First, let's understand the given relationships between the angles:
1. Supplementary angles [tex]\(A\)[/tex] and [tex]\(B\)[/tex] imply that [tex]\[ A + B = 180^\circ \][/tex]
2. Complementary angles [tex]\(B\)[/tex] and [tex]\(C\)[/tex] imply that [tex]\[ B + C = 90^\circ \][/tex]
3. The measure of angle [tex]\(B\)[/tex] is 10 degrees less than three times the measure of angle [tex]\(C\)[/tex]:
[tex]\[ B = 3C - 10 \][/tex]
Now, we need to determine the measure of each angle.
### Step 1: Find the Measure of Angle [tex]\(C\)[/tex]
Since [tex]\( B \)[/tex] and [tex]\( C \)[/tex] are complementary:
[tex]\[ B + C = 90^\circ \][/tex]
We substitute [tex]\( B = 3C - 10 \)[/tex] into the above equation:
[tex]\[ (3C - 10) + C = 90 \][/tex]
Combine like terms:
[tex]\[ 3C + C - 10 = 90 \][/tex]
[tex]\[ 4C - 10 = 90 \][/tex]
Solve for [tex]\( C \)[/tex]:
[tex]\[ 4C = 100 \][/tex]
[tex]\[ C = 25^\circ \][/tex]
### Step 2: Find the Measure of Angle [tex]\(B\)[/tex]
Using [tex]\( C = 25^\circ \)[/tex] in the expression for [tex]\(B\)[/tex]:
[tex]\[ B = 3C - 10 \][/tex]
[tex]\[ B = 3(25) - 10 \][/tex]
[tex]\[ B = 75 - 10 \][/tex]
[tex]\[ B = 65^\circ \][/tex]
### Step 3: Find the Measure of Angle [tex]\(A\)[/tex]
Angles [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are supplementary:
[tex]\[ A + B = 180^\circ \][/tex]
Using [tex]\( B = 65^\circ \)[/tex]:
[tex]\[ A + 65 = 180 \][/tex]
[tex]\[ A = 180 - 65 \][/tex]
[tex]\[ A = 115^\circ \][/tex]
### Step 4: Calculate the Sum of the Measures of Angles [tex]\(A\)[/tex] and [tex]\(C\)[/tex]
Finally, we need the sum of angles [tex]\(A\)[/tex] and [tex]\(C\)[/tex]:
[tex]\[ A + C = 115^\circ + 25^\circ \][/tex]
[tex]\[ A + C = 140^\circ \][/tex]
Therefore, the sum of the measures of angles [tex]\(A\)[/tex] and [tex]\(C\)[/tex] is:
[tex]\[ \boxed{140} \][/tex]
