Triangle DEF is shown below. Use the drop-down menus to explain how to determine the angle measures of any triangle that is similar to triangle DEF. Click the arrows to choose an answer from each menu. A triangle similar to DEF will have an angle that measures degrees, an angle that measures Choose… degrees, and an angle that measures degrees.\
The correct answer and explanation is:
To determine the angle measures of any triangle similar to triangle DEF, we need to understand the basic properties of similar triangles.
- Angle Measures in Similar Triangles: Two triangles are said to be similar if they have the same shape but may differ in size. This means that corresponding angles in similar triangles are congruent (equal in measure), and the corresponding sides are proportional.
- Explanation of Angle Measures: For example, if triangle DEF has angle ∠D=30∘\angle D = 30^\circ, ∠E=60∘\angle E = 60^\circ, and ∠F=90∘\angle F = 90^\circ, then any triangle that is similar to triangle DEF will have the same angle measures. This is because the angles in similar triangles are always congruent. Step 1: Identifying Angles in Triangle DEF In triangle DEF, let’s assume we have:
- ∠D=30∘\angle D = 30^\circ
- ∠E=60∘\angle E = 60^\circ
- ∠F=90∘\angle F = 90^\circ
- Angle Measures in a Similar Triangle:
- A triangle similar to DEF will also have an angle that measures 30° (corresponding to ∠D\angle D).
- The second angle will measure 60° (corresponding to ∠E\angle E).
- The third angle will measure 90° (corresponding to ∠F\angle F).
- Conclusion: Therefore, for a triangle similar to triangle DEF, the angles will always measure 30°, 60°, and 90°. The similarity of the triangles guarantees that the angles are congruent, which means the angle measures will not change, no matter the scale or size of the triangle.
In summary, to determine the angle measures of any triangle similar to triangle DEF, we simply match the corresponding angles from triangle DEF.