Problem Analysis:
The task involves identifying and solving problems related to
similar figures. Similar figures are geometric shapes that have the same shape but not necessarily the same size. The key properties of similar figures are:
1. Corresponding angles are equal.
2. Corresponding sides are proportional.
From the image, we can see two examples provided:
1. Example 1: A triangle with labeled angles and sides.
2. Example 2: Another triangle with labeled angles and sides.
The goal is to determine if the triangles are similar and, if so, to find the scale factor or any missing measurements.
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Step-by-Step Solution:
####
Example 1:
The first example shows two triangles with corresponding angles marked as equal. The triangles are:
- Triangle on the left: Angles labeled as \( \angle A \), \( \angle B \), and \( \angle C \).
- Triangle on the right: Angles labeled as \( \angle D \), \( \angle E \), and \( \angle F \).
The problem states that the triangles are similar because their corresponding angles are equal:
- \( \angle A = \angle D \)
- \( \angle B = \angle E \)
- \( \angle C = \angle F \)
Since the corresponding angles are equal, the triangles are similar by the
Angle-Angle (AA) similarity criterion.
####
Example 2:
The second example shows two triangles with labeled side lengths:
- Triangle on the left: Sides labeled as 3 cm, 4 cm, and 5 cm.
- Triangle on the right: Sides labeled as 6 cm, 8 cm, and 10 cm.
To determine if these triangles are similar, we check if the corresponding sides are proportional. We compare the ratios of the corresponding sides:
\[
\frac{\text{Side 1 (left)}}{\text{Side 1 (right)}} = \frac{3}{6} = \frac{1}{2}
\]
\[
\frac{\text{Side 2 (left)}}{\text{Side 2 (right)}} = \frac{4}{8} = \frac{1}{2}
\]
\[
\frac{\text{Side 3 (left)}}{\text{Side 3 (right)}} = \frac{5}{10} = \frac{1}{2}
\]
Since all three ratios are equal (\( \frac{1}{2} \)), the triangles are similar by the
Side-Side-Side (SSS) similarity criterion. The scale factor between the two triangles is \( \frac{1}{2} \).
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Final Answer:
1.
Example 1: The triangles are similar because their corresponding angles are equal.
2.
Example 2: The triangles are similar because their corresponding sides are proportional, with a scale factor of \( \frac{1}{2} \).
\[
\boxed{\text{Similar by AA criterion (Example 1), Similar by SSS criterion with scale factor } \frac{1}{2} \text{ (Example 2)}}
\]
Parent Tip: Review the logic above to help your child master the concept of geometry similar figures worksheet answers.
