Question

In isosceles triangle FGH, FG is congruent to GH. If Elijah draws a line segment from the vertex G to the midpoint of FH, labeling the midpoint K, what can he prove about triangles FGK and HGK?

A= The triangle are neither congruent nor similar
B= The triangles are congruent, but not similar
C= The triangles are similar, but not congruent
D= The triangles are both congruent and similar

Answer 1

B= The triangles are congruent, but not similar

Answer 2

B) The triangles are congruent, but not similar.

Explanation:

Givens

• 
  `\triangle FGH` is isosceles.
• 
  `FG \cong GH`.

In the image attached you can observe what Elijah did. The vertical line goes from vertex G to the midpoint K. When this happens, the resulting right triangles are congruents, that is, because we already know that FG is congruent to GH, GK is common for both triangles and the pair of angles in the base of the isosceles are congruent, by definiton.

Therefore, the triangles are congruent, but no similar. Choice B is correct.

Remember, when we talk about similarity, it refers to proportional corresponding sides of the triangles. When we talk about congruence, it refers to the exact equivalence between corresponding sides of both triangles. That's the difference. So, as you can deduct, a triangle cannot be similar and congruent at the same time.