees

Degrees are a unit of measurement used to express angles. One full rotation is 360 degrees, and angles are measured in fractions of this full rotation. The script frequently uses degrees to quantify angles, such as a 55-degree angle whose supplement is sought, or a 64-degree, 26-minute, 37-second angle whose complement is calculated.

💡Minutes and Seconds

In the measurement of angles, minutes and seconds are smaller units than degrees, used to express fractions of a degree. There are 60 minutes in a degree and 60 seconds in a minute. The script demonstrates the conversion and subtraction of angles in degrees, minutes, and seconds, such as finding the complement of a 64-degree, 26-minute, 37-second angle.

💡Linear pair

A linear pair refers to two angles that are adjacent and form a straight line, summing up to 180 degrees. The concept is used in the script to explain how angles ABD and CBD form a linear pair, being supplementary and adding up to 180 degrees.

💡Practice problems

Practice problems are exercises designed to help viewers apply and understand the concepts taught in the video. The script includes several practice problems that involve finding supplements and complements of angles, using both degrees and degrees, minutes, and seconds, as well as solving for unknown angles in various geometric scenarios.

💡Ratio

A ratio is a way of comparing quantities by division, often expressed as a:b. In the script, the ratio is used to solve a problem where two complementary angles exist in a 2 to 3 ratio, helping to determine the measure of the larger angle.

💡Elimination method

The elimination method is a technique used in algebra to solve systems of equations by adding or subtracting equations to eliminate one variable. In the script, this method is used to find the measures of two complementary angles where one angle is 20 degrees larger than the other.

💡Substitution

Substitution is an algebraic method where one variable is expressed in terms of another and then replaced in the equations to solve for the unknowns. The script demonstrates the use of substitution to solve for the measure of the smaller angle when one supplementary angle is 12 more than three times the value of the other.

💡Challenge problem

A challenge problem is a more difficult exercise designed to test and deepen understanding of a concept. In the script, the challenge problem involves finding the measure of an angle when its supplement is four times its complement, requiring the setup and solution of a system of equations.

Highlights

Complementary angles are defined as angles that add up to 90 degrees.

In a right triangle, if one angle is known, the complementary angle can be determined to sum up to 90 degrees.

Supplementary angles are pairs of angles that add up to 180 degrees.

Linear pairs are angles that form a straight line and their sum is 180 degrees.

The supplement of a 55-degree angle is calculated by subtracting 55 from 180, resulting in a 125-degree angle.

To find the complement of an angle in degrees, minutes, and seconds, convert 90 degrees into DMS and perform subtraction.

When one angle is 20 degrees larger than the other and they are complementary, solving the system of equations yields the measures of both angles.

If two angles are in a 2:3 ratio and complementary, the larger angle can be found by dividing 90 by the sum of the ratio numbers.

For supplementary angles, if one is 12 degrees more than three times the other, substitution can be used to find the measure of the smaller angle.

The challenge problem involves finding the supplement of an angle when it is four times the complement, which requires setting up and solving a system of equations.

The solution to the challenge problem shows that the angle measures 60 degrees, with the complement being 30 degrees and the supplement 120 degrees.

Mental math techniques are demonstrated for subtracting large numbers and dividing by 5.

Borrowing minutes and seconds when performing subtraction with angles in DMS notation is explained.

The elimination method is used to solve for two variables representing complementary angles differing by 20 degrees.

The ratio of 2:3 is applied to find the measures of complementary angles in a triangle.

Substitution is an alternative method to elimination for solving systems of equations involving supplementary angles.

The video concludes with a comprehensive explanation of solving geometry problems related to complementary and supplementary angles.

Transcripts

in this video we're going to talk about

complementary and supplementary angles

so what exactly

is

a complementary angle

complementary angles are angles

that add up to 90.

so consider this right triangle

let's call it triangle abc

now if angle a is 30

then angle c has to be 60.

so notice that angle a and angle c

are complementary to each other

angle c is the complement of angle a

any time you have two angles that add up

to 90

the two angles are complementary to each

other

now the next term you need to be

familiar with are supplementary angles

supplementary angles

add up to 180 as opposed to 90.

so let's call this a

b

c

and d

angle abd

let's say it's 120.

so what is the value of angle cbd

if you know abd and cbd are

supplementary

cbd has to be 60

because these two angles have to add up

to

180. it turns out that

they also form a linear pair means we

have a straight line with two angles

it forms the linear pair which also adds

up to 180.

now let's work on some practice problems

let's start with this one find the

supplement of a 55 degree angle

supplementary angles we know adds up to

180 so to find the other supplement it's

going to be 180 minus 55.

so if you want to subtract as mentally

you could break up 55 into 50 and 5.

now 180 minus 50 think of 18 minus 5

that's 13. so 180 minus 50 is 130 and

130 minus 5

is 125.

so that's the supplement of a 55 degree

angle it's a 125 degree angle

number two

what is the complement of a 64 degree

26 minutes 37 seconds angle

now to find the complement

we need to subtract

that angle from 90.

so how can we do this

if we're doing 90-64 that would be a

piece of cake

but we have an angle in dms degrees

minutes and seconds

so how can we perform the operation

well let's change 90 into a dms value

so first what we're going to do is we're

going to borrow a 1 from 90.

so 90 is equivalent

to 89 degrees

and 1 degree is 60 minutes

so 89 degrees in 60 minutes

is basically equivalent to 90 degrees

now we do have a value in the seconds

column

so therefore we need to borrow a minute

from 60.

so 89 degrees and 60 minutes is

equivalent

to 89 degrees

59 minutes

and 60 seconds

so now we could take this value

and subtract it by that number

minus sixty-four

that's twenty-five

fifty-nine minus twenty-six we know nine

minus six is three five minus two is

three

and sixty minus thirty-seven

that's 23.

so the answer is 25 degrees

33 minutes

and 23 seconds

so that's the complement

of this angle

number three

one of two complementary angles

is 20 degrees larger than the other

what is the measure of the two angles

so let's say the two angles let's give

it a letter angle a and angle b

if they're complementary to each other

that means that they add up to 90

degrees

now one of the angles is 20 degrees

larger than the other one

so that means

a

is

b plus 20.

so let's say if b is

30 a would be 50 if b is 40 a will be

60.

so they differ by 20.

so we have two equations

and we have two variables how can we

find the value of a and b

so what i'm going to do is subtract both

sides by b

in this equation

if i do so i'm going to have a

minus b

is equal to 20.

so now i have a system of two equations

and i'm going to solve it by the

elimination method also known as the

addition method

so if we add the two equations b and

negative b adds up to zero

a plus a is two a

ninety plus twenty is one ten

so to find the value of a i need to

divide both sides by two

a hundred divided by two is fifty and

ten divided by two is five

so 110 divided by two is fifty-five

so that's the value of angle a

now to find angle b

we know that these two angles have to

add up to 90.

so b is going to be 90

minus 55.

90 minus 50 is 40.

40 minus 5 is 35.

so angle b is 35

so now we have the measure

of the two angles

so 55 plus 35 adds up to 90 and 55 is 20

units higher than 35

let's try this one number four

two complementary angles

exists in a two to three ratio

what is the measure of the larger angle

so let's draw a picture we can use the

right triangle

as an example just like we did in the

beginning of this video

and let's say angle b is the right angle

which means that angle a and c they have

to be complementary to each other

all three angles of a triangle must

always add to 180.

so if angle b is 90

a and c have to add up to 90 which means

a and c are complements of each other

now we could say that angle a is 2x and

angle c is 3x

and in that way they exist in a 2 to 3

ratio

so angle a plus angle c have to add up

to 90.

and angle a is 2x angle c is 3x

and so 2x plus 3x is 5x

so now we need to divide

90 by 5.

now if you wish to perform the division

mentally here's what you can do think of

90 as being 50 plus 40

and we need to divide each number by 5.

50 divided by 5 is 10

40 divided by 5 is 8.

so 90 divided by 5

is 10 plus 8 or 18.

so that's the value of x

but our goal

is to find the measure

of the larger angle

which is angle c and that's 3x

so that's going to be 3 times 18

which if you don't know what 3 times 18

is

you can break up 18 into being 10 plus

8.

3 times 10 is 30

3 times 8 is 24

and 30 plus 24 is 54.

so that's 3 times 18.

so angle c

is 54 degrees

which means that angle a

is 2x 2 times 18 that's 36

and as you can see these two numbers

36 plus 54

adds up to 90.

but the answer to the problem the

measure of the larger angle

is 54.

number five

one of two supplementary angles

is 12 more

than three times the value of the other

calculate the measure of the smaller

angle

so let's say x

is the first angle and y is the second

one

the fact that these two angles are

supplementary

means that they add up to 180.

now one of the two angles is 12 more

than three times the other

so three times the other let's say if

x is the other one

that will be three x and then it's

twelve more than that so three x plus

twelve

our goal is to calculate the measure of

the smaller angle

so we're going to find the value of x

and y

and then we can see that y is going to

be larger angle x is a small angle so

our goal is to find x

now in this problem i'm going to solve

using substitution instead of

elimination

so i'm going to replace y

with 3x plus 12.

so now we have the expression x

plus 3x plus 12

is equal to 180

and x plus 3x

is 4x

next we need to subtract both sides by

12. 180 minus 12

is 168

and then we could divide

both sides by four

16 divided by 4 is 4 and 8 divided by 4

is 2.

so 168 divided by 4

is 42

and so that's the measure of the smaller

angle to find the measure of the larger

angle take this number and plug it into

this expression

or you could do 180 minus 42

which is going to be 138

so that's the measure of the larger

angle but the answer to the problem

is 42

which is what we wanted

challenge problem number six

the supplement of an angle

is four times the complement of the

angle

find the measure of the supplement of

the angle

so feel free to pause the video and work

on this problem

now we need to realize is that there's

three variables

the first one

is the measure of the angle

the second one is the measure of the

complement

of the angle

and the last one

is the measure of the supplement

of the angle

so what we need to do is

assign three variables

to these three quantities

let's say a is the measure of the angle

c is for the complement s is for the

supplement

so you can keep track of what's

what now the first sentence says the

supplement

of an angle is four times the complement

of the angle so we can say that s

is equal to four c

s is four times the value of the

complement

now there are two other equations that

we can write

if you want to solve a system of three

variables you need three equations

now the complement of angle a

is going to be 90 minus 8 because a plus

c has to add up to 90.

and the supplement

and the angle have to add up to 180 so

the supplement is going to be 180 minus

the angle

so now we have the three equations that

we need

so let's replace s

with 180 minus a

and let's replace c with 90 minus a

so now we can have a single equation

that contains

only one variable

so now all we got to do is just some

math

let's distribute the 4 to 90 minus a

so 4 times 90

that's 360 because 4 times 9 is 36 and

then we have 4 times negative a which is

negative 4a

and now let's subtract both sides by 180

and simultaneously let's add 4a

to both sides

so on the left we have negative a plus

4a which is 3a

and 360 minus 180 that's 180.

so next let's divide both sides by three

so a

is 180 divided by three

now 18 divided by three is six so 180

divided by three is 60.

so the angle has a measure

of 60 degrees so i'm going to put that

here a is equal to 60.

the complement is 90 minus 60 so the

complement is 30.

the supplement is 180 minus 60

which is 120.

and we can see that our answers are

correct because the supplement is indeed

four times the value of the complement

thirty times four is one twenty

and so that's the end of this video

hopefully this gives you a good idea of

how to solve geometry problems that are

associated with complementary and

supplementary angles

you