Question

for the triangles below, use the tools to move the solid triangle exactly onto the dashed one. then answer the parts below. (a) find the following ratios. write your answers as fractions in simplest form. \\(\frac{ce}{wx} = \\) \\(\frac{ed}{xy} = \\) \\(\frac{cd}{wy} = \\) (b) complete the congruence statements below. \\(\angle w \cong\\) (choose one) \\(\angle x \cong\\) (choose one) \\(\angle y \cong\\) (choose one) (c) select the true statement. \\(\bigcirc\\) \\(\triangle wxy\\) and \\(\triangle ced\\) are similar. \\(\bigcirc\\) \\(\triangle wxy\\) and \\(\triangle ced\\) are not similar.

Explanation:

Step1: Calculate \( \frac{CE}{WX} \)

CE = 4, WX = 16. So \( \frac{4}{16} = \frac{1}{4} \).

Step2: Calculate \( \frac{ED}{XY} \)

ED = 8, XY = 32. So \( \frac{8}{32} = \frac{1}{4} \).

Step3: Calculate \( \frac{CD}{WY} \)

CD = 6, WY = 24. So \( \frac{6}{24} = \frac{1}{4} \).

Step4: Determine congruent angles (b)

Since the triangles have proportional sides (from part a, ratio \( \frac{1}{4} \)), they are similar (and congruent via scaling? Wait, no, similar with ratio 1/4. Wait, but the angles should be congruent. So \( \angle W \cong \angle C \), \( \angle X \cong \angle E \), \( \angle Y \cong \angle D \) (assuming the triangles are similar, so corresponding angles).

Step5: Determine similarity (c)

Since the ratios of corresponding sides are equal (\( \frac{1}{4} \) for all), by SSS similarity, \( \triangle WXY \) and \( \triangle CED \) are similar. So the first option is true.

Answer:

s:
(a) \( \frac{CE}{WX} = \frac{1}{4} \), \( \frac{ED}{XY} = \frac{1}{4} \), \( \frac{CD}{WY} = \frac{1}{4} \)
(b) \( \angle W \cong \angle C \), \( \angle X \cong \angle E \), \( \angle Y \cong \angle D \) (assuming the correspondence)
(c) \( \triangle WXY \) and \( \triangle CED \) are similar.

(Note: For part (b), the exact angle correspondence depends on the triangle labeling, but with the side ratios, the corresponding angles should match the order of the sides. So CE corresponds to WX, ED to XY, CD to WY, so vertices C-E-D correspond to W-X-Y, so angles: \( \angle W \cong \angle C \), \( \angle X \cong \angle E \), \( \angle Y \cong \angle D \))