Question

unit 3 exam on lessons 3.1 to 3.8 (28 points)

1. triangles def and ghi are shown in the accompanying diagram. using only the information

marked/labeled in the diagrams, which of the following statements is correct?
(1) \\( \triangle def \sim \triangle ghi \\) by aa~
(2) \\( \triangle def \sim \triangle ghi \\) by sss~
(3) \\( \triangle def \sim \triangle ghi \\) by sas~
(4) \\( \triangle def \\) and \\( \triangle ghi \\) are not similar.

Explanation:

Step1: Check SSS~ similarity

For SSS~, the ratios of corresponding sides should be equal.
In \(\triangle DEF\), sides are \(19,19,25\). In \(\triangle GHI\), sides are \(95,95,126\).
\(\frac{19}{95}=\frac{1}{5}\), \(\frac{25}{126}
eq\frac{1}{5}\). So SSS~ (option 2) is incorrect.

Step2: Check SAS~ similarity

For SAS~, two sides in proportion and included angle equal.
No information about included angles (except in \(\triangle GHI\) we have \(\angle I = 49^{\circ}\), but no corresponding angle in \(\triangle DEF\) with side - side - angle relation). So SAS~ (option 3) is incorrect.

Step3: Check AA~ similarity

For AA~, two angles should be equal. We only know \(\angle I = 49^{\circ}\) in \(\triangle GHI\) and no angles in \(\triangle DEF\) are given.

Answer:

(4) \(\triangle DEF\) and \(\triangle GHI\) are not similar.