Angle Addition Postulate explained with examples
Summary
TLDRThe 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
🔍 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.
📐 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.
🧮 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.
Mindmap
Keywords
💡Angle Addition Postulate
💡Segment Addition Postulate
💡Vertex
💡Interior Point
💡Ray
💡Angle Measure
💡Equation
💡Variable
💡Algebraic Expression
💡Check
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
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