1. Name the similar triangles for the following shape. Write the letters so that equal angles appear in corresponding order.
2. For each pair of similar triangles, list all the pairs of corresponding angles and write the ratios of the corresponding sides.
1. First, we can identify the equal angles based on their position within the shape:
\(\angle \text{Q} = \angle \text{U}\)
\(\angle \text{QSR} = \angle \text{UST}\)
\(\triangle \text{QRS} \approx \triangle \text{UTS}\)
2. We can now identify the corresponding angles:
\(\angle \text{Q} = \angle \text{R}\)
\(\angle \text{QSR} = \angle \text{UST}\)
\(\angle \text{T} = \angle \text{R}\)
\(\cfrac{\text{QR}}{\text{UT}} = \cfrac{\text{RS}}{\text{TS}} = \cfrac{\text{QS}}{\text{US}}\)
1. Name the similar triangles for the following shape. Write the letters so that equal angles appear in corresponding order.
2. For each pair of similar triangles, list all the pairs of corresponding angles and write the ratios of the corresponding sides.
1. First, we can identify the equal angles based on their position within the shape:
\(\angle \text{E} = \angle \text{E}\)
\(\angle \text{A} = \angle \text{DCE}\)
\(\triangle \text{ABE} \approx \triangle \text{DE}\)
2. We can now identify the corresponding angles:
\(\angle \text{A} = \angle \text{DCE}\)
\(\angle \text{B} = \angle \text{CDE}\)
\(\angle \text{E} = \angle \text{E}\)
\(\cfrac{\text{AB}}{\text{CD}} = \cfrac{\text{BE}}{\text{DE}} = \cfrac{\text{AE}}{\text{CE}}\)
1. Name the similar triangles for the following shape. Write the letters so that equal angles appear in corresponding order.
2. For each pair of similar triangles, list all the pairs of corresponding angles and write the ratios of the corresponding sides.
1. First, we can identify the equal angles based on their position within the shape:
We can also determine the value of \(x\) using the Pythagorean theorem. This will allow us to determine the ratio of the 2 triangles within the shape.
From the diagram above, we can identify \(a=x\), \(b = 9\), and \(c = 20.1\):
\(a^2 + b^2 = c^2\)
\(x^2 + 9^2 = 20.1^2\)
\(x^2 = 404.01 - 81\)
\(\sqrt{x^2} = \sqrt{323.01}\)
\(x = 17.97 \approx 18\)
We can now determine the ratios with \(x\) as the opposite side of the small triangle and the adjacent side of the large triangle:
\(\text{ratio}_1: \cfrac{9}{18} = \cfrac{1}{2}\)
\(\text{ratio}_2: \cfrac{18}{36} = \cfrac{1}{2}\)
Therefore, we can determine that \(\triangle \text{MCD} \approx \triangle \text{KCM}\) as they share a common angle and common ratio.
2. We can now identify the corresponding angles:
\(\angle \text{D} = \angle \text{KMC}\)
\(\angle \text{DCM} = \angle \text{MCK}\)
\(\angle \text{CMD} = \angle \text{K}\)
\(\cfrac{\text{MC}}{\text{KC}} = \cfrac{\text{CD}}{\text{CM}} = \cfrac{\text{MD}}{\text{KM}}\)