Question

the triangles can be proven similar by applying the similarity criteria for

Explanation:

⚡ Using what you learned: similarity criteria for triangles (aa, sss, sas)

Step 1: Identify vertical angles

The intersecting lines \(TB\) and \(UC\) intersect at point \(D\). This creates a pair of vertical angles:

$$ \angle TDU \cong \angle BDC $$

Step 2: Compare ratios of corresponding sides

We check the ratios of the sides forming these angles in \(\triangle TDU\) and \(\triangle BDC\):

$$ \frac{UD}{CD} = \frac{30}{154} \approx 0.1948 $$

$$ \frac{TD}{BD} = \frac{55}{84} \approx 0.6548 $$

Since these do not match, we check the alternative pairing:

$$ \frac{UD}{BD} = \frac{30}{84} = \frac{5}{14} $$

$$ \frac{TD}{CD} = \frac{55}{154} = \frac{5}{14} $$

Since the ratios are equal:

$$ \frac{UD}{BD} = \frac{TD}{CD} $$

Step 3: Apply similarity criterion

Two pairs of corresponding sides are proportional, and their included angles are congruent (\(\angle TDU \cong \angle BDC\)). Therefore, the triangles are similar by the Side-Angle-Side (SAS) similarity criterion.

Answer:

SAS