1. **Stating the problem:** We are given seven angles labeled $m\angle 1$ through $m\angle 7$ with specific degree measures. The problem involves understanding how these angle measures were determined in two adjacent triangles sharing a common vertex. 2. **Recall the triangle angle sum rule:** The sum of the interior angles in any triangle is always $$180^\circ$$. 3. **Analyze the left triangle:** It has angles $m\angle 1 = 35^\circ$, $m\angle 2 = 65^\circ$, and $m\angle 3 = 29^\circ$. Let's check if these add up to 180°: $$35^\circ + 65^\circ + 29^\circ = 129^\circ$$ This sum is less than 180°, so there must be an additional angle at the bottom left corner given as 80° (from the description). Adding this: $$129^\circ + 80^\circ = 209^\circ$$ This suggests the 80° is not part of this triangle's interior angles but possibly an exterior or adjacent angle. 4. **Analyze the shared vertex angles:** Angles $m\angle 4 = 115^\circ$ and $m\angle 5 = 165^\circ$ are at the shared vertex. Since these two angles are adjacent and on a straight line, their sum should be 180°: $$115^\circ + 165^\circ = 280^\circ$$ This is more than 180°, so likely these angles are not linear pairs but part of different triangles or configurations. 5. **Analyze the right triangle:** It has angles $m\angle 5 = 165^\circ$, $m\angle 6 = 36^\circ$, and $m\angle 7 = 144^\circ$. Check their sum: $$165^\circ + 36^\circ + 144^\circ = 345^\circ$$ This is more than 180°, so these cannot be interior angles of a triangle. 6. **Conclusion:** The given angle measures likely come from a more complex geometric configuration involving adjacent triangles and possibly exterior angles or overlapping angles. The key rule used is that the sum of interior angles in a triangle is 180°, and adjacent angles on a straight line sum to 180°. The angles were found by applying these rules and possibly subtracting or adding angles to find unknown measures. **Final note:** To find each angle, the process involves using the triangle sum property and linear pair property, then solving for unknown angles by subtraction or addition.