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12.2 geometric proofs.notebook December 02, 2016 Friday 12/2/16 Construct a formal proof to prove the following: 1) Given: 1/3(3x 12)1 = 2x Prove : x = 5 Steps Reasons 1/3(3x 12)1 = 2x Given x 4 1 = 2x distributive x 5 = 2x combine like terms 5 = x subtraction x = 5 symmetric 1 12.2 geometric proofs.notebook December 02, 2016 Given 2 substitution subtraction 12 12/4 x = 3 Given subtraction 3x+88=28 3x = 6 Dividing x = 2 2 12.2 geometric proofs.notebook December 02, 2016 DEFINITIONS: Congruence: if two object are congruent, then their measures EX: AB BC, then AB=BC. If AB = 20, BC = 20 Congruent Segments: If two segments are congruent, then their measures are equal EX: AB CD, then AB = CD. If AB = 15 feet, then CD = 15 feet Congruent Angles: If two angles are congruent, then their measures are equal EX: Angle 1 Angle 2, then Angle 1=Angle 2 If angle 1 = 30, then angle 2 = 30 Midpoint: any point that cuts a segment into two congruent halves EX: A B C, If AB BC, then AB=BC and B is the midpoint Vertical Angles: Two pairs of congruent opposite angles are formed when two lines intersect 1=3 2 4=2 1 3 4 Complementary Angles: Angles whose sum equals 90 degrees 80 10 Supplementary angles: Angles whose sum equals 180 degrees EX: 100 80 Perpendicular Lines: two lines that intersect at right angles 90 POSTULATES: Ruler Postulate: The distance between any two point can be measured Protractor Postulate: If point A is exterior to a given segment CB, then the angle between A and CB can be measured EX: C B A Segment Addition Postulate: If B is between A and C, then AB+BC = AC EX: A B C IF AB = 5 and BC = 10, AC = 15 Angle Addition Postulate: IF P lies interior to Angle RST, then RSP+PST=RST EX: IF Angle 1 = 20, and Angle 2 = 50 1 A Angle A = 70 2 POINTS, LINES, AND PLANES POSTULATES: 1) Through any two points there is exactly one line 2) A line contains at least two points 3) Two lines intersect at EXACTLY one point EX: 4) Linear Pair Postulate: If two angles form a linear pair, then they are supplementary 5) Two planes intersect at exactly ONE line 3 12.2 geometric proofs.notebook December 02, 2016 4 12.2 geometric proofs.notebook December 02, 2016 DEFINITIONS: Congruence: If two objects are congruent, then their measures are equal EX: AB BC, AB = BC (Definition of congruence) Congruence of Segments: If two segments are congruent, then their measures are equal EX: A B C D IF AB = 5, CD = 5 Congruent Angles: If two angles are congruent then their measures are equal Angle A = Angle B, then Angle A B Midpoint: The middle of a segment; divided into two congruent halves EX: A B C AB BC, then B is the midpoint Angle Bisector: A segment that bifurcates (cuts in half) any angle creating two congruent interior angles b EX: segment ad bisects angle bac. bad, and dac are congruent a d c Vertical Angles: pair of congruent opposite angles formed by the intersection of two lines 3 1=2 1 3=4 2 4 Complementary Angles: angles whose sum is 90 degrees EX: 10 80 Supplementary: angles whose sum is 180 degrees EX: 100 80 Perpendicular Lines: two lines that intersect at right angles EX: 90 POSTULATES: Ruler Postulate: that you can measure the distance between any two points Protractor Postulate: Give any point that is exterior to a given line segment, you can measure the angle between that point EX: A B C Segment addition postulate: If B is between A and C, then AB+BC=AC. EX: A 5 B 10 C AB + BC = 15 Angle addition postulate: If B is interior to angle ACD, then ACB + BCD = ACD EX: 3 1 2 Angle 1+2 = 3 5 12.2 geometric proofs.notebook December 02, 2016 DEFINITIONS: Congruence if two objects are congruent, their measures are equal EX: AB BC, therefore AB = BC (Definition of Congruence) Congruent Segments: If two segments are congruent, then their measures are equal A B If AB = CD, then AB CD. If AB = 10, then CD = 10 C D Congruent Angles: If two angles are congruent, then their measures are equal Angle 1 Angle 2. If Angle 1 = 60, then Angle 2 = 60 Midpoint: Point that bifurcates (cut in half) a segment into two equal halves EX: A B C If AB = BC, then B is the midpoint. Angle Bisector: segment that bifurcates any angle into two congruent halves bd is an angle bisector: a abd = dbc d b c Vertical Angles: pairs of congruent angles formed by two intersecting lines 2 1 = 3 3 1 2 = 4 4 Complementary Angles: angles whose sum is 90 degrees EX: 10 80 Supplementary Angles: angles whose sum is 180 degrees EX: 80 100 Perpendicular Lines: two lines that intersect at a right angle 90 POSTULATES: Ruler Postulate: the distance between any two points can be measured EX: Given: segment AB = 10 inches Prove: AB = 10 inches (The distance can be measured) Protractor Postulate: given any point exterior to a segment can be measured between 0180 degrees EX: A B C Segment Addition Postulate: If B is between A and C, then AB+BC=AC EX: A 5 B 10 C If AB=5 and BC = 10, AC=15 Given: AB=5 and BC = 10 Prove: AC = 15 Steps Reasons AB=5 and BC = 10 Given AB+BC=AC Segment Addition 5+10=AC Substitution 15=AC Addition AC=15 Symmetric Angle Addition Postulate: If P is interior to RST, then RSP+PST=RST R Angle RSP+PST=RST P S T 6 12.2 geometric proofs.notebook December 02, 2016 Definitions: Congruence If two objects are congruent, then their measures are equal EX: Given: AB = BC Prove: AB BC (Definition of Congruence) Congruent Segments IF two segments are congruent, then their measures are equal EX: A B B C IF AB BC and AB = 10 feet, BC = 10 feet (Congruent segments) Congruent Angles IF two angles are congruent, then their measures are equal EX: Angle A Angle B and Angle A = 20, then Angle B = 20 Midpoint Cuts a segment into two equal segments EX: A C B IF AC CB, then C is the MIDPOINT!!! Angle Bisector Segment that cuts any angle into two congruent halves IF Segment BD bisects Angle A ABC, then Angles ABD and B DBA are congruent D C Vertical Angles Any two lines intersecting form two pairs of congruent angles c a a=b d c=d b Complementary Angles Angles whose sum equals 90 degrees Angle 1 = 50, Angle 2 = 40 50 1 (1 + 2 = 90) 40 2 Supplementary angles Angles whose sum equals 180 degrees 100 80 Perpendicular Lines two lines that intersect forming 4 right angles 90 POSTULATES: Ruler Postulate: Any two points, the distance between them can be measured Protractor Postulate: If a point is exterior to a line segment, then the angle between that point and the segment can be measured EX: A B C 7 12.2 geometric proofs.notebook December 02, 2016 Definitions: Congruence If two objects are congruent, then their measures are equal ex: Given: AB = BC Prove: AB BC (Definition of congruence) Congruent Segments If two segments are congruent, then their measures are equal Congruent Angles If two angles are congruent, then their measures are equal Midpoint The middle of a line segment creating two congruent segments Angle Bisector a line segment that cuts an angle into two congruent angles (Measures are equal) Vertical Angles Opposite congruent angles formed by intersecting lines or segments 3 2 1 4 Complementary Angles angle whose sum is 90 degrees 10 80 Supplementary angles angle whose sum is 180 degrees 100 80 Perpendicular lines Two lines that intersect forming 4 right angles 90 POSTULATES Segment addition postulate If B is between A and C, then AB + BC = AC A B C AB + BC = AC Angle Addition Postulate If P is in the interior of RST, then RSP + PST = RST 40 RST = 60 20 Ruler Postulate Any two points on a line can be measured. Protractor Postulate Any point exterior to a line segment can be measured between 0 180 degrees C A B POINTS, LINES, AND PLANES POSTULATE: 1) Through any two points, there is EXACTLY one line 2) A line contains at least 2 points 3) Two line intersect at EXACTLY one point 8 12.2 geometric proofs.notebook December 02, 2016 9 12.2 geometric proofs.notebook December 02, 2016 IB4E: Write a 2 column proof for the following: Given: 1/3(3x12) 1 = 2x Prove: x = 5 Steps Reasons 10 12.2 geometric proofs.notebook December 02, 2016 11