Video Transcript
In the following figure, find the
measure of all the angles in triangle 𝐴𝐵𝐶.
In this question, we’re given a
figure containing expressions for the measure of the internal angles in a triangle
𝐴𝐵𝐶 in terms of an unknown 𝑥. We need to determine the measures
of all three of these angles. To do this, we can start by
recalling that the sum of all of the internal angles’ measures in a triangle is 180
degrees. Therefore, we have that the measure
of angle 𝐴 plus the measure of angle 𝐵 plus the measure of angle 𝐶 is 180
degrees. We can substitute the given
expressions for the angle measures into this equation and remove the units to obtain
that 𝑥 plus one plus three 𝑥 minus one plus three 𝑥 is equal to 180.
To solve this equation for 𝑥, we
want to isolate 𝑥 on one side of the equation. To do this, let’s combine the like
terms to get seven 𝑥 and then calculate that one minus one is zero. So, we have seven 𝑥 equals
180. We can then solve for 𝑥 by
dividing both sides of the equation by seven. We get that 𝑥 equals 180 over
seven. We cannot simplify this any further
since seven and 180 share no common factors above one.
We can use this value of 𝑥 to find
the measures of the three internal angles by substituting them into the given
expressions for the measure and evaluating. First, we substitute 𝑥 equals 180
over seven into the expression for the measure of angle 𝐴. We get the measure of angle 𝐴
equals 180 over seven plus one. We can evaluate this expression by
rewriting one as seven over seven. We get that the measure of angle 𝐴
is 187 over seven degrees.
We can follow the same process for
the measure of angle 𝐵. We have that the measure of angle
𝐵 equals three times 180 over seven minus one. We can then evaluate this
expression by multiplying the numerator by three and rewriting one as seven over
seven. We get that the measure of angle 𝐵
is 533 over seven degrees. Finally, substituting 𝑥 equals 180
over seven into the expression for the measure of angle 𝐶 and evaluating gives us
that the measure of angle 𝐶 is 540 over seven degrees.
It can be useful to check that the
sum of these angle measures is 180 degrees. If we added these measures
together, we would obtain 187 plus 533 plus 540 all over seven, which is 180
degrees. Hence, the measure of angle 𝐴 is
187 over seven degrees. The measure of angle 𝐵 is 533 over
seven degrees. And the measure of angle 𝐶 is 540
over seven degrees.
