In triangle DEF the measure of angle D is 46° and the measure fangle E is 104°. In triangle RsT. the measure of angle R is46° and the measure of angle S is 104°. Which of the following additional pieces of information is needed to determinewhether triangle DEF is similar to triangle RS7?

To determine whether two triangles are similar, we need to establish that their corresponding angles are congruent (angle-angle similarity criterion) or that their sides are proportional.

### Given Information:
- In triangle DEF:
  - \( \angle D = 46^\circ \)
  - \( \angle E = 104^\circ \)

- In triangle RST:
  - \( \angle R = 46^\circ \)
  - \( \angle S = 104^\circ \)

### Steps:

1. **Third Angle**: 
   Using the Triangle Angle Sum Theorem, the sum of the interior angles of a triangle is always \( 180^\circ \). We can find the third angle in each triangle by subtracting the sum of the known angles from \( 180^\circ \).

   For triangle DEF:
   \[
   \angle F = 180^\circ - 46^\circ - 104^\circ = 30^\circ
   \]
   For triangle RST:
   \[
   \angle T = 180^\circ - 46^\circ - 104^\circ = 30^\circ
   \]

2. **Conclusion about Similarity**:
   Now we know:
   - \( \angle D = \angle R = 46^\circ \)
   - \( \angle E = \angle S = 104^\circ \)
   - \( \angle F = \angle T = 30^\circ \)

   Since all corresponding angles are congruent, we can conclude that triangles DEF and RST are similar **by the angle-angle (AA) similarity criterion**.

### Additional Information Needed:
No additional information is needed to determine whether triangle DEF is similar to triangle RST because the angles alone establish similarity by the AA criterion.

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you prove triangles are congruent, not just similar?
2. What is the Side-Angle-Side (SAS) criterion for triangle similarity?
3. How do proportional sides affect triangle similarity?
4. Can two triangles be similar if only two angles are known to be congruent?
5. How would the similarity criteria change if the triangle was not a Euclidean triangle?

**Tip:** In similarity, it's important to remember that only angles are necessary for determining similarity, while sides are needed for congruence.

Explore how to determine whether triangles DEF and RST are similar using the angle-angle (AA) similarity criterion. Understand the Triangle Angle Sum Theorem and its application to find missing angles in triangles. Suitable for Grades 7-9 students learning geometry.

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