Notice that when the value of m∠B is minimum, the value of m∠C will be maximum.
C
c
Notice that when the value of m∠C is minimum, the value of m∠B will be maximum.
A
a
m∠B+m∠C=159
B
b
0 < m∠C ≤ 68
C
c
90 < m∠B ≤ 158
Practice makes perfect
a
We have been told that △ ABC is an obtuse triangle with m∠A=21 and ∠C is an acute angle. In order to find m∠B+m∠C, we will use the Triangle Angle-Sum Theorem.
The sum of the measures of the angles of a triangle is 180.
We illustrate the theorem as the following.
m ∠A+m ∠B+m ∠C=180
Applying the theorem, we can find m∠B+m∠C as the following.
21+m∠B+m∠C=180 ⇒ m∠B+m∠C=159
b
Considering the given information and the result that we found in the previous part, we can make the following conclusions. Remember that since â–³ ABC is an obtuse triangle, one of its angle must be an obtuse angle.
Conclusion I: & 90< m∠B < 180
Conclusion II: & 0 < m∠C < 90
Conclusion III: & m∠B+m∠C=159The value of m∠B affects the value of m∠C. When the value of m∠B is minimum, the value of m∠C will be maximum. In this case, the lower bound of m ∠B is 90. Let's find the upper bound of m ∠C using Conclusion III.
As a result, the range of whole numbers for m∠C can be written as the following. Notice that m∠C can take the value of 68.
0 < m∠C < 69
c
As in Part B, the value of m∠C also affects the value of m∠B. If m ∠C takes its minimum value, m∠B takes its maximum value. In this case, the lower bound of m∠C is 0. Let's find the upper bound of m∠B.
As a result, the range of whole numbers for m∠B can be written as the following. Notice that m∠B cannot take the value of 159 because that would mean m∠C=0.
90 < m∠B < 159
