Which Pairs Of Polygons Are Similar Select Each Correct Answer

Which Pairs of Polygons are Similar? Selecting the Correct Answers

Similar polygons are a fundamental concept in geometry, crucial for understanding scaling, proportions, and various applications in fields like architecture, engineering, and computer graphics. This article dives deep into the concept of similar polygons, providing a comprehensive guide to identifying pairs of similar polygons and offering practice problems to solidify your understanding.

Understanding Similarity in Polygons

Two polygons are considered similar if they meet two essential criteria:

1. Corresponding Angles are Congruent: This means that the angles in the same relative position in both polygons have the same measure. For example, if we have two quadrilaterals, their corresponding angles (angle A in polygon 1 and angle A' in polygon 2, angle B and B', etc.) must all be equal.

2. Corresponding Sides are Proportional: The ratio of the lengths of corresponding sides must be constant. This constant ratio is called the scale factor. If the ratio of the lengths of corresponding sides is 2:1, it means that one polygon is twice as large as the other.

Important Note: Simply having congruent angles or proportional sides is not sufficient to prove similarity. Both conditions must be met.

Identifying Similar Polygons: A Step-by-Step Approach

Let's outline a systematic approach to determining whether two polygons are similar:

1. Identify Corresponding Angles: First, carefully examine the given polygons. Look for a way to pair up the angles such that they are in the same relative position within each polygon. Label the angles consistently (e.g., A, B, C, D for one polygon and A', B', C', D' for the other). This step is crucial for establishing correspondence.

2. Check for Congruent Corresponding Angles: Measure the angles (if values are not provided). All corresponding angles must be equal for the polygons to be similar.

3. Identify Corresponding Sides: Once you've established the correspondence of angles, identify the corresponding sides. These are the sides that lie between the same corresponding angles. For instance, side AB corresponds to side A'B', side BC to B'C', and so on.

4. Calculate the Ratio of Corresponding Sides: Find the ratio of the lengths of each pair of corresponding sides. For example, calculate AB/A'B', BC/B'C', CD/C'D', and so forth.

5. Check for Constant Ratio: If the ratio calculated in step 4 is constant for all pairs of corresponding sides, then the polygons are similar. This constant ratio is the scale factor.

Examples of Similar Polygons

Let's illustrate with examples. Assume we have the following measurements for two quadrilaterals:

Quadrilateral ABCD:

• AB = 4 cm
• BC = 6 cm
• CD = 8 cm
• DA = 10 cm
• ∠A = 70°
• ∠B = 110°
• ∠C = 90°
• ∠D = 90°

Quadrilateral A'B'C'D':

• A'B' = 2 cm
• B'C' = 3 cm
• C'D' = 4 cm
• D'A' = 5 cm
• ∠A' = 70°
• ∠B' = 110°
• ∠C' = 90°
• ∠D' = 90°

Analysis:

1. Corresponding Angles: ∠A = ∠A' = 70°, ∠B = ∠B' = 110°, ∠C = ∠C' = 90°, ∠D = ∠D' = 90°. All corresponding angles are congruent.

2. Corresponding Sides and Ratio:

• AB/A'B' = 4/2 = 2
• BC/B'C' = 6/3 = 2
• CD/C'D' = 8/4 = 2
• DA/D'A' = 10/5 = 2

The ratio of corresponding sides is constant (2).

Conclusion: Since both conditions (congruent corresponding angles and proportional corresponding sides) are met, quadrilaterals ABCD and A'B'C'D' are similar.

Examples of Non-Similar Polygons

Consider two triangles:

Triangle ABC:

• AB = 5 cm
• BC = 7 cm
• CA = 9 cm
• ∠A = 50°
• ∠B = 60°
• ∠C = 70°

Triangle A'B'C':

• A'B' = 10 cm
• B'C' = 14 cm
• C'A' = 18 cm
• ∠A' = 50°
• ∠B' = 70°
• ∠C' = 60°

Analysis:

While the corresponding sides are proportional (the scale factor is 2), the corresponding angles are not congruent. ∠B ≠ ∠B'. Therefore, these triangles are not similar.

Practice Problems

1. Two squares: Are two squares always similar? Explain your answer.

2. Two rectangles: Are two rectangles always similar? Explain your answer.

3. Two regular pentagons: Are two regular pentagons always similar? Explain your answer.

4. Given two triangles with corresponding angles 30°, 60°, and 90°. If the sides of one triangle are 3, 4, and 5, and the sides of the other triangle are 6, 8, and 10. Are these triangles similar?

5. Two hexagons have congruent corresponding angles. The ratio of the lengths of their corresponding sides is 3:1. Are these hexagons similar?

Advanced Concepts and Applications

The concept of similar polygons extends beyond simple geometric shapes. It's a cornerstone for:

• Scale Drawings: Architects and engineers use similarity to create scaled models and blueprints of buildings and structures.

• Computer Graphics: Image scaling and resizing in digital graphics rely heavily on the principles of similar polygons.

• Trigonometry: Many trigonometric concepts, such as similar triangles used in solving problems related to heights and distances, rely on the concept of similarity.

Conclusion

Understanding similar polygons is crucial for numerous applications in geometry and related fields. By systematically checking for congruent corresponding angles and proportional corresponding sides, you can confidently determine whether two polygons are similar. Remember, both conditions must be satisfied. Through practice and a thorough understanding of the underlying principles, you can master this important geometrical concept. The practice problems provided offer a good starting point for solidifying your understanding. Continue exploring the fascinating world of geometry, and you will find that the applications of similar polygons are vast and far-reaching.