Asked by D4VD
Analyzing Possible Angles
Sally states that a triangle can only have one obtuse or right angle. However, a triangle must have at least two acute angles. Is Sally correct? Explain.
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Answered by
GPT-5 mini
AI
Yes — Sally is correct.
Reason: let the triangle's interior angles be A, B, C (all > 0 and A + B + C = 180°).
- If two angles were ≥ 90° then A + B ≥ 180°, leaving C ≤ 0°, which is impossible for a triangle. So a triangle cannot have two right or two obtuse angles; it has at most one right or obtuse angle.
- If one angle is right (90°) or obtuse (>90°), the sum remaining for the other two angles is < 90° or ≤ 90°, so each of the other two must be < 90° (acute). If no angle is ≥ 90°, then all three are acute. Thus every triangle has at least two acute angles.
Examples: acute triangle 60°, 60°, 60°; right triangle 90°, 60°, 30°; obtuse triangle 120°, 40°, 20°.
Reason: let the triangle's interior angles be A, B, C (all > 0 and A + B + C = 180°).
- If two angles were ≥ 90° then A + B ≥ 180°, leaving C ≤ 0°, which is impossible for a triangle. So a triangle cannot have two right or two obtuse angles; it has at most one right or obtuse angle.
- If one angle is right (90°) or obtuse (>90°), the sum remaining for the other two angles is < 90° or ≤ 90°, so each of the other two must be < 90° (acute). If no angle is ≥ 90°, then all three are acute. Thus every triangle has at least two acute angles.
Examples: acute triangle 60°, 60°, 60°; right triangle 90°, 60°, 30°; obtuse triangle 120°, 40°, 20°.
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