Question

\\(\delta yeg\\) and \\(\delta pom\\) are acute triangles. \\(\angle g \cong \angle m\\). select all the statements that will prove \\(\delta yeg\\) is always similar to \\(\delta pom\\).

a. \\(\frac{eg}{om} = \frac{yg}{pm}\\)

b. \\(\angle e \cong \angle o\\)

c. \\(\frac{eg}{om} = \frac{ye}{po}\\)

d. \\(\overline{yg} \cong \overline{pm}\\) and \\(\overline{ye} \cong \overline{po}\\)

Explanation:

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

Step 1: Analyze the given information

We are given two acute triangles, \(\Delta YEG\) and \(\Delta POM\), with one pair of congruent angles:

$$ \angle G \cong \angle M $$

To prove that \(\Delta YEG \sim \Delta POM\), we need to satisfy one of the triangle similarity criteria:

• AA (Angle-Angle): Two pairs of corresponding angles are congruent.
• SAS (Side-Angle-Side): Two pairs of corresponding sides are proportional, and their included angles are congruent.
• SSS (Side-Side-Side): All three pairs of corresponding sides are proportional.

Step 2: Evaluate Option A

Option A states:

$$ \frac{EG}{OM} = \frac{YG}{PM} $$

• The sides involved in the ratio \(\frac{EG}{OM}\) are \(EG\) (from \(\Delta YEG\)) and \(OM\) (from \(\Delta POM\)).
• The sides involved in the ratio \(\frac{YG}{PM}\) are \(YG\) (from \(\Delta YEG\)) and \(PM\) (from \(\Delta POM\)).
• The angle included between sides \(EG\) and \(YG\) is \(\angle G\).
• The angle included between sides \(OM\) and \(PM\) is \(\angle M\).
• Since we are given \(\angle G \cong \angle M\), this ratio satisfies the SAS Similarity Criterion (two pairs of proportional sides with the congruent included angle between them).

Therefore, Option A is correct.

Step 3: Evaluate Option B

Option B states:

$$ \angle E \cong \angle O $$

• We are already given \(\angle G \cong \angle M\).
• If we also have \(\angle E \cong \angle O\), we have two pairs of congruent corresponding angles.
• This satisfies the AA Similarity Criterion.

Therefore, Option B is correct.

Step 4: Evaluate Option C

Option C states:

$$ \frac{EG}{OM} = \frac{YE}{PO} $$

• The sides involved are \(EG\) and \(YE\) from \(\Delta YEG\), and \(OM\) and \(PO\) from \(\Delta POM\).
• The angle included between \(EG\) and \(YE\) is \(\angle E\).
• The angle included between \(OM\) and \(PO\) is \(\angle O\).
• We are given \(\angle G \cong \angle M\), which is not the included angle for these sides. This represents an "SSA" relationship, which does not guarantee similarity.

Therefore, Option C is incorrect.

Step 5: Evaluate Option D

Option D states:

$$ \overline{YG} \cong \overline{PM} \text{ and } \overline{YE} \cong \overline{PO} $$

• Congruence of two pairs of sides along with one pair of congruent angles (\(\angle G \cong \angle M\)) does not guarantee similarity (this is an SSA configuration, which does not prove congruence or similarity).

Therefore, Option D is incorrect.

Answer:

• a. \(\frac{EG}{OM} = \frac{YG}{PM}\)
• b. \(\angle E \cong \angle O\)