In a triangle ABC, the measure of angle A is fifteen less than twice the measure of angle B. The measure of angle C equals the sum of the measures of angle A and angle B. What is the measure of angle B?

2 Answers
Oct 31, 2015

If the reference to "fifteen" means #15^@#
then #B=35^@#.

If the refrence to "fifteen" means #15# radians
then #B=pi/6+5# radians.

Explanation:

We are told explicitly that
[1]#color(white)("XXXX")A+15 = 2B#
[2]#color(white)("XXXX")C=A+B#
and, implicitly (assuming the triangle lies in a Cartesian plane)
[3]#color(white)("XXXX")A+B+C=180 [pi]#
#color(white)("XXXX")#note: I will use degrees as the default assumption but place radian measures in square brackets when it differs.

Arranging these into standard form:
[4]#color(white)("XXXX")A-2B=-15#
[5]#color(white)("XXXX")A+B-C=0#
[6]#color(white)("XXXX")A+B+C=180color(white)("XXX") [pi]#

Adding [5] and [6]
[7]#color(white)("XXXX")2A+2B=180color(white)("XXX") [pi]#
Multiplying [4] by 2
[8]#color(white)("XXXX")2A-4B=-30#
Subtracting [8] from [7]
[9]#color(white)("XXXX")6B = 210color(white)("XXX") [pi+30]#
Dividing by 6
[10]#color(white)("XXXX")B=35color(white)("XXX") [pi/6+5]#

#B=35^@#

Explanation:

First of all, let's translate this problem into mathematical language. I suppose that we are working with degree measures of angles (so that writing "fifteen" you meant "fifteen degrees").
#A=2B-15^@#
#C=A+B#
We also have to "decode" another crucial information: these are angles in a triangle, so their sum equals #180^@#:
#A+B+C=180^@#

We want to find the measure of the angle #B# and we have 3 equations and 3 unknowns.
The measure of the angle #C# is very easy to find, so we compute it to reduce the number of variables involved: since #A+B=C#, we take the third equation and substitute the second one
#180^@=A+B+C=(A+B)+C=C+C=2C#
So dividing both sides by #2#, we get that #C=90^@# is a right angle and the second equation turns to #A+B=90^@#, meaning that #A=90^@-B#.

Now we substitute this last result in the first equation:
#2B-15^@=A=90^@-B#
and we get an equation where #B# is the only unknown
#2B-15^@=90^@-B#
Let's solve this:
#B+(2B-15^@)=(90^@-B)+B# (sum #B# on both sides)
#3B-15^@=90^@#
#15^@+(3B-15^@)=(90^@)+15^@# (sum #15^@# on both sides)
#3B=105^@#
#(3B)/3=(105^@)/3# (divide both sides by #3#)
#B=35^@#

Note: The only angle left is #A# and the computation of #A# can be done using one of the 3 former equations. We get that #A=55^@#.