Question

complete the two-column proof.

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$$\begin{tabular}{|l|r|} \\hline statements & reasons \\\\ \\hline 1) \\(\\delta ghi\\) & given \\\\ \\(\\overline{of} \\parallel \\overline{gi}\\) & \\\\ \\hline 2) \\(\\angle igh \\cong \\angle foh\\) and & corresponding angles theorem \\\\ \\(\\angle gih \\cong \\angle ofh\\) & \\\\ \\hline 3) \\(\\delta ghi \\sim \\delta ohf\\) & \\\\ \\hline 4) \\(\\frac{gh}{oh} = \\frac{ih}{fh}\\) & \\\\ \\hline 5) \\(gh = go + oh\\) & \\\\ \\(hi = if + fh\\) & \\\\ \\hline 6) \\(\\frac{(go + oh)}{oh} = \\frac{(if + fh)}{fh}\\) & substitution property of equality \\\\ \\hline 7) \\(\\frac{go}{oh} + \\frac{oh}{oh} = \\frac{if}{fh} + \\frac{fh}{fh}\\) & separation of fractions \\\\ \\hline 8) \\(\\frac{go}{oh} + 1 = \\frac{if}{fh} + 1\\) & simplify \\\\ \\hline 9) \\(\\frac{go}{oh} = \\frac{if}{fh}\\) & subtraction property of equality \\\\ \\hline \\end{tabular}$$

Explanation:

Identificar el criterio de semejanza para el paso 3

$$ \angle IGH \cong \angle FOH \text{ y } \angle GIH \cong \angle OFH \implies \Delta GHI \sim \Delta OHF \text{ por AA Similarity} $$

Identificar la razón de proporcionalidad para el paso 4

$$ \Delta GHI \sim \Delta OHF \implies \frac{GH}{OH} = \frac{IH}{FH} \text{ por la definición de triángulos semejantes (Corresponding sides of similar triangles are proportional)} $$

Identificar el postulado de suma de segmentos para el paso 5

$$ GH = GO + OH \text{ y } HI = IF + FH \text{ por el Segment Addition Postulate} $$

Answer:

Question 3

AA Similarity

Question 4

Corresponding sides of similar triangles are proportional

Question 5

Segment Addition Postulate