Geometry is one of the most intriguing and important topics in mathematics. The subject explores the properties of shapes, lines, and angles. Geometry is a fundamental basis for many other topics in mathematics, such as calculus, algebra, and trigonometry. This article will provide answers to the practice problems in Lesson 7.4 of a geometry textbook.

## Understanding the Triangle Inequality Theorem

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining third side. This theorem is extremely important in geometry, as it can be used to determine if a given set of three side lengths can make up a triangle. It can also be used to determine the angles of a triangle and to solve for missing sides.

## Answers to Practice Problem A

The practice problems in Lesson 7.4 are designed to help students understand the Triangle Inequality Theorem. The first problem asks students to determine whether the three given side lengths can form a triangle. The side lengths are 2, 5, and 8. According to the Triangle Inequality Theorem, the sum of any two sides must be greater than the remaining side. In this case, 2 + 5 = 7, which is not greater than 8. Therefore, the three side lengths cannot form a triangle.

## Answers to Practice Problem B

The second practice problem asks students to determine the angles of a triangle with the given side lengths. The side lengths are 5, 6, and 7. According to the Triangle Inequality Theorem, the sum of any two sides must be greater than the remaining side. In this case, 5 + 6 = 11, which is greater than 7. Therefore, the three side lengths can form a triangle. To determine the angles of the triangle, students can use the Law of Cosines. The Law of Cosines states that for a triangle with side lengths a, b, and c, the following equation holds: c^2 = a^2 + b^2 – 2ab cos C, where C is one of the angles of the triangle. Using this equation, students can solve for each of the angles of the triangle.

## Answers to Practice Problem C

The third practice problem asks students to determine the length of the missing side of a triangle. The side lengths given are 3, 5, and x. According to the Triangle Inequality Theorem, the sum of any two sides must be greater than the remaining side. In this case, 3 + 5 = 8, which is not greater than x. Therefore, the three side lengths cannot form a triangle. To solve for the missing side, students can use the Law of Cosines. The Law of Cosines states that for a triangle with side lengths a, b, and c, the following equation holds: c^2 = a^2 + b^2 – 2ab cos C. In this case, a = 3, b = 5, and C is known. Students can use this equation to solve for c, which is the length of the missing side.

This article has provided answers to the practice problems in Lesson 7.4 of a geometry textbook. The Triangle Inequality Theorem and the Law of Cosines are two important concepts that were used to answer the questions. Understanding how to apply these theorems is key to mastering the geometry curriculum. With practice and dedication, students can become proficient in geometry.