4. Do you think thar the previous RATIOS wil always be the same for THE SAME ANGLE SIZES although the triangles may differ in size? Explain your answer. [3]

Yes, the ratios of the sides of triangles will always be the same for the same angle sizes, regardless of the size of the triangles. This is due to the concept of similar triangles. ### Explanation: 1. **Similar Triangles**: Triangles that have the same angle sizes are called similar triangles. According to the properties of similar triangles, the corresponding angles are equal, and the lengths of corresponding sides are in proportion. 2. **Angle-Angle (AA) Criterion**: If two triangles have two angles that are equal, then the third angle must also be equal (since the sum of angles in a triangle is always 180 degrees). Therefore, if two triangles have the same angle sizes, they are similar. 3. **Side Ratios**: For similar triangles, the ratio of the lengths of corresponding sides is constant. This means that if triangle \( A \) has sides \( a_1, b_1, c_1 \) and triangle \( B \) has sides \( a_2, b_2, c_2 \), then the following ratios hold true: \[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \] This ratio is known as the scale factor. ### Conclusion: Thus, regardless of the size of the triangles, as long as the angles remain the same, the ratios of the sides will always be consistent. This is a fundamental property of triangles in Euclidean geometry.