Question

determine whether each set of triangles is similar. justify your reasoning.

1. △abc and △dec
2. △xyz and △pqr

Explanation:

Problem 7: $\triangle ABC$ and $\triangle DEC$

Step1: Identify Common Angle

$\angle C$ is common to both $\triangle ABC$ and $\triangle DEC$, so $\angle C = \angle C$ (reflexive property).

Step2: Identify Corresponding Angles

$\angle B$ and $\angle DEC$ are marked as equal (both have the same angle marking), so $\angle B = \angle DEC$.

Step3: Apply AA Similarity Criterion

By the Angle - Angle (AA) similarity criterion, if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. Since $\angle C=\angle C$ and $\angle B = \angle DEC$, $\triangle ABC \sim \triangle DEC$ by AA similarity.

Problem 8: $\triangle XYZ$ and $\triangle PQR$

Step1: Check for Included Angle and Proportional Sides

First, check if the included angles are equal. $\angle Y$ in $\triangle XYZ$ and $\angle Q$ in $\triangle PQR$ are marked as equal (both have the same angle marking).

Step2: Calculate Ratios of Sides

Calculate the ratios of the sides adjacent to the equal angles. For $\triangle XYZ$ and $\triangle PQR$, the sides adjacent to $\angle Y$ and $\angle Q$ are:

• For $\triangle XYZ$: $XY = 10$, $YZ=12$
• For $\triangle PQR$: $PQ = 6$, $QR = 8$

Calculate the ratios: $\frac{XY}{PQ}=\frac{10}{6}=\frac{5}{3}$ and $\frac{YZ}{QR}=\frac{12}{8}=\frac{3}{2}$. Since $\frac{5}{3}
eq\frac{3}{2}$, the sides are not proportional.

Step3: Determine Similarity

Since the ratios of the sides adjacent to the equal included angle are not equal, $\triangle XYZ$ and $\triangle PQR$ are not similar.

Final Answers

1. $\triangle ABC$ and $\triangle DEC$ are similar by AA similarity (common $\angle C$ and $\angle B=\angle DEC$).
2. $\triangle XYZ$ and $\triangle PQR$ are not similar as the ratios of sides adjacent to the equal included angle are not equal.

For the answer format as per the question's requirement (if we consider the first part (problem 7) as the main focus with the given diagram):

Answer:

(for problem 7):
$\triangle ABC$ and $\triangle DEC$ are similar by the Angle - Angle (AA) similarity criterion (common $\angle C$ and $\angle B=\angle DEC$).

(If we consider problem 8):