Question

the two triangles below are similar.
complete the similarity statement.
$\triangle hij \sim \triangle \square$
find the ratio of a side length in the first triangle to its corresponding side length in the second triangle.
simplify your answer and write it as a proper fraction, improper fraction, or whole number.

Explanation:

Part 1: Completing the Similarity Statement

To determine the similarity statement, we match the angles of the triangles. In $\triangle HIJ$, the angles are $29^\circ$ (at $I$), $104^\circ$ (at $H$), and $47^\circ$ (at $J$). In the second triangle, the angles are $29^\circ$ (at $P$), $104^\circ$ (at $R$), and $47^\circ$ (at $Q$). So, we match the vertices by their corresponding angles: $H$ (104°) corresponds to $R$ (104°), $I$ (29°) corresponds to $P$ (29°), and $J$ (47°) corresponds to $Q$ (47°). Thus, $\triangle HIJ \sim \triangle RPQ$.

Step1: Identify corresponding sides

From the similarity statement $\triangle HIJ \sim \triangle RPQ$, we find corresponding sides. For example, $HJ = 10$ in $\triangle HIJ$ and $RQ = 2$ in $\triangle RPQ$ (or other corresponding pairs like $IJ = 20$ and $PQ = 4$, or $HI = 15$ and $RP = 3$).

Step2: Calculate the ratio

Take the ratio of a side from $\triangle HIJ$ to its corresponding side in $\triangle RPQ$. Using $HJ = 10$ and $RQ = 2$, the ratio is $\frac{10}{2} = 5$. (Or using $IJ = 20$ and $PQ = 4$, $\frac{20}{4} = 5$; or $HI = 15$ and $RP = 3$, $\frac{15}{3} = 5$.)

Answer:

$\triangle HIJ \sim \triangle RPQ$

Part 2: Finding the Ratio of Side Lengths