Triangle Sum & Exterior Angle Theorem Worksheet Answer Key
Table of Contents :
Understanding the Triangle Sum and Exterior Angle Theorem is fundamental in geometry. These theorems help us solve various problems related to triangles, whether in academic settings or real-world applications. Here, we'll break down the concepts associated with these theorems and provide an answer key to a worksheet that revolves around these essential principles.
What is the Triangle Sum Theorem? 🏺
The Triangle Sum Theorem states that the sum of the interior angles of a triangle is always 180 degrees. This is a crucial concept as it lays the groundwork for many geometrical proofs and problems.
The Formula
For any triangle, if the angles are denoted as (A), (B), and (C), the Triangle Sum Theorem can be expressed mathematically as:
[ A + B + C = 180^\circ ]
Example
Consider a triangle with angles of 50° and 70°. To find the third angle (C):
[ C = 180^\circ - (50^\circ + 70^\circ) = 180^\circ - 120^\circ = 60^\circ ]
Thus, the angles of the triangle are 50°, 70°, and 60°.
What is the Exterior Angle Theorem? 🌐
The Exterior Angle Theorem states that an exterior angle of a triangle is equal to the sum of the two opposite interior angles. This theorem provides a way to find unknown angles in situations where the interior angles are known.
The Formula
If angle (D) is an exterior angle, and angles (A) and (B) are the opposite interior angles, the theorem can be expressed as:
[ D = A + B ]
Example
If we have a triangle with interior angles (A = 40^\circ) and (B = 60^\circ), the exterior angle (D) at one vertex is calculated as follows:
[ D = 40^\circ + 60^\circ = 100^\circ ]
This means the exterior angle measures 100°.
Triangle Sum & Exterior Angle Theorem Worksheet 📝
To practice applying these theorems, you can work on a worksheet designed around different triangle configurations. Below is a simplified version of what the worksheet might look like:
Sample Problems
| Problem No. | Given Angles | Find Angle (C) | Find Exterior Angle (D) |
|---|---|---|---|
| 1 | 30°, 60° | ||
| 2 | 45°, 45° | ||
| 3 | 70°, 50° | ||
| 4 | 80°, 50° |
Solving the Worksheet
-
Problem 1: Given angles are 30° and 60°.
- Find Angle (C): [ C = 180^\circ - (30^\circ + 60^\circ) = 90^\circ ]
- Exterior Angle (D): [ D = 30^\circ + 60^\circ = 90^\circ ]
-
Problem 2: Given angles are 45° and 45°.
- Find Angle (C): [ C = 180^\circ - (45^\circ + 45^\circ) = 90^\circ ]
- Exterior Angle (D): [ D = 45^\circ + 45^\circ = 90^\circ ]
-
Problem 3: Given angles are 70° and 50°.
- Find Angle (C): [ C = 180^\circ - (70^\circ + 50^\circ) = 60^\circ ]
- Exterior Angle (D): [ D = 70^\circ + 50^\circ = 120^\circ ]
-
Problem 4: Given angles are 80° and 50°.
- Find Angle (C): [ C = 180^\circ - (80^\circ + 50^\circ) = 50^\circ ]
- Exterior Angle (D): [ D = 80^\circ + 50^\circ = 130^\circ ]
Answer Key 📜
Here's a summary of the answers for the worksheet problems:
<table> <tr> <th>Problem No.</th> <th>Angle C (°)</th> <th>Exterior Angle D (°)</th> </tr> <tr> <td>1</td> <td>90°</td> <td>90°</td> </tr> <tr> <td>2</td> <td>90°</td> <td>90°</td> </tr> <tr> <td>3</td> <td>60°</td> <td>120°</td> </tr> <tr> <td>4</td> <td>50°</td> <td>130°</td> </tr> </table>
Conclusion
Mastering the Triangle Sum and Exterior Angle Theorem is essential for anyone looking to excel in geometry. With practice worksheets and a solid understanding of the theorems, you can tackle various problems involving triangles. Remember that the sum of angles in any triangle always equals 180° and that the exterior angle is always the sum of the opposite interior angles. Happy studying! ✨