Question

given:
cf = df
\\( \overline { f c } \perp \overline { a e } \\)
\\( \overline { f d } \perp \overline { b e } \\)
prove:
\\( \angle 1 = \angle 2 \\)
based on the given information, which of the following lines would be in this proof?
triangle fce congruent to triangle fde by hl.
triangle fce congruent to triangle fde by la.
triangle acf congruent to triangle bdf by hl.

Explanation:

Step1: Analyze the given information

We know that \(CF = DF\) (hypotenuse for right - triangles \(FCE\) and \(FDE\)). Also, \(FC\perp AE\) and \(FD\perp BE\), so \(\angle FCE=\angle FDE = 90^{\circ}\). The hypotenuse - leg (\(HL\)) congruence criterion states that if the hypotenuse and a leg of one right - triangle are congruent to the hypotenuse and a leg of another right - triangle, then the two right - triangles are congruent. Here, \(FE\) is the common leg for right - triangles \(FCE\) and \(FDE\).

Step2: Check congruence criteria

For right - triangles \(FCE\) and \(FDE\):

• Hypotenuse: \(CF = DF\) (given)
• Leg: \(FE=FE\) (common side)

By the \(HL\) (Hypotenuse - Leg) congruence criterion for right - triangles, \(\triangle FCE\cong\triangle FDE\)

Step3: Analyze other options

• For \(LA\) (Leg - Angle) criterion: We are not given an angle (other than the right - angle) in the problem statement to use the \(LA\) criterion for \(\triangle FCE\) and \(\triangle FDE\).
• For \(\triangle ACF\) and \(\triangle BDF\): We have no information about the sides \(AC\) and \(BD\) (to use \(HL\) for these two triangles)

Answer:

Triangle FCE congruent to Triangle FDE by HL.