Question

are the two triangles congruent? if so, write the congruence statement
choose the correct answer below
a. yes, ( \triangle bcdcong\triangle gfe ) by the hl theorem.
b. yes, ( \triangle bcdcong\triangle gef ) by the hl theorem
c. yes, ( \triangle bcdcong\triangle egf ) by the hl theorem.
d. yes, ( \triangle bcocong\triangle efg ) by the hl theorem.
e. no, the triangles are not congruent.

Explanation:

Step1: Recall the HL (Hypotenuse - Leg) congruence theorem

The HL theorem 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.
In right - triangle \(BCD\) and right - triangle \(GFE\):

• The hypotenuse \(BD = GE=15\)
• The leg \(DC = GF = 12\)

Step2: Check the correspondence of vertices

For congruent triangles, the vertices must correspond. In \(\triangle BCD\) and \(\triangle GFE\), when we use the HL theorem, the right angles are \(\angle C\) and \(\angle F\).
The hypotenuse \(BD\) in \(\triangle BCD\) corresponds to the hypotenuse \(GE\) in \(\triangle GFE\), and the leg \(DC\) in \(\triangle BCD\) corresponds to the leg \(GF\) in \(\triangle GFE\). So, \(\triangle BCD\cong\triangle GFE\)

Answer:

A. Yes, \(\triangle BCD\cong\triangle GFE\) by the HL theorem.