Unit 6 Similar Triangles Homework 5 Answer Key

Unit 6 Similar Triangles Homework 5: A Comprehensive Guide to Solving Problems

This comprehensive guide delves into the solutions for Unit 6, Homework 5, focusing on similar triangles. We'll break down the core concepts, offer step-by-step solutions to various problem types, and provide extra tips and tricks to master this crucial geometry topic. Remember, understanding the underlying principles is key to success, not just memorizing answers.

Understanding Similar Triangles

Before we dive into the specific homework problems, let's solidify our understanding of similar triangles. Two triangles are considered similar if:

• Their corresponding angles are congruent (equal). This means that the angles in one triangle have the same measure as the corresponding angles in the other triangle.
• Their corresponding sides are proportional. This means that the ratio of the lengths of corresponding sides is constant. This constant ratio is often referred to as the scale factor.

We use the symbol ~ to denote similarity. For example, if triangle ABC is similar to triangle DEF, we write ΔABC ~ ΔDEF.

Identifying Similar Triangles

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

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

Solving Problems from Unit 6, Homework 5: Example Problems and Solutions

This section will walk through example problems similar to those found in Unit 6, Homework 5, illustrating various approaches and problem-solving strategies. Since I don't have access to your specific homework assignment, I'll create representative problems covering the range of concepts typically found in such a unit.

Problem 1: Using AA Similarity

Problem: Triangle ABC has angles A = 50°, B = 60°, and C = 70°. Triangle DEF has angles D = 50°, E = 60°, and F = 70°. Are triangles ABC and DEF similar? Explain your reasoning.

Solution:

Yes, triangles ABC and DEF are similar by AA Similarity. Both triangles have angles A = D = 50° and B = E = 60°. Since two angles are congruent in both triangles, the third angle must also be congruent (70° in both cases). This satisfies the AA Similarity postulate. Therefore, ΔABC ~ ΔDEF.

Problem 2: Using SSS Similarity

Problem: Triangle PQR has sides PQ = 6, QR = 8, and RP = 10. Triangle XYZ has sides XY = 3, YZ = 4, and ZX = 5. Are triangles PQR and XYZ similar? Explain your reasoning.

Solution:

Let's check the ratios of corresponding sides:

• PQ/XY = 6/3 = 2
• QR/YZ = 8/4 = 2
• RP/ZX = 10/5 = 2

Since the ratios of all corresponding sides are equal (2), the triangles are similar by SSS Similarity. Therefore, ΔPQR ~ ΔXYZ with a scale factor of 2.

Problem 3: Using SAS Similarity

Problem: Triangle JKL has sides JK = 9, KL = 12, and angle K = 75°. Triangle MNO has sides MN = 6, NO = 8, and angle N = 75°. Are triangles JKL and MNO similar? Explain your reasoning.

Solution:

Let's examine the ratio of the sides and the included angle:

• JK/MN = 9/6 = 1.5
• KL/NO = 12/8 = 1.5
• Angle K = Angle N = 75°

Since the ratio of two corresponding sides (JK/MN and KL/NO) is equal (1.5), and the included angle is congruent (75°), the triangles are similar by SAS Similarity. Therefore, ΔJKL ~ ΔMNO with a scale factor of 1.5.

Problem 4: Finding Missing Side Lengths

Problem: Given that ΔABC ~ ΔDEF, AB = 12, BC = 15, and DE = 8. Find the length of EF.

Solution:

Since the triangles are similar, the ratios of corresponding sides are equal. Therefore:

AB/DE = BC/EF

12/8 = 15/EF

Cross-multiplying:

12 * EF = 15 * 8

12 * EF = 120

EF = 120/12

EF = 10

Therefore, the length of EF is 10.

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's denote the height of the tree as 'h'. Then:

h/20 = 6/4

Cross-multiplying:

4h = 120

h = 30

Therefore, the tree is 30 feet tall.

Advanced Concepts and Problem-Solving Strategies

This section introduces more advanced concepts and strategies that might be relevant to more challenging problems in your homework.

Indirect Measurement

Similar triangles are frequently used for indirect measurement, as illustrated in the previous problem involving the tree and its shadow. This technique is used when direct measurement is difficult or impossible.

Proportions and Ratios

Mastering proportions and ratios is crucial for working with similar triangles. Practice simplifying ratios and solving for unknown variables in proportional equations.

Geometric Mean

The geometric mean is a useful tool when dealing with proportions in similar triangles, particularly in right-angled triangles. Understanding how to apply the geometric mean can simplify certain calculations.

Tips for Success

• Draw diagrams: Always draw clear diagrams to visualize the problem and label all given information.
• Identify corresponding parts: Carefully identify corresponding angles and sides in similar triangles.
• Write proportions: Set up correct proportions to solve for unknown quantities.
• Check your work: Always review your calculations and ensure that your answers make sense in the context of the problem.
• Practice regularly: Consistent practice is key to mastering similar triangles. Work through numerous examples and problems to build your skills and confidence.

Conclusion

Mastering similar triangles is a crucial skill in geometry. By understanding the fundamental concepts, applying the appropriate postulates and theorems, and practicing regularly, you can confidently solve a wide range of problems. Remember to break down complex problems into smaller, manageable steps and always double-check your work. With dedicated effort and consistent practice, you'll successfully complete Unit 6, Homework 5, and build a strong foundation in this important geometric concept. Remember that this is a guide, and you should always consult your textbook and teacher for specific instructions and clarifications.