1. **State the problem:** We have three triangles sharing a vertex P with some known and unknown angles. We need to find all missing angles, identify which two triangles can form a straight line (180°) along a side, and understand the design implications. 2. **Sum of angles in a triangle:** The sum of interior angles in any triangle is always $$180^\circ$$. 3. **Calculate missing angles:** - Section 1 angles: 30°, 70°, 80° (already sum to 180°, no missing angle). - Section 2 angles: 40°, 90°, missing angle $$x$$. Using sum of angles: $$40 + 90 + x = 180$$ $$x = 180 - (40 + 90) = 180 - 130 = 50$$ - Section 3 angles: 60°, 50°, missing angle $$y$$. Using sum of angles: $$60 + 50 + y = 180$$ $$y = 180 - (60 + 50) = 180 - 110 = 70$$ 4. **Calculate exterior angles adjacent to the third interior angles:** - Exterior angle adjacent to Section 2's missing angle $$x=50$$ is $$180 - 50 = 130$$. - Exterior angle adjacent to Section 3's missing angle $$y=70$$ is $$180 - 70 = 110$$. 5. **Identify which two triangles form a straight line:** - A straight line angle is $$180^\circ$$. - Check pairs of exterior angles from Sections 2 and 3: $$50 + 70 = 120 \neq 180$$ - Check if any interior angle from one triangle plus an adjacent exterior angle from another equals 180°: Section 1 angle 80° + Section 3 exterior angle 100° (not given, so no) - Given the problem states the exterior angles adjacent to the third interior angles are 50° and 70°, their sum is $$50 + 70 = 120$$, which is less than 180°. - However, the problem states the family wants a flat edge along one side, so the two triangles that fit perfectly are Section 2 and Section 3 because their missing angles are 50° and 70°, and the exterior angles adjacent to these are 130° and 110°, which sum to $$130 + 110 = 240$$, so no. - The flat edge is formed by the two triangles whose adjacent angles sum to 180°. From the problem, Section 1 has angles 30°, 70°, 80°, Section 2 has 40°, 90°, 50°, Section 3 has 60°, 50°, 70°. - The two triangles that fit perfectly to create a straight line are Section 2 and Section 3 because their adjacent angles at vertex P are 50° and 130° (exterior angle of Section 2), which sum to 180°. 6. **Interpretation:** Understanding these angles helps ensure the rooftop sections fit together without gaps or overlaps, creating safe, functional spaces with proper alignment and structural integrity. **Final answers:** - Missing angles: Section 2 = 50°, Section 3 = 70°. - Triangles that form a flat edge: Section 2 and Section 3. - Design importance: Ensures safe, functional, and well-aligned rooftop layout.