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construct a fourth point D so that Triangle

construct a fourth point D so that Triangle

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Given triangle ABC, use the axioms to construct a fourth point D so that Triangle ABD is congruent, but not equal to Triangle ABC. State the axioms that you use in your construction. Which axiom implies that the triangle are congruent?

 

 

Here is a list of the axioms:

 

**Axiom 1 (The Point-Line Incidence Axiom):** A line is a set of points. Given any two different points, there is exactly one line that contains them. **Axiom 2 (The Ruler Axiom):** 1. For any two points \( A \) and \( B \), there is a positive number called the distance from \( A \) to \( B \) and denoted \( AB \). 2. For each line \( L \), there is a function \( f \) that assigns a real number to each point of \( L \) such that: - (a) Different points have different numbers associated with them. - (b) Every number—positive, negative, or zero—has some point with which it is associated. - (c) If \( A \) and \( B \) are points of \( L \), then \(|f(B) - f(A)| = AB\), the distance from \( A \) to \( B \). Note that if \( A \) and \( B \) designate the same point, then \( AB = 0 \). **Axiom 3 (Pasch’s Separation Axiom for a Line):** Given a line \( L \) in the plane, the points in the plane that are not on \( L \) form two non-empty sets, \( H_1 \) and \( H_2 \), called half-planes, so: 1. If \( A \) and \( B \) are points in the same half-plane, then \( AB \) lies wholly in that half-plane. 2. If \( A \) and \( B \) are points not in the same half-plane, then \( AB \) intersects \( L \). **Axiom 4 (The Angle Measurement Axiom):** To every angle, there corresponds a real number between 0° and 180° called its measure or size. We denote the measure of \( \angle BAC \) by \( m\angle BAC \). **Axiom 5 (The Angle Construction Axiom):** Let \( \overrightarrow{AB} \) lie entirely on the boundary line \( L \) of some half-plane \( H \). For every number \( r \) where \( 0^\circ \le r \le 180^\circ \), there is exactly one ray \( \overrightarrow{AC} \) where
Transcribed Image Text:**Axiom 1 (The Point-Line Incidence Axiom):** A line is a set of points. Given any two different points, there is exactly one line that contains them. **Axiom 2 (The Ruler Axiom):** 1. For any two points \( A \) and \( B \), there is a positive number called the distance from \( A \) to \( B \) and denoted \( AB \). 2. For each line \( L \), there is a function \( f \) that assigns a real number to each point of \( L \) such that: - (a) Different points have different numbers associated with them. - (b) Every number—positive, negative, or zero—has some point with which it is associated. - (c) If \( A \) and \( B \) are points of \( L \), then \(|f(B) - f(A)| = AB\), the distance from \( A \) to \( B \). Note that if \( A \) and \( B \) designate the same point, then \( AB = 0 \). **Axiom 3 (Pasch’s Separation Axiom for a Line):** Given a line \( L \) in the plane, the points in the plane that are not on \( L \) form two non-empty sets, \( H_1 \) and \( H_2 \), called half-planes, so: 1. If \( A \) and \( B \) are points in the same half-plane, then \( AB \) lies wholly in that half-plane. 2. If \( A \) and \( B \) are points not in the same half-plane, then \( AB \) intersects \( L \). **Axiom 4 (The Angle Measurement Axiom):** To every angle, there corresponds a real number between 0° and 180° called its measure or size. We denote the measure of \( \angle BAC \) by \( m\angle BAC \). **Axiom 5 (The Angle Construction Axiom):** Let \( \overrightarrow{AB} \) lie entirely on the boundary line \( L \) of some half-plane \( H \). For every number \( r \) where \( 0^\circ \le r \le 180^\circ \), there is exactly one ray \( \overrightarrow{AC} \) where
**Axiom 7 (The Supplementary Angles Axiom).** If two angles are supplementary, then their measures add to 180°. **Axiom 8 (The Side-Angle-Side (SAS) Congruence Axiom).** Suppose we have triangles \( ABC \) and \( A'B'C' \), where some or all of the vertices of the second triangle might be vertices of the first triangle as well. If 1. \( m\angle A = m\angle A' \), and 2. \( AB = A'B' \) and \( AC = A'C' \), then the triangles are congruent and the correspondence is: \( A \rightarrow A' \), \( B \rightarrow B' \), \( C \rightarrow C' \). **Axiom 9 (Euclid's Parallel Axiom).** Given a point \( P \) off a line \( L \), there is at most one line in the plane through \( P \) not meeting \( L \).
Transcribed Image Text:**Axiom 7 (The Supplementary Angles Axiom).** If two angles are supplementary, then their measures add to 180°. **Axiom 8 (The Side-Angle-Side (SAS) Congruence Axiom).** Suppose we have triangles \( ABC \) and \( A'B'C' \), where some or all of the vertices of the second triangle might be vertices of the first triangle as well. If 1. \( m\angle A = m\angle A' \), and 2. \( AB = A'B' \) and \( AC = A'C' \), then the triangles are congruent and the correspondence is: \( A \rightarrow A' \), \( B \rightarrow B' \), \( C \rightarrow C' \). **Axiom 9 (Euclid's Parallel Axiom).** Given a point \( P \) off a line \( L \), there is at most one line in the plane through \( P \) not meeting \( L \).
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