Book cover for McDougal Littell Jurgensen Geometry : Student Edition 2000

McDougal Littell Jurgensen Geometry : Student Edition 2000

Ray C. Jurgensen, Richard G. Brown

ISBN #9780395977279

1st Edition

2,712 Questions

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20,674 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter emphasizes the role of inequalities in geometry, covering both foundational concepts such as the triangle inequality as well as advanced proof techniques including indirect proofs through contrapositives and inverses. Students learn not only to prove geometric statements indirectly but also to apply these concepts in practical contexts such as mapping and technology-enhanced explorations, reinforcing the interconnected nature of mathematical theory and real-world applications.

Learning Objectives

1

Understand and apply various types of inequalities in geometric contexts, including triangle inequalities.

2

Learn and utilize indirect proof techniques such as proving via contrapositives and inverses.

3

Analyze and solve problems related to one and two triangle inequalities.

4

Explore applications of geometric inequalities in real-world scenarios such as cartography and non-Euclidean geometries.

5

Develop proficiency in integrating computer technology and key explorations to reinforce geometric concepts.

Key Concepts

CONCEPT

DEFINITION

Geometric Inequality

A statement about the relative size or order of two geometric quantities, typically related to lengths, angles, or areas.

Indirect Proof

A proof technique that establishes a statement by assuming the opposite and then demonstrating a contradiction.

Contrapositive

A method of indirect proof where the contrapositive of a given statement (if not Q then not P) is proved instead of the direct implication.

Inverse

The logical statement formed by negating both the hypothesis and the conclusion of the original statement; it is not logically equivalent to the original statement in all cases.

Triangle Inequality

A fundamental inequality stating that in any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

Non-Euclidean Geometry

A type of geometry that relaxes or alters Euclid's parallel postulate, leading to different geometric properties and inequalities.

Example Problems

Example 1

Some information about the diagram is given. Tell whether the other statements can be deduced from what is given. (Write yes or no.) Given: Point $Y$ lies between points $X$ and $Z$. a. $X Y=\frac{1}{2} X Z$ b. $X Z=X Y+Y Z$ c. $X Z>X Y$ d. $Y Z>X Y$ e. $X Z>Y Z$ f. $X Z>2 X Y$ CAN'T COPY THE GRAPH

Example 2

Some information about the diagram is given. Tell whether the other statements can be deduced from what is given. (Write yes or no.) Given: Point $B$ lies in the interior of $\angle A O C$. a. $m \angle 1=m \angle 2$ b. $m \angle A O C=m \angle 1+m \angle 2$ c. $m \angle A O C>m \angle 1$ d. $m \angle A O C>m \angle 2$ e. $m \angle 1>m \angle 2$ f. $m \angle A O C>90$ CAN'T COPY THE GRAPH

Example 3

Some information about the diagram is given. Tell whether the other statements can be deduced from what is given. (Write yes or no.) Given: $\square A B C D ; A C>B D$ a. $A B>A D$ b. $A M>M C$ c. $D M=M B$ d. $A M>M B$ CAN'T COPY THE GRAPH

Example 4

Some information about the diagram is given. Tell whether the other statements can be deduced from what is given. (Write yes or no.) Given: $m \angle R V S=m \angle R S V=65$ a. $R T>R S$ b. $R T>R V$ c. $R S>S T$ d. $V T<R S$ CAN'T COPY THE GRAPH

Example 5

Some information about the diagram is given. Tell whether the other statements can be deduced from what is given. (Write yes or no.) When some people are given that $j>k$ and $l>m$, they carelessly conclude that $j+k>I+m .$ Find values for $j, k, l,$ and $m$ that show this conclusion is false. CAN'T COPY THE GRAPH
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Step-by-Step Explanations

QUESTION

How can one prove a geometric statement using the indirect proof method via the contrapositive?

STEP-BY-STEP ANSWER:

Step 1: Identify the original statement in the form 'If P, then Q'.
Step 2: Write the contrapositive of the statement, which is 'If not Q, then not P'.
Step 3: Assume not Q is true and use known geometric properties and inequalities to logically deduce not P.
Step 4: Demonstrate that assuming not Q inevitably leads to not P, thus proving the contrapositive.
Step 5: Conclude that, since the contrapositive is true, the original statement 'If P, then Q' must also be true.
Final Answer: The indirect proof via the contrapositive confirms the validity of the original geometric statement.

Indirect Proof (Contrapositive)

QUESTION

Given two sides of a triangle with lengths a and b, how do you determine the range of possible lengths for the third side c?

STEP-BY-STEP ANSWER:

Step 1: Start with the triangle inequality theorem which states that the sum of any two sides must be greater than the third side.
Step 2: Write the inequalities: a + b > c, a + c > b, and b + c > a.
Step 3: Solve the first inequality for c: c < a + b.
Step 4: Solve the second inequality for c: c > b - a (assuming a ≤ b, use absolute value |a - b| to account for order).
Step 5: Combine the results to conclude that c must satisfy |a - b| < c < a + b.
Final Answer: The third side c must be greater than the absolute difference of a and b and less than the sum of a and b.

Triangle Inequality

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Common Mistakes

  • Confusing the contrapositive with the inverse, despite their distinct logical properties.
  • Neglecting the necessity of all three triangle inequalities when analyzing triangles.
  • Assuming that the triangle inequality applies only to Euclidean geometries without exploring its variations in non-Euclidean contexts.
  • Overlooking the role of absolute values when computing differences in lengths, leading to incorrect ranges for the third side.