1. **Problem Statement:** We need to analyze a triangle with an exterior angle formed by extending one side, a line inside the triangle creating two interior angles, and at least five labelled angles with three given angle measurements. 2. **Key Concepts:** - The exterior angle of a triangle is equal to the sum of the two opposite interior angles. - The sum of interior angles in any triangle is always $180^\circ$. - A line inside the triangle (like a median or segment) can create two interior angles whose sum relates to the triangle's angles. 3. **Setup:** - Let the triangle be $\triangle ABC$. - Extend side $BC$ beyond $C$ to point $D$ to form exterior angle $\angle ACD$. - Draw a segment inside the triangle from vertex $A$ to side $BC$ at point $E$, creating two interior angles $\angle BAE$ and $\angle EAC$. 4. **Given Angles:** - $\angle ABC = 50^\circ$ - $\angle BAC = 40^\circ$ - $\angle ACD$ (exterior angle) is given as $70^\circ$ 5. **Find:** The remaining angles and verify the relationships. 6. **Calculations:** - Since $\angle ACD$ is an exterior angle at vertex $C$, it equals the sum of the two opposite interior angles: $$\angle ACD = \angle BAC + \angle ABC = 40^\circ + 50^\circ = 90^\circ$$ - But given $\angle ACD = 70^\circ$, this suggests a different configuration or a typo; assuming the problem wants to illustrate the exterior angle theorem, we proceed with the theorem. - The interior angle at $C$ is: $$\angle ACB = 180^\circ - (\angle ABC + \angle BAC) = 180^\circ - (50^\circ + 40^\circ) = 90^\circ$$ - The segment $AE$ inside the triangle creates two angles $\angle BAE$ and $\angle EAC$ such that: $$\angle BAE + \angle EAC = \angle BAC = 40^\circ$$ 7. **Summary:** - Five labelled angles: $\angle ABC = 50^\circ$, $\angle BAC = 40^\circ$, $\angle ACB = 90^\circ$, $\angle BAE$, and $\angle EAC$. - Exterior angle $\angle ACD = 90^\circ$ (by theorem). - The line inside the triangle splits $\angle BAC$ into two parts. **Final answer:** The exterior angle equals the sum of the two opposite interior angles, and the interior segment divides one angle into two parts summing to the original angle.