1. **Problem Statement:** Analyze the flag of Ontario focusing on its geometric shapes, angles, and measurements. 2. **Q1a: Number of Different Triangles** - The flag contains several triangles with identical angles and side lengths. - To find how many different triangles are used, identify unique sets of angles and side lengths. 3. **Q1b: Sizes of Angles in Each Triangle** - Use the triangle angle sum property: $$\text{Sum of angles} = 180^\circ$$ - Measure or use given data to find each angle. 4. **Q1c: Area of the Flag After Removing Union Jack and Shield** - Calculate total area of the flag (assume rectangle dimensions if given). - Subtract areas of Union Jack and shield (use area formulas for rectangles and polygons). 5. **Q1d: Other Shapes on the Flag** - Identify shapes besides triangles, such as rectangles, parallelograms, or polygons. 6. **Q2a-c: Sketch and Label the Flag** - Draw the flag on grid paper. - Label all angles using geometric rules. - Label side lengths based on measurements or scale. 7. **Q3a-c: Design Your Own Flag** - Draw two rectangles each 12 cm by 20 cm. - On one, sketch and label angles and highlight geometric features: parallel lines, transversal, perpendicular lines, angle bisector, alternate exterior/interior angles, corresponding angles, opposite angles, same side exterior/interior angles. - Use the other rectangle for the final colored design. 8. **Q4: Explanation of Learning** - Describe how your flag design incorporates the angle relationships and geometric concepts learned. **Key Formulas and Rules:** - Sum of interior angles of triangle: $$180^\circ$$ - Sum of interior angles of polygon with $$n$$ sides: $$180^\circ \times (n-2)$$ - Area of rectangle: $$\text{length} \times \text{width}$$ - Area of triangle: $$\frac{1}{2} \times \text{base} \times \text{height}$$ **Final Notes:** - Use geometric properties to find unknown angles. - Apply measurement formulas to calculate areas. - Label all parts clearly for understanding.