Question

Proving triangles congruent : Given: FGH and FJH are right triangles, GH JH

Prove: FGH ≅ FJH

Answer 1

Triangles FGH and FJH are proven congruent by the HL Congruence Theorem, which applies because they are right triangles with a shared hypotenuse FH and a congruent leg GH.

Step-by-step explanation:

To prove that triangles FGH and FJH are congruent, we will use the properties of right triangles and congruence postulates or theorems such as the HL (Hypotenuse-Leg) Congruence Theorem or SSS (Side-Side-Side). Since both triangles are given as right triangles, they have one right angle each.

We are also given that GH is congruent to JH, which suggests that they share the hypotenuse FH. Now, by the HL Congruence Theorem, which states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent. Therefore, we have all the necessary components (right angles, shared hypotenuse FH, leg GH) to conclude that triangles FGH and FJH are congruent.

Answer 2

To prove triangle congruency for right triangles FGH and FJH, the RHS criterion is used, showing both have a right angle, share side FH as the hypotenuse, and have equal sides GH and JH.

Step-by-step explanation:

To prove that the right triangles FGH and FJH are congruent, we can use the Right Angle-Hypotenuse-Side (RHS) criterion.

Given that both triangles are right-angled, by definition they each have a 90-degree angle at H.

Furthermore, it's given that GH equals JH, meaning that both triangles share the hypotenuse FH as a common side.

The third congruency criterion is satisfied because both GH and JH are equal in length, serving as the side opposite to the right angle in their respective triangles.

Therefore, by the RHS criterion, we have angle FHG equal to angle FJH (both 90 degrees), FH as a common side, and side GH equal to side JH.

Consequently, triangle FGH is congruent to triangle FJH (FGH ≅ FJH).