Question

complete the proof that $overline{gh} cong overline{ij}$.

(image of triangle $khi$ with $g$ on $kh$, $j$ on $ki$, $overline{gj} parallel overline{hi}$, and angles at $h$ and $i$ marked congruent)

	statement	reason
2	$angle h cong angle i$	given
3	$angle jgk cong angle h$	corresponding angles theorem
4	$angle gjk cong angle i$	corresponding angles theorem
5	$angle jgk cong angle i$	transitive property of congruence
6	$angle jgk cong angle gjk$	transitive property of congruence
7	$gk = jk$
8	$hk = ik$	isosceles triangle theorem
9	$hk = gh + gk$	additive property of length
10	$ik = ij + jk$	additive property of length
11	$gh + gk = ik$	transitive property of equality
12	$gh + gk = ij + jk$	transitive property of equality
13	$gh + jk = ij + jk$	substitution
14	$gh = ij$	properties of addition, subtraction, multiplication, and division
15	$overline{gh} cong overline{ij}$	definition of congruence

Explanation:

Step1: Analyze congruent angles

From steps 3, 2, and 5, we have $\angle JGK \cong \angle H$ and $\angle H \cong \angle I$, so $\angle JGK \cong \angle I$.

Step2: Apply isosceles triangle theorem

In $\triangle K G J$, if $\angle JGK \cong \angle I$ (and $\angle I \cong \angle KJG$ from step 4), then the sides opposite these congruent angles are equal.

Step3: Match to the statement

The statement $GK = JK$ follows because the angles opposite these sides ($\angle KJG$ and $\angle JGK$) are congruent, so this is the Isosceles Triangle Theorem (converse: if two angles of a triangle are congruent, the sides opposite them are congruent).

Answer:

Isosceles Triangle Theorem (Converse)