Question

name: daniel tello period: 3 proving triangle similarity hw determine whether the triangles are similar. if so, write a similarity statement and name the postulate or theorem you used. if not, explain. 1. 2. 4.

Explanation:

Step1: Check the ratio of corresponding sides

For the first pair of triangles (assuming the sides are \( \frac{3}{6}=\frac{1}{2} \), \( \frac{7}{14}=\frac{1}{2} \) (if we assume some correspondence based on the figure's structure, and since \( \angle QUT=\angle RUS \) (vertical angles are equal). By the Side - Angle - Side (SAS) similarity theorem.

Step2: For the second pair of triangles \( \triangle ABC \) and \( \triangle PNO \)

\( \frac{12}{8} = 1.5\), \( \frac{30}{24}=1.25\), \( \frac{36}{20}=1.8\). Since the ratios of the corresponding sides are not equal.

Step3: For the third pair of triangles \( \triangle EFH \) and \( \triangle IGH \)

\( \frac{25}{50}=\frac{1}{2}\), \( \frac{20}{40}=\frac{1}{2}\), and \( \angle EFG=\angle HGI \) (common angle). By the Side - Angle - Side (SAS) similarity theorem.

Step4: For the fourth pair of triangles \( \triangle JKL \) and \( \triangle XYZ \)

In \( \triangle JKL \), using the angle - sum property of a triangle (\( \angle J=180^{\circ}-48^{\circ}-76^{\circ}=56^{\circ} \)). In \( \triangle XYZ \), using the angle - sum property (\( \angle Y = 180^{\circ}-63^{\circ}-76^{\circ}=41^{\circ} \)). Since the angles are not equal.

Answer:

1. The triangles are similar. Similarity statement: \( \triangle QUT\sim\triangle RUS \), by the SAS (Side - Angle - Side) similarity theorem.
2. The triangles are not similar because the ratios of their corresponding sides (\( \frac{AB}{PN},\frac{BC}{NO},\frac{AC}{PO} \)) are not equal.
3. The triangles are similar. Similarity statement: \( \triangle EFH\sim\triangle IGH \), by the SAS (Side - Angle - Side) similarity theorem.
4. The triangles are not similar because their corresponding angles are not equal.