Inequalities In One Triangle Worksheet Answers Explained
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In a triangle, inequalities play a vital role in understanding the relationships between its sides and angles. These inequalities help us determine the possible lengths of a triangle's sides given certain conditions, and also assist in solving various problems in geometry. In this article, we will explain the essential concepts behind triangle inequalities, provide a thorough breakdown of a worksheet on the topic, and present the answers in an easy-to-understand manner.
Understanding Triangle Inequalities
Triangle Inequality Theorem
The Triangle Inequality Theorem states that for any triangle with sides of lengths (a), (b), and (c):
- (a + b > c)
- (a + c > b)
- (b + c > a)
This theorem essentially indicates that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Understanding this theorem is fundamental for solving problems related to triangle inequalities.
Types of Triangle Inequalities
- Side Length Inequalities: These inequalities help us find the possible lengths of a triangle's sides based on given values.
- Angle-Side Inequalities: The larger angle in a triangle is always opposite the longer side. Conversely, the smaller angle is opposite the shorter side.
Worksheet Breakdown
The worksheet on triangle inequalities often includes a variety of problems that require applying the above concepts. Below, we provide sample problems along with their explanations and answers.
Sample Problems
Here’s an outline of what a worksheet might look like, along with answers that explain each concept clearly:
| Problem Number | Problem Description | Explanation & Answer |
|---|---|---|
| 1 | If one side of a triangle is 5, and another is 7, what is the possible range for the third side? | According to the Triangle Inequality Theorem: <br> 1st inequality: (5 + 7 > c \Rightarrow c < 12) <br> 2nd inequality: (5 + c > 7 \Rightarrow c > 2) <br> 3rd inequality: (7 + c > 5 \Rightarrow c > -2) <br> Thus, (2 < c < 12) |
| 2 | Given angles of 30° and 60°, what can we say about the third angle? | The sum of angles in a triangle is always 180°: <br> (30° + 60° + c = 180° \Rightarrow c = 90°) <br> Thus, the third angle is 90°. |
| 3 | If the lengths of two sides of a triangle are 8 and 3, determine if the third side can be 5. | Applying the theorem: <br> 1st inequality: (8 + 3 > 5 \Rightarrow 11 > 5) (True) <br> 2nd inequality: (8 + 5 > 3 \Rightarrow 13 > 3) (True) <br> 3rd inequality: (3 + 5 > 8 \Rightarrow 8 > 8) (False) <br> Therefore, it cannot be 5. |
| 4 | If (x) is the length of one side and the other two sides are 7 and 10, find the range for (x). | Apply Triangle Inequality: <br> 1st inequality: (7 + 10 > x \Rightarrow x < 17) <br> 2nd inequality: (7 + x > 10 \Rightarrow x > 3) <br> 3rd inequality: (10 + x > 7 \Rightarrow x > -3) <br> Thus, (3 < x < 17) |
| 5 | In triangle ABC, if angle A is greater than angle B, what can we conclude about the sides a and b? | According to angle-side inequality: If (A > B), then (a > b). This means the side opposite to angle A (side a) is longer than the side opposite to angle B (side b). |
Important Notes
"Always ensure that the values for sides are non-negative and that they adhere to the triangle inequality rules to form a valid triangle."
By following these examples, students can gain a comprehensive understanding of how to apply the Triangle Inequality Theorem effectively.
Conclusion
Understanding inequalities in triangles is crucial for solving geometric problems. The Triangle Inequality Theorem not only provides a framework for determining possible side lengths but also for grasping the relationships between sides and angles. When you work through problems systematically and apply these inequalities, you'll become proficient at analyzing any triangle, ensuring a solid foundation in geometry.
With practice and the knowledge shared in this article, you will find that solving triangle inequality problems becomes more intuitive. Keep these principles in mind as you work through your worksheet, and you'll be well on your way to mastering triangle inequalities!