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Chapter Analysis
Intermediate24 pages • EnglishQuick Summary
Chapter 7, 'A Tale of Three Intersecting Lines', introduces fundamental concepts about triangle construction. It explains the triangle inequality theorem which dictates that the sum of any two-side lengths must be greater than the third side. Various methods and examples demonstrate how to construct triangles given certain side lengths and angles, exploring cases when triangles cannot exist due to geometric constraints. The chapter also touches on the classification of triangles based on side lengths and angles.
Key Topics
- •Triangle Inequality Theorem
- •Construction of Triangles
- •Types of Triangles
- •Angle Sum Property
- •Classification by Sides and Angles
Learning Objectives
- ✓Understand the triangle inequality theorem and use it to determine the possibility of triangle existence.
- ✓Construct triangles given a set of side lengths and angles.
- ✓Explore different types of triangles and their properties.
- ✓Apply the angle sum property in triangle-related problems.
- ✓Differentiate triangles based on angles and side lengths.
Questions in Chapter
Check if a triangle exists for each of the following set of lengths: (a) 1, 100, 100 (b) 3, 6, 9 (c) 1, 1, 5 (d) 5, 10, 12.
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Does there exist an equilateral triangle with sides 50, 50, 50? In general, does there exist an equilateral triangle of any sidelength? Justify your answer.
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For each of the following, give at least 5 possible values for the third length so there exists a triangle having these as sidelengths: (a) 1, 100 (b) 5, 5 (c) 3, 7.
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For each of the following angles, find another angle for which a triangle is (a) possible, (b) not possible. Find at least two different angles for each category: (a) 30° (b) 70° (c) 54° (d) 144°.
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Determine which of the following pairs can be the angles of a triangle and which cannot: (a) 35°, 150° (b) 70°, 30° (c) 90°, 85° (d) 50°, 150°.
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Additional Practice Questions
Explain what the triangle inequality theorem is and why it is important in constructing triangles.
mediumAnswer: The triangle inequality theorem states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. It is crucial as it ensures that the three sides can actually form a closed shape, a triangle.
What potential problems might you encounter when drawing triangles with large side lengths using just a ruler?
mediumAnswer: When using a ruler for large side lengths, inaccuracies in measuring angles and the lengths themselves might result due to ruler limitations, leading to non-intersecting sides or inaccurately shaped triangles.
Describe why triangles with side lengths that do not satisfy the triangle inequality cannot exist.
easyAnswer: Triangles with side lengths not satisfying the inequality imply that one side is too long relative to the sum of the other two, preventing the sides from forming a closed, three-sided figure.
If a triangle has all angles measured as 60 degrees, what special properties does it have?
easyAnswer: A triangle with all angles measuring 60 degrees is known as an equilateral triangle, having all sides of equal length and considered highly symmetrical.
In a scalene triangle, which has all different side lengths, what can you say about its angles?
easyAnswer: In a scalene triangle, because all sides are of different lengths, all angles will also differ in measure and there are no congruent angles.