Collinearity, Betweenness, and Assumptions, level 4.
In this final video we will go over 4 examples that involve angle measurements.
We will be making assumptions from diagrams in order to solve these problems.
Let's start with the first example.
Find the measure of angle ABC.
In this problem we are provided with a diagram and are asked to find the measurement of angle
ABC.
We can determine the measurement of this angle by finding the value of the variable X and
then we substitute this value into the algebraic expression of the angle and simplify.
We first need to set up a geometric relation that will allow us to solve for the variable
X. Notice that points A, B and D are collinear this means that they lie in the same line.
This also means that this line represents a straight angle.
Recall that straight angles measure 180 degrees.
Also notice that this straight angle is formed by two angles in this case angle ABC and angle
CBD.
So the sum of the measures of angle ABC plus angle CBD must be equal to 180 degree since
both angles form a straight angle.
We can now substitute the algebraic expressions for each angle into this geometric relation
doing that we obtain the following equation.
Now it is just a matter of simplifying and collecting like terms.
Solving for the unknown variable we obtain x equals 42.
We now have to substitute this value into the algebraic expression representing the
measurement of angle ABC doing that and simplifying we obtain 134 degrees and this is our final
answer.
Let's try the next example.
Angle ABC is a right angle.
The ratio of the measures of angle ABD and angle DBC is 3 to 2.
Find the measure of angle ABD.
In this problem we are provided with a diagram of a right angle that is formed by two smaller
angles.
We are given a ratio between the two smaller angles and are asked to determine the measurement
of angle ABD.
In order to determine the measurement of angle ABD we need to figure out the combination
of angles that would add up to 90 degrees and have a ratio of 3 to 2.
In other words the measurement of angle ABD plus the measurement of angle DBC should equal
the measurement of angle ABC which is a right angle so the measurements should add to 90
degrees.
Also the ratio between the measurement of angle ABD and the measurement of angle DBC
should equal 3 over 2.
We can systematically go through various angles and determine what combination of angles meets
these two constraints or we can solve the problem by using algebra.
We can write an equivalent ratio by multiplying the fraction 3 over 2 by x over x.
This way we can now replace the measure of angle ABD with 3x and replace the measure
of angle DBC with 2x.
We then use these algebraic expressions and substitute them into the geometric relation,
doing that we obtain the following equation.
Now it is just a matter of simplifying the equation and solving for x, doing that we
obtain x equals 18.
We then use this value of x to find the measure of angle ABD doing that we obtain 54 degrees
and this is our final answer.
Let's move along to the next example.
The measure of angle 1, angle 2 and angle 3 are in the ratio 1 to 3 to 2.
Find the measure of each angle.
Similar to the previous problem we are provided with a diagram with various angles.
We are given the ratio of the measurement between all 3 angles.
We are asked to find the measurement of all three angles.
Let's first determine the geometric relation of the problem.
Notice that angle 1, angle 2 and angle 3 form a straight angle.
So we can set up the following geometric relation, the measure of angle 1 plus the measure of
angle 2 plus the measure of angle 3 is equal to 180 degree.
Next we need to determine the algebraic relations between the angles, since we know that the
ratio between the measures of the angles is 1 to 3 to 2 if we multiply all the values
of this ratio by x we should obtain an equivalent ratio.
By doing this we can now assign an algebraic expression to each of the three angles.
We can now use these algebraic expressions and substitute them into the geometric relation
doing that we obtain the following.
Now it is just a matter of simplifying the expression and solving for X, doing that we
obtain X equals 30.
The last step is to substitute this value of X into the algebraic expression of each
angle doing that we obtain 30, 90 and 60 degrees and this is our final answer.
Alright let's end the video by going over the final example.
Given the measure of angle 1 equals 2x plus 40, the measure of angle 2 equals 2y plus
40, the measure of angle 3 equals x plus 2y, find the measure of angle 1, the measure of
angle 2 and the measure of angle 3.
In this problem we are provided with a diagram and are given algebraic relations for 3 angles.
We are asked to determine the measurement of all 3 angles.
For this problem we are first going to determine the geometric relations between the various
angles.
Notice that angle 1 and angle 2 form a straight angle so we can set up the following geometric
relation the measurement of angle 1 plus the measurement of angle 2 is equal to 180.
In a similar fashion angle 2 and angle 3 also form a straight angle so we can set up the
following geometric relation the measurement of angle 2 plus the measurement of angle 3
equals 180 degrees.
Now that we have these two geometric relations let's go ahead and substitute the measurement
of the angles with the given algebraic expressions.
Doing that we obtain the following.
Let's go ahead and simplify both of these equations by collecting like terms and keeping
the variables on the left side of the equation and any constants on the right side of the
equation doing that we obtain the following.
We are now face to face with a system of linear equations so we will use the substitution
method to solve this system.
Solving for x in the first equation we obtain the following.
Now we go ahead and substitute this expression into the second equation.
From here it is just a matter of simplifying and collecting like terms, we then solve for
the variable y doing that we obtain y equals 30.
Next we take this value for y and substitute it into the first equation this allows us
to solve for the variable x doing that we obtain x equals 20.
The last step is to evaluate the algebraic expression for each of the 3 angles we do
this by substituting the values for x and y.
Doing that and simplifying we obtain 80 degrees for the measurement of angle 1, 100 degrees
for the measurement of angle 2 and 80 degrees for the measurement of angle 3 and this is
our final answer.
In our next series of videos we will start learning how to write simple proofs.
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