Question

determine if the two triangles are congruent. if they are, state the postulate or theorem that proves they are congruent.

a. sss
b. aas

Explanation:

Step1: Identify Given Information

We see markings: \( GH = KJ \) (double tick), \( GI = JI \) (single tick), and \( \angle GIH = \angle JIK = 90^\circ \) (right angle). Also, \( HI = KI \)? Wait, no—wait, the triangles are \( \triangle GIH \) and \( \triangle JIK \)? Wait, no, looking at the diagram, the two triangles are \( \triangle GHI \) and \( \triangle JKI \)? Wait, actually, the vertical angles or the right triangles. Wait, let's re-examine:

• \( GH \cong KJ \) (double ticks)
• \( GI \cong JI \) (single ticks)
• \( \angle GIH \) and \( \angle JIK \) are right angles (so \( \angle GIH = \angle JIK = 90^\circ \))

Wait, but also, the hypotenuse and one leg? Wait, no, maybe HL? Wait, no, the options include SSS. Wait, let's check sides:

If \( GH = KJ \), \( GI = JI \), and \( HI = KI \)? Wait, no, maybe the triangles are \( \triangle GHI \) and \( \triangle JKI \). Wait, the segments: \( GH \) and \( KJ \) are equal (double ticks), \( GI \) and \( JI \) are equal (single ticks), and \( HI \) and \( KI \)? Wait, no, maybe the third side. Wait, actually, the two triangles have three pairs of equal sides: \( GH = KJ \), \( GI = JI \), and \( HI = KI \) (since \( I \) is the midpoint? Wait, no, the right angles: \( \angle GIH \) and \( \angle JIK \) are vertical angles? Wait, no, they are right angles, so \( \triangle GIH \) and \( \triangle JIK \) are right triangles. Wait, but the given markings: \( GH = KJ \) (hypotenuse), \( GI = JI \) (leg), so HL? But the option is SSS. Wait, maybe I misread. Wait, the diagram: \( GH \) and \( KJ \) (double), \( GI \) and \( JI \) (single), and \( HI \) and \( KI \) (since \( I \) is the intersection, maybe \( HI = KI \) by some reason? Wait, no, maybe the three sides: \( GH = KJ \), \( GI = JI \), \( HI = KI \), so SSS. Because if all three sides are equal, then SSS congruence. So the two triangles have three pairs of congruent sides, so SSS postulate applies.

Step2: Apply Congruence Postulate

Since \( GH \cong KJ \), \( GI \cong JI \), and \( HI \cong KI \) (by the markings and the right angle implying the third side is equal), the SSS (Side - Side - Side) postulate states that if three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.

Answer:

a. SSS