Question

When mapping ∆ D(-5,0)E(-3,0)F(-3,4) onto ∆ J(1,2)K(1,4)L(5,4) to prove they are congruent, which of the following identifies a pair of corresponding parts that are also congruent?

a. D ≅ L
b. F ≅ J
c. DE ≅ JK
d. DF ≅ KL

Answer 1

To determine which pairs of parts are congruent when mapping triangle ∆ DEF onto triangle ∆ JKL, we should analyze the vertices and the sides of both triangles.

1. **Vertices of ∆ DEF:**
- D(-5, 0)
- E(-3, 0)
- F(-3, 4)

2. **Vertices of ∆ JKL:**
- J(1, 2)
- K(1, 4)
- L(5, 4)

Next, we can calculate the lengths of the sides in each triangle to compare corresponding parts.

### Lengths of sides in ∆ DEF:
- **DE:** The length between points D(-5,0) and E(-3,0) is calculated by:
\[
DE = |-5 - (-3)| = | -5 + 3 | = 2
\]

- **EF:** The length between points E(-3,0) and F(-3,4) is:
\[
EF = |0 - 4| = 4
\]

- **DF:** The length between points D(-5,0) and F(-3,4) is calculated using the distance formula:
\[
DF = \sqrt{(-3 - (-5))^2 + (4 - 0)^2} = \sqrt{(2)^2 + (4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]

### Lengths of sides in ∆ JKL:
- **JK:** The length between points J(1,2) and K(1,4) is:
\[
JK = |2 - 4| = 2
\]

- **KL:** The length between points K(1,4) and L(5,4) is:
\[
KL = |1 - 5| = 4
\]

- **JL:** The length between points J(1,2) and L(5,4) is:
\[
JL = \sqrt{(5 - 1)^2 + (4 - 2)^2} = \sqrt{(4)^2 + (2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}
\]

### Corresponding Parts:
Now we can compare:
- DE (2) corresponds to JK (2).
- EF (4) corresponds to KL (4).
- DF (2√5) corresponds to JL (2√5).

Now, let's analyze the options given:
a. D ≅ L (not true, as they have different coordinates).
b. F ≅ J (not true, as they have different coordinates).
c. DE ≅ JK (this is true; both have length 2).
d. DF ≅ KL (not true, DF ≅ JL).

### Conclusion:
The correct answer is:
**c. DE ≅ JK**