## Introduction

As a website operator, one of the most crucial aspects of your role is to provide high-quality educational resources for your website visitors. Students and teachers alike are always on the lookout for reliable instructional materials that can help them acquire new knowledge and improve their skills. In this article, we will provide an answer key for the Isosceles and Equilateral Triangles Worksheet 4-6, a resource that can be used for geometry instruction.

## Understanding the Isosceles and Equilateral Triangles Worksheet 4-6

The Isosceles and Equilateral Triangles Worksheet 4-6 is designed to help students learn about the properties and characteristics of these two types of triangles. The worksheet consists of six questions, each with several sub-questions, aimed at testing the students’ understanding of the topic.

In question one, students are required to identify examples of isosceles triangles and equilateral triangles from a given set of figures. Question two asks them to identify the base and height of an isosceles triangle, while question three requires them to apply the Pythagorean theorem to calculate the length of the hypotenuse of a right triangle with sides of 5 and 12 units.

Question four is about applying the knowledge of the properties of equilateral triangles to solve a problem. In question five, students are required to use their understanding of symmetry and the properties of isosceles triangles to solve a problem about a kite.

Finally, question six deals with solving a real-life problem using the concepts of isosceles and equilateral triangles. Students are required to calculate the area of an equilateral triangle and find the perimeter of a trapezoid using the Pythagorean theorem.

## Answer Key

1. a) Isosceles triangle: figure D; Equilateral triangle: figure B.

b) Isosceles triangle: figure E; Equilateral triangle: figure A.

c) Isosceles triangle: figure G; Equilateral triangle: figure C.

2. Base: 6 units; Height: 4 units.

3. Hypotenuse = √(5² + 12²) = 13 units.

4. Each angle of an equilateral triangle is 60 degrees. Therefore, the sum of the angles of the given triangle is 180 degrees, and each angle must be 60/3 = 20 degrees.

5. Since triangle AED is isosceles, and AE = ED, then angle AED must be equal to angle EAD. Similarly, angle ADE is equal to angle DEA. Also, angles ADB and DCB are equal since they are vertical angles. Therefore, angles AED, EAD, ADE, and DEA must add up to 360-2×124=112 degrees. Hence, angle ADB plus angle DCB must be equal to (180- 112)/2 = 34 degrees.

6. a) The area of an equilateral triangle with side length 4 cm is (4²√3) /4 = 4√3 square cm.

b) Let AB = x, BC = y, and AD = CD = 6 cm. Using the Pythagorean theorem, we have:

x² – y² = 36 (eq. 1)

x + y = 12 (eq. 2)

Solving for y in (eq. 2), we obtain y = 12 – x. Substituting this into (eq.1), we have:

x² – (12-x)² = 36

Expanding the equation, we get:

24x – 72 = 0

Solving for x, we get x = 3. Therefore, y = 9. Finally, using the formula for the perimeter of a trapezoid, we have:

Perimeter = AB + BC + AD + CD = 3 +9 +6 +6 = 24 cm.

## Conclusion

The Isosceles and Equilateral Triangles Worksheet 4-6 provides a set of challenging and engaging questions that can help students expand their knowledge and develop their problem-solving skills. As a website operator, providing answer keys for such educational resources can serve as a valuable tool for teachers and learners alike. By doing so, you are making a significant impact on the lives of students by contributing to their academic success.