2 1 IfThen Statements Converses CONDITIONAL STATEMENTS are

2 -1 If-Then Statements; Converses

CONDITIONAL STATEMENTS are statements written in if-then form. The clause following the “if” is called the hypothesis and the clause following “then” is called the conclusion.

Examples • If it rains after school, then I will give you a ride home. • If you make an A on your test, then you will get an A on your report card.

CONVERSE is formed by interchanging the hypothesis and the conclusion.

Examples False Converses • If Bill lives in Texas, then he lives west of the Mississippi River. • If he lives west of the Mississippi River, then he lives in Texas

Counterexample • An example that shows a statement to be false • It only takes one counterexample to disprove a statement

Biconditional • A statement that contains the words “if and only if” • Segments are congruent if and only if their lengths are equal.

2 -2 Properties from Algebra

Addition Property • If a = b, and c = d, • then a + c = b + d

Subtraction Property • If a = b, and c = d, • then a - c = b - d

Multiplication Property • If a = b, • then ca = bc

Division Property • If a = b, and c 0 • then a/c = b/c

Substitution Property • If a = b, then either a or b may be substituted for the other in any equation (or inequality)

Reflexive Property • a = a

Symmetric Property • If a = b, then b = a

Transitive Property • If a = b, and b = c, then a =c

Distributive Property • a(b + c) = ab + ac

Properties of Congruence

Reflexive Property • DE • D D

Symmetric Property • If DE FG, then FG DE • If D E, then E D

Transitive Property • If DE FG, and FG JK, then DE JK • If D E, and E F, then D F

2 -3 Proving Theorems

Midpoint of a Segment – is the point that divides the segment into two congruent segments

THEOREM 2 -1 Midpoint Theorem If a point M is the midpoint of AB, then AM = ½AB and MB=½AB

BISECTOR of ANGLE– is the ray that divides the angle into two adjacent angles that have equal measure.

THEOREM 2 -2 Angle Bisector Theorem If BX is the bisector of ABC, then: m ABX = ½m ABC and m XBC = ½ m ABC

2 -4 Special Pairs of Angles

COMPLEMENTARY two angles whose measures have the sum 90º J 39º 51º K

SUPPLEMENTARY two angles whose measures have the sum 180º H 133º G 47º

VERTICAL ANGLES– two angles whose sides form two pairs of opposite rays.

THEOREM 2 -3 Vertical angles are congruent

2 -5 Perpendicular Lines

Perpendicular Lines– two lines that intersect to form right angles ( 90° angles)

2 -4 THEOREM If two lines are perpendicular, then they form congruent adjacent angles.

2 -5 THEOREM If two lines form congruent adjacent angles, then the lines are perpendicular.

2 -6 THEOREM If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary

2 -6 Planning a Proof

Parts of a Proof 1. A diagram that illustrates the given information 2. A list, in terms of the figure, of what is given 3. A list, in terms of the figure, of what you are to prove 4. A series of statements and reasons that lead from the given information to the statement that is to be proved

2 -7 THEOREM If two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent.

2 -8 THEOREM If two angles are complements of congruent angles (or of the same angle), then the two angles are congruent.

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