Question

hi ⊥ gi, gi ⊥ gj, and gj ≅ hi. complete the proof that δgjk ≅ δihk. (image of triangles hik and gjk with a table of statements and reasons) statement reason 1 hi ⊥ gi given 2 gi ⊥ gj given 3 gj ≅ hi given 4 ∠g ≅ ∠i all right angles are congruent 5 ∠gkj ≅ ∠hki blank 6 δgjk ≅ δihk blank

Explanation:

Step1: Analyze ∠GKJ and ∠HKI

∠GKJ and ∠HKI are vertical angles. Vertical angles are formed when two lines intersect, and they are always congruent.

Step2: Determine Reason for ∠GKJ ≅ ∠HKI

The reason for ∠GKJ ≅ ∠HKI is that vertical angles are congruent.

Step3: Prove Triangle Congruence

For triangle \( \triangle GJK \) and \( \triangle IHK \), we have:

• \( \angle G \cong \angle I \) (all right angles congruent),
• \( \angle GKJ \cong \angle HKI \) (vertical angles congruent),
• \( \overline{GJ} \cong \overline{HI} \) (given).

By the Angle - Angle - Side (AAS) congruence criterion, \( \triangle GJK \cong \triangle IHK \).

Answer:

For step 5, the reason is "Vertical angles are congruent". For step 6, the reason is "AAS (Angle - Angle - Side) Congruence Criterion".