Geometry 5.4 Practice B Answers

Geometry 5.4 Practice B Answers: A Comprehensive Guide to Understanding Similar Triangles

This article provides a comprehensive walkthrough of the answers for a typical Geometry 5.4 Practice B worksheet focusing on similar triangles. We'll delve into the core concepts of similarity, explore various problem-solving techniques, and offer detailed explanations to solidify your understanding. Understanding similar triangles is crucial for further advancements in geometry and related fields. This guide is designed to not only provide the answers but also enhance your problem-solving skills and build a strong foundation in this geometric concept.

Introduction to Similar Triangles and Key Concepts

Before diving into the practice problems, let's refresh our understanding of similar triangles. Two triangles are considered similar if their corresponding angles are congruent (equal) and their corresponding sides are proportional. This means that one triangle is essentially a scaled version of the other. The symbol ~ represents similarity. For example, if triangle ABC ~ triangle DEF, it signifies that:

• ∠A ≅ ∠D, ∠B ≅ ∠E, and ∠C ≅ ∠F (Congruent angles)
• AB/DE = BC/EF = AC/DF (Proportional sides)

Several postulates and theorems help us determine if two triangles are similar. The most common include:

• AA Similarity (Angle-Angle): If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
• SAS Similarity (Side-Angle-Side): If two sides of one triangle are proportional to two sides of another triangle, and the included angles are congruent, then the triangles are similar.
• SSS Similarity (Side-Side-Side): If three sides of one triangle are proportional to three sides of another triangle, then the triangles are similar.

Mastering these postulates and theorems is essential for solving similarity problems effectively.

Practice Problem Walkthroughs and Solutions

Since a specific Practice B worksheet isn't provided, we will work through example problems that represent the typical types of questions found in a Geometry 5.4 Practice B assignment on similar triangles. These examples cover a range of difficulty levels and problem-solving techniques.

Example Problem 1: Using AA Similarity

• Problem: In ∆ABC, ∠A = 50° and ∠B = 60°. In ∆DEF, ∠D = 50° and ∠E = 60°. Are ∆ABC and ∆DEF similar? Justify your answer.

• Solution: Yes, ∆ABC ~ ∆DEF by AA similarity. Since ∠A = ∠D = 50° and ∠B = ∠E = 60°, two angles of ∆ABC are congruent to two angles of ∆DEF. The third angles (∠C and ∠F) must also be congruent because the sum of angles in a triangle is always 180°. Therefore, the triangles are similar.

Example Problem 2: Using SAS Similarity

• Problem: Triangle ABC has sides AB = 6, BC = 8, and AC = 10. Triangle DEF has sides DE = 3, EF = 4, and DF = 5. ∠B = ∠E = 90°. Are the triangles similar? Justify your answer.

• Solution: Yes, ∆ABC ~ ∆DEF by SAS similarity. Let's check the ratios of corresponding sides:

  • AB/DE = 6/3 = 2
  • BC/EF = 8/4 = 2
  • AC/DF = 10/5 = 2

The ratios of the corresponding sides are equal (all equal to 2), and the included angles (∠B and ∠E) are both 90°. Therefore, the triangles are similar by SAS similarity.

Example Problem 3: Using SSS Similarity

• Problem: Triangle PQR has sides PQ = 12, QR = 15, and PR = 18. Triangle XYZ has sides XY = 4, YZ = 5, and XZ = 6. Are the triangles similar? Justify your answer.

• Solution: Yes, ∆PQR ~ ∆XYZ by SSS similarity. Let's check the ratios of corresponding sides:

  • PQ/XY = 12/4 = 3
  • QR/YZ = 15/5 = 3
  • PR/XZ = 18/6 = 3

Since all three ratios of corresponding sides are equal (all equal to 3), the triangles are similar by SSS similarity.

Example Problem 4: Finding Missing Side Lengths in Similar Triangles

• Problem: ∆ABC ~ ∆DEF. AB = 8, BC = 12, AC = 16, and DE = 4. Find the lengths of EF and DF.

• Solution: Since the triangles are similar, the ratios of their corresponding sides are equal. We can set up proportions to find the missing side lengths:

  • AB/DE = BC/EF => 8/4 = 12/EF => EF = 6
  • AB/DE = AC/DF => 8/4 = 16/DF => DF = 8

Example Problem 5: Real-world Application of Similar Triangles

• Problem: A tree casts a shadow of 20 feet. At the same time, a 6-foot-tall person casts a shadow of 4 feet. How tall is the tree?

• Solution: We can use similar triangles to solve this problem. The tree and its shadow form one triangle, and the person and their shadow form a similar triangle. Let 'h' represent the height of the tree. We can set up a proportion:

  • h/20 = 6/4
  • h = (6 * 20) / 4
  • h = 30 feet

Therefore, the tree is 30 feet tall.

Example Problem 6: More Complex Similarity Problem

• Problem: In the diagram, lines AB and CD are parallel. Given that AE = 6, EB = 9, and AD = 8, find the length of EC.

• Solution: Since AB || CD, we have similar triangles ∆ADE ~ ∆CBE by AA similarity (alternate interior angles are congruent). We can set up a proportion using the corresponding sides:

  • AE/EC = AD/CB
  • 6/EC = 8/CB

We need another relationship. Note that AE/EB = AD/CB, forming another proportion:

* 6/9 = 8/CB
* CB = 12

Now substitute this value back into our first proportion:

* 6/EC = 8/12
* EC = 9

Therefore, the length of EC is 9.

Explanation of Key Mathematical Principles

The solutions above rely on the fundamental principles of ratios and proportions. Understanding how to set up and solve proportions is crucial for mastering similar triangles. Remember that:

• A ratio is a comparison of two quantities.
• A proportion is a statement that two ratios are equal. Proportions can be solved using cross-multiplication.

Frequently Asked Questions (FAQ)

• Q: What is the difference between congruent and similar triangles?

• A: Congruent triangles have the same size and shape, meaning all corresponding sides and angles are congruent. Similar triangles have the same shape but not necessarily the same size; their corresponding angles are congruent, but their corresponding sides are proportional.

• Q: Can I use AA similarity if I only know one angle is congruent?

• A: No. AA similarity requires two pairs of congruent angles.

• Q: What if the ratios of corresponding sides are not equal?

• A: If the ratios of corresponding sides are not equal, the triangles are not similar.

• Q: Are all equilateral triangles similar?

• A: Yes. All equilateral triangles have angles of 60°, 60°, and 60°, satisfying the AA similarity criterion. Their sides are proportional because all sides are equal within each equilateral triangle.

Conclusion

Understanding similar triangles is a cornerstone of geometry. By mastering the concepts of AA, SAS, and SSS similarity, and practicing with various problems, you can confidently solve complex geometric problems. Remember to carefully analyze the given information, identify corresponding parts, set up proportions correctly, and solve for the unknown values. Through diligent practice and a strong understanding of the underlying principles, you'll excel in tackling similar triangle problems and build a solid foundation for more advanced geometric concepts. This comprehensive guide has provided you with the tools and understanding needed to confidently approach and solve problems involving similar triangles. Remember to continue practicing and reviewing these concepts to solidify your knowledge and skills.