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Paragraph proofs

Compared with two-column proofs, paragraph proofs:

Allow for more kinds of logical arguments
Afford greater expressiveness
Are much more common in “real” mathematics
Are not a license to skip steps

Example (Sec. 4.4 Sample prob. p. 186)

Given: $▵ A B C$ is isosceles, with $\overline{A B} ≅ \overline{A C}$ . $E$ is the midpoint of $\overline{B C}$ .

Prove: $\overline{A E} ⊥ \overline{B C}$ .

Two-column proof:

	$\overline{B E} ≅ \overline{E C}$	Def. midpoint
	$\overline{A E} ⊥ \overline{B C}$	Equidistance theorem

Paragraph proof:

Since $E$ is the midpoint of $\overline{B C}$ , by the definition of midpoint $\overline{B E} ≅ \overline{E C}$ . Since $\overline{A B} ≅ \overline{A C}$ and $\overline{B E} ≅ \overline{E C}$ , the equidistance theorem tells us that $\overline{A E} ⊥ bis \overline{B C}$ , so in particular $\overline{A E} ⊥ \overline{B C}$ .

Example (Sec. 4.4 problem 13, p. 189)

Given: $\overline{W Y}$ and $\overline{X Z} ⊥ bis$ each other.

Prove: $\overline{W X} ≅ \overline{X Y} ≅ \overline{Y Z} ≅ \overline{Z W}$

Paragraph proof:

Since $\overline{W Y} ⊥ bis \overline{X Z}$ , the equidistance theorem tells us that $\overline{W X} ≅ \overline{Z W}$ and $\overline{X Y} ≅ \overline{Y Z}$ . Similarly, since $\overline{X Z} ⊥ bis \overline{W Y}$ , the equidistance theorem tells us that $\overline{W X} ≅ \overline{X Y}$ and $\overline{Y Z} ≅ \overline{Z W}$ . Using transitivity to combine those congruences, $\overline{W X} ≅ \overline{X Y} ≅ \overline{Y Z} ≅ \overline{Z W}$ .