GEO.1.6

Summary

The video explains various angle relationships, including vertical, adjacent, complementary, and supplementary angles. It covers how to identify these angles using examples and visual aids, emphasizing key concepts like congruence and linear pairs. The instructor also provides tips on distinguishing between complementary and supplementary angles, using memorable tricks. The video concludes with exercises on solving for unknown angles using algebra, applying the angle relationships discussed. Viewers are encouraged to practice on their own and check their answers.

Takeaways

• 🔍 Vertical angles are congruent angles located across from each other on intersecting lines, like angle 1 and angle 2.
• 🎯 Adjacent angles are two angles that are next to each other and share a common side, such as angle 1 and angle 2.
• 🎓 Complementary angles are two angles whose sum equals 90 degrees, for example, angle 1 and angle 2 summing up to 90 degrees.
• 📏 Supplementary angles are two angles whose sum is 180 degrees, like one angle being 135 degrees and another 45 degrees.
• 🔗 Linear pairs are adjacent and supplementary angles that form a straight line, with their sum always being 180 degrees.
• 📐 Vertical angles are congruent, meaning they have equal measures, like two angles each measuring 112 degrees.
• 🧮 Complementary angles add up to 90 degrees; for example, if one angle is 68 degrees, the other must be 22 degrees.
• 🔄 A linear pair is different from supplementary angles because they must be adjacent and supplementary, not just add up to 180 degrees.
• 📝 Angle relationships can be used to find unknown angle measures by applying the properties of vertical, complementary, and supplementary angles.
• 📘 For linear pairs or complementary angles, setting up equations using their properties helps in solving for unknown angles.

Q & A

What are vertical angles?

Vertical angles are two angles that are directly across from each other on intersecting lines. They are always congruent, meaning they have the same measure.

How can you remember what vertical angles are?

You can remember vertical angles by imagining an 'X' shape. The angles directly across from each other through the vertex are the vertical angles.

What are adjacent angles?

Adjacent angles are two angles that are next to each other and share a common side.

What are complementary angles?

Complementary angles are two angles whose sum is 90 degrees.

How can you distinguish between complementary and supplementary angles?

Complementary angles sum to 90 degrees, while supplementary angles sum to 180 degrees. A helpful way to remember is that complementary angles are the 'right' thing to do (90 degrees forms a right angle).

What is a linear pair of angles?

A linear pair consists of two adjacent angles that are supplementary. They form a straight line when combined, adding up to 180 degrees.

Do supplementary angles have to be adjacent to each other?

No, supplementary angles do not have to be adjacent. They can be separate, but their angle measures must add up to 180 degrees.

How do you find the measure of an unknown angle using complementary angles?

To find the measure of an unknown angle using complementary angles, subtract the measure of the known angle from 90 degrees. For example, if one angle is 68 degrees, the other would be 90 - 68 = 22 degrees.

What does it mean for angles to be bisected?

When an angle is bisected, it is divided into two equal angles. If a line bisects an angle, the two resulting angles are congruent.

How do you solve for an unknown angle in a linear pair if given one angle's measure?

If you know one angle in a linear pair, you can find the other by subtracting the known angle's measure from 180 degrees since the angles are supplementary. For example, if one angle is 135 degrees, the other is 180 - 135 = 45 degrees.

Outlines

00:00

📐 Introduction to Angle Relationships

This paragraph introduces the concept of angle relationships, starting with vertical angles. It explains that vertical angles are formed when two lines intersect and are always congruent. An example is provided using four angles, where pairs of opposite angles are congruent. The paragraph then transitions to adjacent angles, which share a common side, with an example illustrating two adjacent angles.

05:01

🎯 Exploring Supplementary and Linear Pair Angles

This paragraph delves into supplementary angles, which add up to 180 degrees, and how they differ from linear pairs. It explains that supplementary angles don’t need to be adjacent, whereas linear pairs are both adjacent and supplementary, forming a straight line. The paragraph ends with an exercise encouraging the reader to identify various types of angles based on the information provided.

10:02

🔍 Solving for Angle Measures Using Relationships

This paragraph focuses on using angle relationships to determine specific angle measures. It walks through examples involving vertical, complementary, and supplementary angles, demonstrating how to calculate unknown angles. The reader is guided through the steps to find angle measures, with explanations on how to set up and solve equations based on the given relationships.

15:07

📏 Understanding and Solving Angle Equations

In this final paragraph, the script continues with exercises on solving angle equations, specifically using the concept of angle bisectors. It explains how to approach problems where angles are bisected, leading to congruent angles that can be solved using algebra. The reader is encouraged to practice on their own and check their work with a provided key.

Keywords

💡Vertical Angles

Vertical angles are pairs of angles formed when two lines intersect, and they are opposite each other. These angles are always congruent, meaning they have equal measures. In the video, the instructor explains vertical angles using an example where angles 1 and 2, as well as angles 3 and 4, are vertical angles, emphasizing that they are across from each other and share the same vertex.

💡Adjacent Angles

Adjacent angles are two angles that share a common side and a common vertex but do not overlap. The video highlights that adjacent angles are next to each other, and the instructor gives an example where angles 1 and 2 are adjacent because they share a side. This concept is important in understanding other angle relationships like linear pairs.

💡Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. The video uses the mnemonic that giving a 'complement' is the 'right' thing to do, associating this with a right angle (90 degrees). For example, if one angle is 68 degrees, the other must be 22 degrees to make the total 90 degrees.

💡Supplementary Angles

Supplementary angles are two angles whose measures add up to 180 degrees. Unlike complementary angles, supplementary angles can either be adjacent (forming a linear pair) or separate from each other. The video explains that if one angle is 135 degrees, the supplementary angle must be 45 degrees to sum to 180 degrees.

💡Linear Pair

A linear pair consists of two adjacent angles whose non-common sides form a straight line, making them supplementary. In the video, the instructor distinguishes linear pairs from general supplementary angles by noting that linear pairs must be adjacent, while supplementary angles do not need to be next to each other.

💡Congruent

Congruent refers to figures or angles that have the same size and shape. In the context of the video, vertical angles are described as always being congruent, meaning their measures are equal. This concept is crucial for solving problems related to angle measures.

💡Angle Bisector

An angle bisector is a line or ray that divides an angle into two congruent angles. The video mentions this concept when discussing how to solve problems involving bisected angles, emphasizing that each resulting angle is equal in measure.

💡Perpendicular

Perpendicular lines are two lines that intersect to form right angles (90 degrees). The video references this when discussing certain angle problems, noting that when lines are perpendicular, the angles formed are complementary as they sum to 90 degrees.

💡Equation Setup

Equation setup involves creating algebraic equations based on known angle relationships to find unknown angle measures. The video demonstrates this process by setting up equations for complementary, supplementary, and vertical angles, reinforcing the importance of understanding these relationships in solving angle problems.

💡Angle Relationships

Angle relationships refer to the different ways angles can relate to each other, such as being vertical, adjacent, complementary, or supplementary. The entire video focuses on teaching these relationships, using examples to show how recognizing these relationships can help in calculating unknown angles.

Highlights

Introduction to angle relationships using a graphic organizer.

Definition and identification of vertical angles as angles across from each other on intersecting lines.

Explanation of why vertical angles are always congruent.

Mnemonic for remembering vertical angles as 'directly across from each other through the vertex'.

Introduction to adjacent angles and their shared common side.

Definition of complementary angles as two angles whose sum is 90 degrees.

Practical example of identifying complementary angles in a diagram.

Definition of supplementary angles as two angles whose sum is 180 degrees.

Memory aid for differentiating between complementary and supplementary angles using the concept of compliments.

Explanation of a linear pair as two adjacent and supplementary angles forming a straight line.

Difference between linear pairs and supplementary angles in terms of adjacency.

Guidance on identifying angle relationships in a given set of angles.

Method for finding angle measures using vertical angle congruency.

Process for solving for angles when given complementary angles.

Approach to calculating angles in a linear pair using the sum of 180 degrees.

Strategy for solving angle measures using algebra and angle relationships.

Example of setting up equations for angles when given perpendicular lines.

Technique for finding angles when given a linear pair and the measures of some angles.

Challenge problem involving finding angles when given a relationship between their measures.

Introduction to using angle bisectors to solve for unknown angles.

Method for solving angles when given they are bisected and their relationship.

Transcripts

so now we're going to look at actual

angle relationships that exist amongst

angles so first we're going to look at

this graphic organizer right here

because we're going to break apart what

they all are okay so we're gonna start

at the top and then we're going to work

our way counterclockwise okay so

vertical angles are two angles across

from each other on intersecting lines so

for example

let me actually draw

me some vertical angles it's right there

okay so I'm going to use this color so

let's say we have one two three and four

that's what I might call them so

vertical angles would be like angle

angle one being congruent to angle two

there across from each other right and

angle three being congruent to angle

four okay they're always across from

each other on intersecting lines they're

always congruent now the reason they're

called vertical if you can imagine is

this is the firt X and they are directly

across from each other through the

vertex okay so that helps you remember

vertical a little bit better so going

counter counterclockwise we're going to

come over here to adjacent angles

okay so adjacent angles the two are two

angles that are next to each other and

share a common side so for example all

right so I got some drawn up there so

make my life easier

so Jason angles like right here

they share a common side so this would

be like angle 1 angle 2 those are

adjacent and this is the side that they

share

all right so now let's move down over

here to complementary angles those are

two angles whose sum is 90 degrees so

let's draw one of them here's an example

of a complementary angle the two angles

whose sum is 90 degrees so like this

would be one

this would be angle wine this would be

equal to 90 degrees so you can say that

angle 1 plus a 2 is equal to 90 degrees

okay next we have

supplementary angles

and those are two angles whose sum is

180 degrees so let me get that drawn so

here we have an example of supplementary

angles so this for example let's say

that this was 135 degrees and that was

45 degrees and then you would end up

with a hundred and eighty degrees okay

now somehow I remember the difference

between complementary and supplementary

supplementary is a little bit harder to

necessary remember so to me the biggest

way or the easiest way to remember the

difference is to remember what

complementary angles are because then

you know that one is 90 you know that

one is 180 so if I know which one is 90

I'm able to figure out the other one is

180 so I think of complementary and I

think of this might be a silly way to

remember but it helps me

I think of complements and they're the

right thing to do to give a person a

compliment so complementary angles are

the right thing to do so they're 90

degrees

it might be silly way but it works now a

linear pair there are two angles that

are adjacent and supplementary and they

form a straight line so basically what I

have down here so I'm an actually

bring that up there as well okay so this

is a linear pair so I would say one two

you could say I have 1 plus angle 2 is

equal to 180 degrees now the difference

between a linear pair and supplementary

is that supplementary angles do not

necessarily

have to be attached so for example

remember what I drew out here

so this is our hundred and thirty five

degree angle

then

this would be

let me make this a little smaller so

that you can see it so here we have them

separated so this is still a hundred and

thirty-five degrees and this is still 45

degrees so in a supplementary angle they

don't actually have to be attached to

each other

they don't even have to be near to each

other but if they equal 180 degrees when

you add their angle measures together

they're supplementary but a linear pair

what's up here they have to be adjacent

to each other and supplementary in order

to be a linear pair

it's now coming on down here identifying

the types of angles pause this and then

see if you can identify the angle

relationships that are taking place

between these angles and then we'll come

back and see how you did

all right so now looking at one

those are vertical angles two are

adjacent and complementary three

adjacent supplementary and a linear pair

remember a linear pair is a linear pair

because it's adjacent in supplementary

so you can't leave those out number four

those are just jacent number five

vertical and complementary look at that

again if you didn't get the

complimentary and six is vertical and

supplementary look that again if you

didn't get supplementary now let's look

at using these angle relationships to

actually find angle measures right so

for a number well I'm just gonna scroll

through them we're gonna look at these

women write down what we have okay for

number one these are vertical angles

vertical angles are congruent so that

must also be a hundred and twelve

degrees number two these are

complementary angles remember complement

so the right thing to do ninety degrees

so just doing 90-68 is going to give you

X being twenty two degrees three those

are supplementary angles it's a linear

pair as well so because they're

supplementary they're 180 degrees so 180

minus 120 four

gives you 56 degrees now when we start

further on it's asking you to think a

little bit more on what you're looking

at right just let's look at number five

we're going to skip number four and go

to five

right so first and foremost you will

notice this 90-degree angle there and

this straight line if this is 90 degrees

then this is 90 degrees so Z is just

going to be 90 minus 72 which is going

to give you 18 degrees right

and now if you look at why

those are vertical so Y is going to be

72 degrees and now finally the next one

I notice is that X is vertical to Z plus

the 90 degrees so 90 plus 18

is 108 degrees okay so hopefully you can

see how we broke that down so try six

through nine on your own and then check

the key

so let's come down to using algebra

let's do a couple these together set up

a few and then you can try the rest on

your own and see how you do it's all

pretty much the same process recognizing

the angle relationships and using those

angle relationships in order to write

your equations from the expressions so

in ten let's see if we're gonna do this

one or set this one I'm looking at P Q T

versus s Q R those are vertical angles

so because of that I'm not going to do

this one with you but you can see that

they're vertical so you just have to set

them equal to each other in order to

solve because vertical angles are

congruent let's look at eleven it's

telling me that a B is perpendicular to

C D that's the first thing it's saying

so when I go over here can I find a B

and C D I'm getting a note

that it's 90 degrees okay and it's

telling me that DCE

gives me that value and then it gives me

ECB so this one

this angle Plus this angle are going to

be equal to 90 degrees solve for that

and then see what you get

twelve let's look at 12 help you break

these down

12 gives me angle K and let's see what

that K n m and M and J so we have M and

J and we want to find the measure of K M

on this well this angle Plus this angle

is equal to how many degrees that's a

linear pair so they're equal to 180

degrees because it's supplementary so I

just have to add those together and set

them equal to 180

now try 13 on your round it's a little

bit more of a challenge and then see how

you do let's see well 14 it's telling me

that they're complementary so if they're

complementary they're equal to 90

degrees

add them together set it equal to 90 15

are supplementary add together set equal

to 180 now these ones are the ones again

that people have the most trouble with

because it doesn't give you actual

numbers it's telling you something so

again I ask myself one end and a one or

two form a linear pair the measure of

angle 2 is 6 more than twice the measure

of angle 1 find the measure of angle 2

looking at number 16 I can't write first

thing I ask myself is what angle does it

tell me nothing about okay

well since the measure of angle 2 is 6

more than twice the measure of angle 1 5

measuring 2 it tells me nothing about

the measure of angle 1 the measure of

angle 1

that's your x-value okay so that's where

your starting and the measure of angle 2

it says it is 6 more than twice the

measure so I have 6 more than twice the

measure so twice the measure would be 2x

right and then if I have 6 more I'm

gonna add 6 to that and it's a linear

pair so I'm gonna add those together and

set them equal to 180

so in 17 let's look at 17 J and K are

complementary

which one does it tell me nothing about

it tells me absolutely nothing about the

measure of angle K so the measure of

angle K is your X and then the measure

of angle J is 18 months than the measure

of angle K and when something is 18 less

if I say I have 18 less than you you

don't do 18 minus whatever you have no

you take whatever you have and you

subtract 18 from it so X minus 18

they're complementary complements are

the right thing to do right angle so add

those together set them equal to 90 and

then solve you can check the key when

you're done so now these ones are using

angle bisectors so let's look at 18 UW

is bisecting so if this is bisecting

that it means that this angle is

congruent to this angle and let me make

that

right and if they're if they're

congruent then I just have to set them

equal to each other let's see if that's

what it's asking me it's giving me tea

eww and it's giving me w u V just set

those equal to each other and solve for

x

and let's see not too much different

over here right for 19 and 20 sews try

those on your own and then check the key

to see how you did