Question

Use the information and the diagram to answer the question.

Information
segment L F is parallel to segment K G

K is the midpoint of segment L H



Two triangles.Short description, Two triangles.,Long description,
Triangle LFK is a right triangle with 90 degree Angle FLK. Side FK is perpendicular to LH. Side LF is the hypotenuse.

Triangle KGH is a right triangle with 90 degree Angle GHK. Side GH is perpendicular to LH. Side KG is the hypotenuse.


Question
Which theorem will prove that triangle L F K is congruent to triangle K G H?

Answer options with 5 options
A.
A S A

B.
A A S

C.
S A S

D.
S S A

E.
S S S

Answer 1

To determine which theorem can be used to prove that triangle LFK is congruent to triangle KGH, we need to analyze the information given.

1. **Triangles and Angles**: We know that angle FLK and angle GHK are both right angles. Therefore, we can say:
\[
\angle FLK \cong \angle GHK \quad \text{(both are 90 degrees)}
\]

2. **Parallel Lines**: Since segment LF is parallel to segment KG and they are both perpendicular to line LH, we can use properties of parallel lines and transversals to confirm that the angles formed by K and G with LH are also congruent.

3. **Midpoint**: K is the midpoint of segment LH, meaning that segment LK is congruent to segment KH. We can represent this with:
\[
LK \cong KH \quad \text{(since K is the midpoint)}
\]

4. **Side Relationships**: We know that FK and GH are both perpendicular to LH and since both triangles share the line segment LF and KG would also be the same or relate through congruence properties.

Putting this all together, we have:

- One pair of congruent angles (both right angles).
- One pair of congruent sides (LK and KH).
- The hypotenuse of triangle LFK (LF) is congruent to the hypotenuse of triangle KGH (KG).

Based on this information, we can see that the triangles can be proven congruent using the **AAS (Angle-Angle-Side)** theorem, because:
- Two angles are congruent (both right angles).
- One side (the side adjacent to the right angle or the side opposite to one of the angles) is congruent.

Thus, the correct answer is:

**B. A A S**