Question

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Explanation:

Identify given geometric information

We have two triangles sharing a vertex \(C\), forming vertical angles \(\angle DCE\) and \(\angle BCA\). The side lengths given for \(\triangle EDC\) are \(DE = 9\) and \(EC = 12\). By the Pythagorean theorem, the hypotenuse \(DC\) is:

$$ DC = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15 $$

For \(\triangle ABC\), the corresponding side lengths are \(BC = 5\) and \(AC = 4\).

Analyze vertical angles

Using the Geometric Proofs concept, the vertical angles \(\angle DCE\) and \(\angle BCA\) are congruent:

$$ \angle DCE \cong \angle BCA $$

Check side proportionality

Using the Proportional Sides concept, we compare the ratios of the sides forming these congruent angles:

$$ \frac{DC}{BC} = \frac{15}{5} = 3 $$

$$ \frac{EC}{AC} = \frac{12}{4} = 3 $$

Since \(\frac{15}{5} = \frac{12}{4} = 3\), the corresponding sides are proportional.

Apply similarity postulate

Using the SAS Similarity postulate, since two pairs of corresponding sides are proportional and their included angles are congruent, the triangles are similar:

$$ \triangle ABC \sim \triangle EDC $$

This matches the second option.

Answer:

• (A) \(\frac{15}{4} = \frac{12}{5} = \frac{9}{3}\) shows the corresponding sides are proportional; therefore, \(\triangle ABC \sim \triangle EDC\) by the SAS Similarity Postulate.
• (B) \(\angle DCE\) is congruent to \(\angle BCA\) by the Vertical Angles Theorem and \(\frac{15}{5} = \frac{12}{4}\) shows the corresponding sides are proportional; therefore, \(\triangle ABC \sim \triangle EDC\) by the SAS Similarity Postulate. (Correct answer)
• (C) \(\angle E\) and \(\angle A\) are right angles; therefore, these angles are congruent since all right angles are congruent. \(\frac{12}{4} = \frac{9}{3}\) shows the corresponding sides are proportional; therefore, \(\triangle ABC \sim \triangle EDC\) by the SSS Similarity Postulate.
• (D) \(\angle DCE\) is congruent to \(\angle CBA\) by the Vertical Angles Theorem and \(\frac{15}{5} = \frac{12}{4}\) shows the corresponding sides are proportional; therefore, \(\triangle ABC \sim \triangle EDC\) by the SSS Similarity Postulate.