Angle Addition Postulate explained with examples

Summary

The video script explains the angle addition postulate, a fundamental concept in geometry. It involves an angle split by a ray into two smaller angles, whose sum equals the original angle's measure. The script uses examples to demonstrate how this postulate aids in solving for unknown angles or setting up equations. It emphasizes the importance of accurate notation, such as using 'm' for measures and including parentheses when degrees are part of algebraic expressions. The examples provided illustrate how to apply the postulate to find missing angle measures and check solutions.

Takeaways

• 📐 The angle addition postulate is a fundamental concept in geometry that deals with the relationship between the measures of angles within a single angle.
• 🔺 It is analogous to the segment addition postulate, but specifically applies to angles rather than line segments.
• 📌 The postulate involves an angle and a point within the interior of that angle, which allows for the formation of two smaller angles within the larger one.
• 👁️ The smaller angles formed by a ray through the vertex of the original angle and the interior point add up to the measure of the original angle.
• ✏️ The notation emphasizes that the sum of the measures of the smaller angles equals the measure of the whole angle, typically denoted with 'm' to represent measure.
• 🔍 The postulate is useful for solving problems involving missing angle measures or for setting up equations in geometry.
• 📘 An example provided in the script illustrates how to use the angle addition postulate to find the measure of an unknown angle when the measures of other angles are known.
• 🔢 The script also includes an algebraic example where variables represent the measures of angles, and the postulate is used to set up and solve an equation.
• 📝 The importance of accurate notation, including the use of parentheses and degree symbols, is highlighted to avoid confusion when dealing with algebraic expressions involving angles.
• 🔄 The process of checking solutions to ensure they are consistent with the angle addition postulate is emphasized as a critical step in problem-solving.

Q & A

What is the angle addition postulate?

The angle addition postulate states that if you have an angle and a point in the interior of that angle, you can draw a ray from the vertex of the angle through the point, creating two smaller angles. The sum of the measures of these two smaller angles equals the measure of the original angle.

How is the angle addition postulate similar to the segment addition postulate?

Both the angle addition postulate and the segment addition postulate share the concept of 'whole equals the sum of its parts.' While the segment addition postulate deals with line segments, the angle addition postulate deals with angles, focusing on how the measures of smaller angles within a larger angle add up to the measure of the whole angle.

What is the significance of naming an angle with three letters in the angle addition postulate?

Naming an angle with three letters ensures that the vertex is always in the middle. This naming convention helps to clearly define the sides of the angle and maintain consistency in the order of the letters when referring to different angles formed within the original angle.

Why is it important to not assume the ray drawn through the angle is an angle bisector?

Assuming the ray drawn through the angle is an angle bisector would imply that it divides the angle into two equal parts. The angle addition postulate does not require this assumption; it simply states that the sum of the measures of the two smaller angles formed by the ray equals the measure of the original angle, regardless of where the ray is located within the angle.

How can the angle addition postulate be used to solve for missing angle measures?

The angle addition postulate can be used to set up equations to solve for missing angle measures by expressing the measure of the whole angle as the sum of the measures of the smaller angles. By knowing the measures of one or more of the smaller angles, you can solve for the unknown angle measure.

What is the correct notation for expressing the measure of an angle in terms of variables?

When expressing the measure of an angle in terms of variables, it's important to use parentheses around the variable to indicate that the entire expression represents a numerical value in degrees. For example, if the measure of an angle is represented by 'x', it should be written as '3x°' to indicate '3 times x degrees'.

How do you verify the correctness of your solution when using the angle addition postulate?

After finding the measure of an unknown angle using the angle addition postulate, you can verify the correctness of your solution by adding the measures of the known angles and the calculated angle to ensure they sum up to the measure of the entire angle.

What is a practical example of using the angle addition postulate to solve for an unknown angle measure?

In the script, an example is given where the measure of angle TM is 12 degrees, and the measure of the entire angle LM is 39 degrees. By using the angle addition postulate, you can set up an equation to solve for the measure of angle LM, which is found to be 27 degrees. This is verified by adding 12 degrees and 27 degrees to get 39 degrees.

Why is it necessary to check your answer after solving for an unknown angle measure?

Checking your answer after solving for an unknown angle measure ensures that the calculations are correct and that the measures of the angles add up correctly according to the angle addition postulate. This verification step helps to confirm that the solution is accurate and adheres to the mathematical principles being applied.

Can the angle addition postulate be used to solve for variables in more complex algebraic expressions involving angles?

Yes, the angle addition postulate can be applied to solve for variables in algebraic expressions involving angles. For instance, if you have expressions like '3x' for one angle and '2x - 6' for another, and you know the sum of these angles equals a certain measure, you can set up and solve an equation to find the value of 'x'.

Outlines

00:00

🔍 Introduction to the Angle Addition Postulate

The video begins with an explanation of the angle addition postulate, comparing it to the segment addition postulate. It introduces the concept of having an angle (named ABC) with a point F inside it. A ray through point F divides the larger angle into two smaller angles. The video emphasizes that this is not necessarily the angle bisector. It provides foundational information about how the smaller angles (ABF and FBC) add up to the larger angle, and the importance of correct notation using 'm' for angle measures.

05:01

📐 Solving for Missing Angles Using the Angle Addition Postulate

In this example, the video demonstrates how to use the angle addition postulate to solve for a missing angle. Given angles TMS and LMT, the goal is to find LMS. The process involves setting up an equation where the two smaller angles add up to the larger one. After substituting the known values, the equation is solved, showing that LMS equals 27 degrees. The solution is then verified by checking if the sum of the smaller angles equals the larger angle.

10:02

🧮 Example with Algebraic Expressions for Angles

The video introduces a more algebraic example, where the angle measures are given in terms of variables. In this scenario, angles MNP and PNR are expressed as 3x and 2x - 6, respectively, with the larger angle MNR equaling 44 degrees. Using the angle addition postulate, the equation is set up, combining the smaller angles to equal the larger one. The variable x is solved as 10, and then plugged back into the expressions for MNP and PNR, yielding 30 degrees and 14 degrees, respectively, confirming the total adds up to 44 degrees.

👋 Conclusion and Final Thoughts

The video wraps up with a brief conclusion, recapping the process of solving for angles using the angle addition postulate. The instructor encourages viewers to practice these types of problems and ensures they understand the steps involved. The goal is to reinforce the key concepts and leave the viewer confident in applying this method to similar problems.

Keywords

💡Angle Addition Postulate

The Angle Addition Postulate is a fundamental concept in geometry that states the measure of an angle can be found by adding the measures of two other angles that sum up to it. In the video, this postulate is used to solve for unknown angles when the measures of other angles within the same geometric figure are known. The script explains that if you have an angle split by a ray into two smaller angles, the sum of the measures of these two smaller angles equals the measure of the original angle.

💡Segment Addition Postulate

The Segment Addition Postulate is mentioned in the script as a comparison to the Angle Addition Postulate. It states that the length of a line segment can be found by adding the lengths of two other segments that form it. While the video's main focus is on angles, the script draws a parallel between how the two postulates operate, emphasizing the idea that 'one part plus another part equals the whole', whether it's a line segment or an angle.

💡Vertex

A vertex is a point where two or more rays or line segments meet. In the context of the video, the vertex is the common point from which the rays of an angle originate. The script uses the vertex to define angles, such as angle ABC, and to illustrate how a ray can be drawn from the vertex through an interior point, dividing the angle into two smaller angles.

💡Interior Point

An interior point, in the context of the video, refers to a point that lies within the boundaries of a geometric figure, in this case, within an angle. The script explains that by choosing an interior point, such as point F, a ray can be drawn from the vertex through this point, effectively dividing the original angle into two smaller angles, which is essential for applying the Angle Addition Postulate.

💡Ray

A ray is a part of a line that starts at a point and extends infinitely in one direction. In the video, a ray is used to divide an angle into two smaller angles by passing through the vertex and an interior point of the angle. The script clarifies that this ray does not necessarily have to be an angle bisector and can pass through any point within the angle.

💡Angle Measure

The measure of an angle is the amount of rotation from one side of the angle to the other, typically expressed in degrees. The script emphasizes the importance of using 'm' with a subscript to denote the measure of an angle, such as m∠ABC, to accurately represent the numerical value of the angle's size.

💡Equation

An equation in the context of the video is a mathematical statement that represents the relationship between the measures of angles. The script demonstrates how to set up equations using the Angle Addition Postulate to solve for unknown angles. For example, the equation m∠ABF + m∠FBC = m∠ABC is used to find the measure of an unknown angle when the measures of the other two angles are known.

💡Variable

In the video, a variable, often represented by a letter like 'x', is used to represent an unknown value that needs to be determined. The script uses variables in equations to solve for the measures of unknown angles. For instance, if the measure of one angle is given as 3x and another as 2x - 6, the variable 'x' is used to express these measures before solving for its value.

💡Algebraic Expression

An algebraic expression is a combination of numbers, variables, and operations (like addition, subtraction, multiplication, and division) that represents a value. In the script, algebraic expressions are used to represent the measures of angles in terms of variables, which are then manipulated to solve for the unknown angles. For example, the expression '3x' is used to denote the measure of an angle in terms of a variable 'x'.

💡Check

Checking in the context of the video refers to the process of verifying the solution to an equation by ensuring that the sum of the known angle measures equals the measure of the whole angle. The script emphasizes the importance of checking to confirm that the calculated values are correct, such as ensuring that the sum of the measures of the smaller angles equals the measure of the larger angle.

Highlights

Introduction to the angle addition postulate and its significance in geometry.

Comparison between the segment addition postulate and the angle addition postulate.

Explanation of how to set up an angle with a point in its interior, creating two smaller angles.

Clarification that the dividing ray does not have to be an angle bisector.

Rule that the measures of the two smaller angles must add up to the measure of the larger angle.

Use of the angle addition postulate to find missing angle measures.

Demonstration of setting up an equation using the angle addition postulate.

Example problem showing how to find the measure of a missing angle.

Emphasis on checking the solution to ensure the angle measures add up correctly.

Introduction to a more algebraic example involving variables for angle measures.

Explanation of the importance of parentheses when using variables with units.

Guidance on setting up and solving an equation with variables for angle measures.

Advice on interpreting the results of an equation involving angle measures.

Method for checking the solution by adding the calculated angle measures.

Conclusion and invitation to the next session.

Transcripts

hi there geometers i am here to talk you

through

the angle addition postulate what it is

what it means how we use it

and give you some examples so first of

all the angle addition postulate assumes

you have an

angle let's go ahead and name my angle

angle abc

okay and it's like if you haven't seen

my video on segment edition postulate

you might want to look at that because

i'm going to make some comparisons here

the segment addition postulate and angle

addition postulate

say very similar things one's about

segments one's about angles

and like the segment addition postulate

it sort of goes with the idea of

between in terms of the setup here's

what i mean

you have an angle and then you have a

point

that is in the interior of the angle

okay like let's call this point

point f okay

and what that does is that sets up a

situation where you could draw a ray

that goes from the vertex of your angle

through point f and that ray

is between the two rays that make up the

original angle

so you pretty much just have one bigger

angle

split into two smaller pieces by this

ray that goes through it

now we are not implying that that ray

goes right through the middle i'm not

saying that's the angle bisector it just

goes

through somewhere it may not be right in

the middle of this angle

and therefore we're not going to make

any assumptions that it is

okay and so what we have is we have this

situation where

the smaller two angles then abf

and fbc remember to name an angle with

three letters we start with one side

go to the vertex and then go out the

other side always have to have the

vertex second

so i could have named this angle a b f

or i could have named it f b a

but b has to be in the middle so a b f

and f b c

[Music]

together make up the entire angle

okay so now what i just wrote is

actually incorrect

it's not incorrect in terms of what

we're saying it's incorrect notation

wise

because what we're really going to talk

about here is it's the

measure of the smaller two angles the

measure of this angle however many

degrees this angle is

and the measure of this angle however

many degrees between here

have to add up to the number of degrees

or the measure of the big angle

so we're going to use little m's to be

accurate

so the measure of one smaller angle plus

the measure of the other smaller angle

equals the measure of the whole bigger

angle and another way we can think of

that

just like segment addition postulate is

that one part

plus the other part equals the whole

thing

the whole angle is the sum of its parts

okay so that can be useful if we're

trying to find

things like missing angle measures or

trying to solve equations so in our

first example

i wrote this through this little angle

well

multiple angles um we have three angles

here we've got the whole big one and

then we've got each of the two smaller

ones

so let's suppose that i am given this

information about these

angles okay so i am told that the

measure of angle t

ms which you could think of let's go

ahead and do this

you could think of it as this angle

right here that's tms

is 12 degrees and l m t

which i'm going to think about like this

this whole angle all the way across

that's l

m t so lmt is the whole big one

is 39 degrees and i'm supposed to find

lms

which is this part right here that's the

leftover part

okay so let me just make a squiggle

these are not congruence markings okay

this is just me trying to show

where that angle is okay so again we've

got part plus part equals whole

and in this particular case the two

smaller parts are the two smaller

angles are and let's go from this side

l m s so the measure of angle l

m s plus s next smaller one the one

that's sort of pinkish here s

m t the measure of angle s m

t is equal to the whole thing which is

l m t

l m t okay

so that is how i can set up my equation

and i'm just going to substitute

in this information that i was given tms

is the same as smt okay you could turn

the angle

name around as long as you keep the

vertex in the middle so this is 12.

lmt is 39

and the one we don't know let's just

call x

we don't know the measure of angle lms

that's what we're being asked for in our

problem

so here's my equation the angle addition

postulate just pretty much

told me how to set up the equation so

that i can find that missing angle

and i'm going to subtract now that i

have an equation i just have to pull out

my equation solving

skills and i will be able to get the

answer

so x is 27 so in other words going back

we've got this was 12 degrees

and this is 27 degrees and then we can

sort of check ourselves here

not really sort of we can check

ourselves here 27 plus 12 has to add up

to the entire angle

and that's 39 and that is correct so my

answer for

x was 27 and in this case x

stood for lms so that was what i was

trying to find

anytime you solve an equation you have

to check and say

is the variable what i was trying to

find or do i have to maybe go back and

plug it in

but here the variable stood for the

entire angle and it was the one that we

were trying to find

so there's nothing left to plug in okay

let's look at another example like that

one that's perhaps a little bit more

algebraic

let's suppose i have this figure

and i am told that i have

this info okay i know that the measure

of angle

m n p m n

p i don't think that's too far over

there we go

m and p is 3x p and

r p n r is 2 x minus 6

and m n r that's the whole big one is

and i can see that these two smaller

angles here add up to the whole or

make up the whole big angle so it will

be an angle addition

postulate situation where the two

smaller parts

make up the entire angle okay um

now i did not write this completely

accurately on purpose because i wanted

to show it to you guys

sometimes this confuses people notice

that i didn't put degree symbols up here

um because it's not just a number if i

wanted to add the degree symbol which is

something i need because we measure

angles and degrees so it will be in

degrees

i really need parentheses now a lot of

teachers aren't going to be mad at you

if you don't put parentheses and you put

a degree symbol there

but i just want you to understand why

sometimes they do that

is they're saying this whole thing is a

number

and it's that many degrees in other

words there's nothing special about

those parentheses they're just there

because

it's kind of technically inaccurate to

put the degree symbol

beside something that's not a number

okay so now we're ready to start

setting up our equation and we use the

angle addition

postulate to set up our equation so we

have

part plus part equals whole in this case

one part is

mnp and just like i said in the segment

addition postulate you don't necessarily

have to write everything i'm writing now

the next part is pnr

and this is really just like if your

teacher asks you to write an equation

to show how it's set up then yeah write

it if you like to write it because it

helps you solve the problem

write it otherwise you don't have to the

whole thing is the measure of

mnr

and now i'm going to substitute this

information i was given mnp

is 3x plus

pnr is 2x minus 6

equals mnr is 44.

so now i have an equation i can solve

and i've got three x's and two x's is

five x's

and i solve my equation just like any

other equation

and i get x is 10. now perfect example

of

here i got the value of my variable but

i'm not sure if that's what's being

asked for i didn't actually tell you

what's being asked for so this is where

however i would go ahead and say is that

what i was

needing do i was i asked to find x or

find the value of the variable

or was i maybe asked something like

find the measure of angle

mnp if i was asked to find the measure

of angle mnp

i need to take this 10 and go put it in

to m and p and i get 3 times x becomes 3

times 10

which is 30 degrees

might have also it could have asked me

for angle pnr

pnr 10 plugged in here would give me 2

times 10 is 20

minus 6 20 minus 6 is 14 degrees

and here again we get to sort of check

ourselves because 30

plus 14 it does add up to 44.

so it all makes sense and we must have

done it right

i hope this helped and i'll see you

again next time