Unit 5 Relationships In Triangles Answer Key

Unit 5: Relationships in Triangles - Answer Key & Comprehensive Guide

This comprehensive guide delves into the key concepts covered in Unit 5: Relationships in Triangles. We will explore various theorems, postulates, and problem-solving techniques related to triangle properties. This guide serves as a virtual answer key, providing explanations and solutions to common problems, ultimately enhancing your understanding of triangle relationships. We'll cover topics such as triangle congruence, similarity, special right triangles, and more.

Understanding Triangle Congruence

Triangle congruence is a fundamental concept in geometry. Two triangles are considered congruent if their corresponding sides and angles are equal. Several postulates and theorems help determine triangle congruence:

1. SSS (Side-Side-Side) Postulate:

• Statement: If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
• Example: If triangle ABC has sides AB = 5, BC = 7, AC = 9, and triangle DEF has sides DE = 5, EF = 7, DF = 9, then triangle ABC ≅ triangle DEF (by SSS).

2. SAS (Side-Angle-Side) Postulate:

• Statement: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
• Example: If in triangle ABC, AB = 6, angle B = 45°, BC = 8, and in triangle DEF, DE = 6, angle E = 45°, EF = 8, then triangle ABC ≅ triangle DEF (by SAS).

3. ASA (Angle-Side-Angle) Postulate:

• Statement: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
• Example: If in triangle ABC, angle A = 60°, AB = 10, angle B = 70°, and in triangle DEF, angle D = 60°, DE = 10, angle E = 70°, then triangle ABC ≅ triangle DEF (by ASA).

4. AAS (Angle-Angle-Side) Theorem:

• Statement: If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.
• Example: If in triangle ABC, angle A = 50°, angle B = 60°, AC = 12, and in triangle DEF, angle D = 50°, angle E = 60°, DF = 12, then triangle ABC ≅ triangle DEF (by AAS).

5. HL (Hypotenuse-Leg) Theorem (Right Triangles Only):

• Statement: If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
• Example: This theorem only applies to right-angled triangles. If two right-angled triangles have congruent hypotenuses and one pair of congruent legs, they are congruent.

Understanding Triangle Similarity

Triangle similarity occurs when two triangles have the same shape but not necessarily the same size. Corresponding angles are equal, and corresponding sides are proportional. Similar to congruence, postulates and theorems establish similarity:

1. AA (Angle-Angle) Similarity Postulate:

• Statement: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
• Example: If angle A = angle D and angle B = angle E, then triangle ABC ~ triangle DEF (by AA).

2. SSS (Side-Side-Side) Similarity Theorem:

• Statement: If the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar.
• Example: If AB/DE = BC/EF = AC/DF, then triangle ABC ~ triangle DEF (by SSS).

3. SAS (Side-Angle-Side) Similarity Theorem:

• Statement: 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.
• Example: If AB/DE = BC/EF and angle B = angle E, then triangle ABC ~ triangle DEF (by SAS).

Special Right Triangles

Certain right triangles possess unique properties that simplify calculations:

1. 45-45-90 Triangles (Isosceles Right Triangles):

• Properties: Two legs are congruent, and the hypotenuse is √2 times the length of a leg.
• Example: If a leg has length 'x', the other leg is also 'x', and the hypotenuse is x√2.

2. 30-60-90 Triangles:

• Properties: The hypotenuse is twice the length of the shorter leg, and the longer leg is √3 times the length of the shorter leg.
• Example: If the shorter leg (opposite the 30° angle) has length 'x', the hypotenuse is 2x, and the longer leg (opposite the 60° angle) is x√3.

Pythagorean Theorem and its Converse

The Pythagorean Theorem is a fundamental concept for right-angled triangles:

• Statement: In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (legs). a² + b² = c², where 'c' is the hypotenuse.
• Converse: If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right-angled triangle.

Problem Solving and Applications

Numerous real-world problems utilize the principles of triangle relationships. Examples include surveying, architecture, engineering, and navigation. Solving these problems involves applying the theorems and postulates discussed above. Here are some examples with solutions:

Example 1: Two triangles, ABC and DEF, have AB = 8, BC = 10, AC = 6, and DE = 4, EF = 5, DF = 3. Are the triangles similar?

Solution: We can use the SSS similarity theorem. Check if the ratios of corresponding sides are equal:

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

Since the ratios are all equal, the triangles are similar (by SSS similarity).

Example 2: A right-angled triangle has legs of length 5 and 12. Find the length of the hypotenuse.

Solution: Use the Pythagorean Theorem: a² + b² = c²

5² + 12² = c² 25 + 144 = c² 169 = c² c = √169 = 13

The hypotenuse is 13 units long.

Example 3: A triangle has sides of length 6, 8, and 10. Is it a right-angled triangle?

Solution: Use the converse of the Pythagorean Theorem. The longest side is 10.

6² + 8² = 36 + 64 = 100 10² = 100

Since the square of the longest side equals the sum of the squares of the other two sides, the triangle is a right-angled triangle.

Example 4: Find the length of the hypotenuse of a 45-45-90 triangle with legs of length 7.

Solution: In a 45-45-90 triangle, the hypotenuse is √2 times the length of a leg.

Hypotenuse = 7√2

Example 5: Find the length of the longer leg of a 30-60-90 triangle with a shorter leg of length 4.

Solution: In a 30-60-90 triangle, the longer leg is √3 times the length of the shorter leg.

Longer leg = 4√3

This comprehensive guide provides a detailed overview of Unit 5: Relationships in Triangles, including key concepts, theorems, postulates, and problem-solving techniques. Remember to practice various problems to solidify your understanding. By mastering these principles, you'll be well-equipped to tackle more complex geometry challenges. Remember that consistent practice and application are crucial for mastering these concepts. Further exploration of advanced geometric principles will build upon this foundation.