b 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
am∠B+m∠C=159
B
b0<m∠C≤68
C
c90<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 △ABC is an obtuse triangle, one of its angle must be an obtuse angle.
The 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.
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