Question

complete the proof that $\triangle stu \sim \triangle wvu$.
(image of triangles with labeled lengths: wu=6, vu=4, su=9, tu=6)
statement | reason

1. $\frac{su}{wu} = \frac{9}{6} = 1.5$ | substitute given values.
2. $\frac{tu}{vu} = \frac{6}{4} = 1.5$ | substitute given values.
3. pick statement | substitute (1,2)
4. $\angle tus \cong \angle vuw$ | pick reason
5. $\triangle stu \sim \triangle wvu$ | pick similarity criterion

Explanation:

Step1: Analyze Statement 3

From steps 1 and 2, we have \(\frac{SU}{WU} = 1.5\) and \(\frac{TU}{VU}=1.5\). So the statement for step 3 should be \(\frac{SU}{WU}=\frac{TU}{VU}\) (since both equal 1.5, they are equal to each other).

Step2: Analyze Reason for Statement 4

\(\angle TUS\) and \(\angle VUW\) are vertical angles. Vertical angles are congruent, so the reason is "Vertical angles are congruent".

Step3: Analyze Similarity Criterion for Step 5

We have two sides in proportion (\(\frac{SU}{WU}=\frac{TU}{VU}\)) and the included angle congruent (\(\angle TUS\cong\angle VUW\)). The similarity criterion for this is SAS (Side - Angle - Side) similarity.

Answer:

1. \(\boldsymbol{\frac{SU}{WU}=\frac{TU}{VU}}\)
2. Reason: Vertical angles are congruent
3. Similarity criterion: SAS (Side - Angle - Side) similarity