Question

complete the proof that \\( \overline { h i } \cong \overline { f k } \\).

Explanation:

Step1: Given

From the problem statement, we know that $\overline{GH}\cong\overline{JK}$ (given in the proof - starting point).

Step2: Given

$\angle FKG\cong\angle IHJ$ (marked in the diagram as equal angles).

Step3: Vertical angles

$\angle HJI$ and $\angle FGK$ are vertical angles, and vertical angles are congruent. So, $\angle HJI\cong\angle FGK$.

Step4: Segment addition postulate

For collinear points $J$, $K$, $G$, we have $GK = JK+GJ$ (by the segment addition postulate which states that if $B$ is between $A$ and $C$, then $AC = AB + BC$).

Step5: Segment addition postulate

For collinear points $G$, $H$, $J$, we have $HJ=GH + GJ$ (by the segment addition postulate).

Step6: Substitution

Since $GH\cong JK$ (from step 1, so $GH = JK$), substituting $JK$ with $GH$ in $GK = JK+GJ$ gives $GK=GH + GJ$.

Step7: Transitive property

From $HJ = GH+GJ$ (step 5) and $GK=GH + GJ$ (step 6), by the transitive property of equality ($a=b$ and $b = c$ implies $a=c$), we have $HJ = GK$.

Step8: ASA (Angle - Side - Angle)

We have $\angle FKG\cong\angle IHJ$ (step 2), $GK = HJ$ (step 7), and $\angle HJI\cong\angle FGK$ (step 3). So, $\triangle FGK\cong\triangle JIH$ by the ASA (Angle - Side - Angle) congruence criterion.

Step9: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since $\triangle FGK\cong\triangle JIH$, then $\overline{HI}\cong\overline{FK}$ (by CPCTC).

Answer:

1. Given
2. Given
3. Vertical angles are congruent
4. Segment addition postulate
5. Segment addition postulate
6. Substitution property (of equality)
7. Transitive property (of equality)
8. ASA (Angle - Side - Angle)
9. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)