When given angle measures of 25°, 65°, and
90°, is it impossible to draw a triangle, possible
to draw only one triangle, or possible to draw
more than one triangle? Explain.
The Correct Answer and Explanation is :
Given the angle measures of 25°, 65°, and 90°, it is possible to draw only one triangle. This conclusion arises from the fundamental properties of triangles and the sum of their interior angles.
Explanation:
- Sum of Angles in a Triangle: One of the primary properties of a triangle is that the sum of its interior angles must equal 180°. In this case, we can calculate the total of the given angles: [
25° + 65° + 90° = 180°
] Since the sum equals 180°, these angles can indeed form a triangle. - Triangle Type: The angles provided also suggest that this triangle is a right triangle, as one of its angles measures 90°. Right triangles have unique properties, such as the presence of a right angle and specific relationships among their sides, governed by the Pythagorean theorem.
- Uniqueness of the Triangle: In Euclidean geometry, if the measures of the angles of a triangle are known, the triangle is determined up to similarity. In other words, if you know the three angle measures, you can only create one unique triangle shape. The sides of the triangle can vary in length, but the angles will always remain constant at 25°, 65°, and 90°. This is a consequence of the Angle-Angle (AA) similarity postulate, which states that if two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
- Conclusion: Since the given angles add up to 180° and form a right triangle, only one triangle can be constructed with these specific angle measures. Thus, it is not only possible to draw a triangle with angles of 25°, 65°, and 90°, but only one distinct triangle can be formed with these angle measures.