1. **State the problem:** We are given a triangle with angles expressed as algebraic expressions: $5x - 11$, $2x + 20$, and $11x - 63$. We need to solve for $x$ and then find the measures of angles 3 and 6 in the figure. 2. **Use the triangle angle sum formula:** The sum of the interior angles of a triangle is always 180 degrees. $$ (5x - 11) + (2x + 20) + (11x - 63) = 180 $$ 3. **Simplify the equation:** $$ 5x - 11 + 2x + 20 + 11x - 63 = 180 $$ Combine like terms: $$ (5x + 2x + 11x) + (-11 + 20 - 63) = 180 $$ $$ 18x - 54 = 180 $$ 4. **Solve for $x$:** Add 54 to both sides: $$ 18x - \cancel{54} + \cancel{54} = 180 + 54 $$ $$ 18x = 234 $$ Divide both sides by 18: $$ \frac{18x}{\cancel{18}} = \frac{234}{\cancel{18}} $$ $$ x = 13 $$ 5. **Find the measures of the three angles in the triangle:** - Angle 1: $5x - 11 = 5(13) - 11 = 65 - 11 = 54^6$ - Angle 2: $2x + 20 = 2(13) + 20 = 26 + 20 = 46^6$ - Angle 7: $11x - 63 = 11(13) - 63 = 143 - 63 = 80^6$ 6. **Find angle 3:** Given $m\angle1 = 71^6$ and $m\angle2 = 29^6$, and the quadrilateral inside the triangle has angles $80^6$, $115^6$, and $61^6$ at corners, angle 3 is adjacent to angle 1 and angle 2. Since angles around a point sum to 360 degrees, and angle 7 is $90^6$, angle 3 can be found by subtracting the known angles from 360: $$ m\angle3 = 360 - (m\angle1 + m\angle2 + m\angle7) = 360 - (71 + 29 + 90) = 360 - 190 = 170^6 $$ 7. **Find angle 6:** Given $m\angle4 = 115^6$ and $m\angle5 = 29^6$, and angle 6 is adjacent to these, angle 6 can be found by subtracting the sum of angles 4 and 5 from 180 (assuming a straight line or supplementary angles): $$ m\angle6 = 180 - (m\angle4 + m\angle5) = 180 - (115 + 29) = 180 - 144 = 36^6 $$ **Final answers:** $$ m\angle3 = 170^6 $$ $$ m\angle6 = 36^6 $$