This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions for Exercise 3, question 1: 1a) Step 1: Calculate the third angle for each triangle. For the first triangle (ABC): A = 180^ - 55^ - 68^ = 57^ The angles are 57^, 55^, 68^. For the second triangle (PQR): R = 180^ - 55^ - 68^ = 57^ The angles are 55^, 68^, 57^. Step 2: Compare the angles. Both triangles have angles 55^, 68^, 57^. Since all corresponding angles are equal, the triangles are similar by the Angle-Angle-Angle (AAA) similarity criterion. Step 3: State the similar triangles in the correct order. Matching angles: A = R = 57^, B = P = 55^, C = Q = 68^. ABC RPQ 1b) Step 1: Compare the angles. The first triangle (ABC) has angles marked with one arc ( A) and two arcs ( B). The second triangle (PQR) has angles marked with one arc ( P) and two arcs ( Q). This indicates that A = P and B = Q. Step 2: Determine similarity. Since two angles of one triangle are equal to two angles of another triangle, their third angles must also be equal ( C = R). Therefore, the triangles are similar by the Angle-Angle (AA) similarity criterion. Step 3: State the similar triangles in the correct order. Matching angles: A = P, B = Q, C = R. ABC PQR 1c) Step 1: Calculate the third angle for each triangle. For the first triangle (ABC): A = 180^ - 70^ - 70^ = 40^ The angles are 40^, 70^, 70^. For the second triangle (PQR): Q = 180^ - 70^ - 70^ = 40^ The angles are 70^, 40^, 70^. Step 2: Compare the angles. Both triangles have angles 40^, 70^, 70^. Since all corresponding angles are equal, the triangles are similar by the Angle-Angle-Angle (AAA) similarity criterion. Step 3: State the similar triangles in the correct order. Matching angles: A = Q = 40^, B = P = 70^, C = R = 70^. ABC QPR 1d) Step 1: Calculate the third angle for each triangle. For the first triangle (ABC): C = 180^ - 69^ - 37^ = 74^ The angles are 69^, 37^, 74^. For the second triangle (PQR): Q = 180^ - 54^ - 57^ = 69^ The angles are 54^, 57^, 69^. Step 2: Compare the angles. The angles of ABC are 69^, 37^, 74^. The angles of PQR are 54^, 57^, 69^. The sets of angles are not identical. Only one angle (69^) is common, but not all three. Step 3: Determine similarity. Since the corresponding angles are not all equal, the triangles are not similar. The triangles are not similar. Send me the next one 📸