Chapter 2
Symmetric Property of Congruence
If Line DE is congruent to Line FG, then Line FG is congruent to Line DE. If Angle D is congruent to Angle E, then Angle E is congruent to Angle D.
Theorem 2-1 Midpoint Theorem
If M is the midpoint of Line AB, then AM=1/2AB and MB=1/2AB.
Theorem 2-2 Angle Bisector Theorem
If Ray BX is the bisector of Angle ABC, then m∠ABX=1/2 m∠ABC and m∠XBC=1/2m∠ABC.
Biconditional
If a conditional and its converse are both true they can be combined into a single statement by using the words "if and only if." A statement that contains the words "if and only if" is called a biconditional. p if and only if q.
Transitive Property of Equaliy
If a=b and b=c, then a=c.
Division Property of Equality
If a=b and c is not = 0, then a/c=b/c.
Addition Property of Equality
If a=b, and c=d, then a+c=b+d.
Subtraction Property of Equality
If a=b, and c=d, then a-c=b-d.
Symmetric Property of Equality
If a=b, then b=a.
Multiplication Property of Equality
If a=b, then ca=cb.
Substitution Property of Equality
If a=b, then either a or b maybe substituted for the other in any equation (or inequality).
Hypothesis
If p, then q. p: Hypotheiss
Conclusion
If p, then q. q: Conclusion
Theorem 2-6
If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary.
Theorem 2-8
If two angles are complements of congruent angles (or of the same angle), then the two angles are congruent.
Theorem 2-7
If two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent.
Theorem 2-4
If two lines are perpendicular, then they form congruent adjacent angles.
Theorem 2-5
If two lines form congruent adjacent angles, then the lines are perpendicular.
Reflexive Property of Equality
a=a
Conditional Statements
A geometry student reads, "If B is between A and C, then AB+BC=AC. This is a example of if-then statements, which are also called conditional statements or simply conditionals.
Transitive Property of Congruence
If Line DE is congruent to Line FG and Line FG is congruent to Line JK, then Line DE is congruent to Line JK. If Angle D is congruent to Angle E and Angle E is congruent to Angle F, then Angle D is congruent to Angle F.
Reflexive Property of Congruence
Line DE is congruent to line DE Angle D is congruent Angle D.
Counterexample
Statement: If Ed lives in Texas, then he lives south of Canada. False converse: If Ed lives south of Canada, then he lives in Texas. An if-then statement is false if an example can be found for which the hypothesis is true and the conclusion is false. Such an example is called a counterexample.
Contrapostive
The contrapositive of a conditional is formed by interchanging and negating both hypothesis and conclusion. If not q, then not p.
Converse
The converse of a conditional is formed by interchanging the hypothesis and the conclusion. If q, then p.
Inverse
The inverse of a conditional is formed by negating the hypothesis and the conclusion. If not p, then not q.
Theorem 2-3
Vertical angles are congruent.
Distributive Property
a(b+c)=ab+ac.