1. **State the problem:** We have a triangle ABC with angles $a$, $b$, and $c$. Given: - $a$ is 13 less than $c$, so $a = c - 13$ - $b$ is 11 less than four times $c$, so $b = 4c - 11$ We need to find the measure of each angle $a$, $b$, and $c$. 2. **Use the triangle angle sum property:** The sum of angles in a triangle is always 180 degrees. $$a + b + c = 180$$ 3. **Substitute the expressions for $a$ and $b$:** $$ (c - 13) + (4c - 11) + c = 180 $$ 4. **Simplify the equation:** $$ c - 13 + 4c - 11 + c = 180 $$ $$ (c + 4c + c) - (13 + 11) = 180 $$ $$ 6c - 24 = 180 $$ 5. **Solve for $c$:** Add 24 to both sides: $$ 6c - 24 + 24 = 180 + 24 $$ $$ 6c = 204 $$ Divide both sides by 6: $$ \cancel{6}c = \frac{204}{\cancel{6}} $$ $$ c = 34 $$ 6. **Find $a$ and $b$ using $c=34$:** $$ a = c - 13 = 34 - 13 = 21 $$ $$ b = 4c - 11 = 4(34) - 11 = 136 - 11 = 125 $$ 7. **Check the sum:** $$ a + b + c = 21 + 125 + 34 = 180 $$ This confirms the solution is correct. **Final answer:** $$ a = 21^\circ, \quad b = 125^\circ, \quad c = 34^\circ $$