In `Delta DEF` and `Delta PQR, DE = DF, angle F = angle P` and ` angle E = angle Q`. The two triangles are

Options

A. Isosceles but not necessarily are
B. Isosceles and congruent
C. Congruent but not isosceles
D. Neither congruent nor isosceles

Answer

To solve the problem, we need to analyze the given information about triangles DEF and PQR. We know the following:

1. DE = DF (two sides of triangle DEF are equal)
2. ∠F = ∠P (angle F in triangle DEF is equal to angle P in triangle PQR)
3. ∠E = ∠Q (angle E in triangle DEF is equal to angle Q in triangle PQR)

We need to determine the relationship between the two triangles.

### Step-by-Step Solution:

**Step 1: Identify Triangle DEF**
- We have triangle DEF with sides DE and DF being equal.
- Since DE = DF, triangle DEF is an isosceles triangle.

**Hint for Step 1:** Remember that in an isosceles triangle, the angles opposite the equal sides are also equal.

**Step 2: Analyze Angles in Triangle DEF**
- Since DE = DF, by the property of isosceles triangles, we have:
- ∠E = ∠F (the angles opposite the equal sides are equal).

**Hint for Step 2:** Use the property of isosceles triangles to relate the angles.

**Step 3: Identify Triangle PQR**
- Now, look at triangle PQR. We know:
- ∠F = ∠P
- ∠E = ∠Q

**Hint for Step 3:** Compare the angles of triangle DEF with those of triangle PQR.

**Step 4: Compare Angles**
- From the above, we have:
- ∠E = ∠Q and ∠F = ∠P.
- Therefore, we can conclude that:
- ∠D = ∠R (since the sum of angles in a triangle is 180°, and the remaining angles must be equal).

**Hint for Step 4:** Use the angle sum property of triangles to find relationships between angles.

**Step 5: Conclude the Relationship**
- Both triangles DEF and PQR have two angles equal (∠E = ∠Q and ∠F = ∠P).
- This means that the triangles are similar by the Angle-Angle (AA) criterion for similarity.

**Hint for Step 5:** Remember that if two angles of one triangle are equal to two angles of another triangle, the triangles are similar.

**Final Conclusion:**
- The two triangles DEF and PQR are similar triangles but not necessarily congruent, as we do not have information about the lengths of the sides.

**Final Answer:** The two triangles are similar but not necessarily congruent (isosceles).