Segment and Angle Proofs

Overview

A proof is a logical, step-by-step explanation showing that the truth of a hypothesis guarantees the truth of a conclusion. This lesson builds proofs from the pieces you already know — definitions, postulates, and the algebraic properties of equality — and presents them in two-column form (statements on the left, reasons on the right) as well as in paragraph form. Most of the work is naming the exact reason that justifies each step: addition, subtraction, and division of equals, substitution, and the Segment Addition Postulate.

Vocabulary

Proof: A logical, step-by-step explanation that shows the truth of a hypothesis guarantees the truth of a conclusion, with every step supported by a definition, postulate, property, or theorem.
Two-Column Proof: A formal proof organized into two columns: statements (what is done or claimed) on the left and reasons (why each statement is correct) on the right.
Paragraph Proof: An informal proof written as a paragraph of sentences; it contains the same statements and reasons as a two-column proof, presented in sentence form. A flow proof presents the same information in boxes and arrows.
Addition Property of Equality: If the same quantity is added to both sides of an equation, the resulting equation is still true: if a = b, then a + c = b + c.
Subtraction Property of Equality: If the same quantity is subtracted from both sides of an equation, the resulting equation is still true: if a = b, then a - c = b - c.
Division Property of Equality: If both sides of an equation are divided by the same nonzero quantity, the resulting equation is still true: if a = b and c is not zero, then a/c = b/c.
Substitution Property: If two quantities are equal, one may be replaced by the other in any expression or equation.
Segment Addition Postulate: If point B is between points A and C on a segment, then AB + BC = AC.

Main Ideas

Every step of a proof needs a reason. The first statement is the Given (the hypothesis), the last statement is what you were asked to Prove (the conclusion), and everything between must be justified.
The algebraic properties of equality do the heavy lifting: you may add, subtract, multiply, or divide both sides of an equation by the same quantity and the equality still holds.
The Segment Addition Postulate turns a picture into an equation: when a point lies between the endpoints of a segment, the two parts add to the whole.
Substitution finishes many proofs: once two expressions are known to equal the same whole, replace one with the other to reach the conclusion.
Two-column, paragraph, and flow proofs carry identical logic — only the presentation differs. Two-column proofs are the most common.

Lesson Notes

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Practice Problems

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