If-Then Statements; Converses 2-1 If-Then Statements; Converses

CONDITIONAL STATEMENTS are statements written in if-then form 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.

Properties from Algebra 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  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

If a point M is the midpoint of AB, then 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.

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

A • X • C B •

Special Pairs of Angles 2-4 Special Pairs of Angles

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

two angles whose measures have the sum 180º 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.

Vertical angles are congruent 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 A diagram that illustrates the given information A list, in terms of the figure, of what is given A list, in terms of the figure, of what you are to prove 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.

THE END