Reasoning and Proof Algebra Properties of Equality

Transcription

1 Reasoning and Proof Algebra Properties of Equality Addition (APE) Multiplication (MPE) Reflexive Substitution If a b and x y, then a x b y. If a b and x y, then ax by. some segment some angle A, AB, AB AB. A A. If a b and b c, then a c.

2 : Intersecting lines l and m with two vertical angles having measures x and y. Prove: x = y Statements Reasons l m 1. x + z = Definition of a straight angle x z y 2. y + z = x + z = y + z Definition of a straight angle Substitution (steps 1 & 2) 4. x = y 4. APE

3 Prove the following conditional: If two angles are congruent and supplementary, then they are right angles. : Prove: A and B are right angles A B and A and B are supplementary. A B Statements m A m B m A m B m A m B 180 o m A m A 180 o 2 m A 180 o m A 90 o m B 90 A and Bare right angles o A and B are supplementary Reasons Definition of congruent angles Def. of supplementary angles Substitution (steps 1 & 3) Simplification MPE Substitution (steps 1 & 6) Definition of a right angle

4 Congruent Triangles Two figures are CONGRUENT if they have exactly the same size and shape. When two figures are congruent, we always list CORRESPONDING LABELS in the same order. Triangles have 6 parts (3 angles and 3 sides). When triangles are congruent, their corresponding parts are also congruent. List the 6 pairs of corresponding parts from the congruent triangles above. AB DE AC DF BC EF BAC EDF CBA FED BCD EFD

5 Congruent Triangles When all six parts of one triangle are congruent to the corresponding 6 parts of another, then the two triangles are congruent. However, what is the minimum amount of information required to conclude that two triangles are congruent? For example, would knowing that two pairs of sides are congruent be enough? Key Concept: If the information you provide is sufficient, then you should be able to create only one unique triangle with the given conditions. Use the straws provided to represent sides of triangles and determine what the minimum conditions are to conclude that two triangles are congruent. Make a list of these conditions.

6 Congruent Triangles Why doesn t Angle, Side, Side work? Reason 1: ASS is a BAD word!! Also, there is a counterexample. The triangle below is not the only one possible with the given conditions. S S S A

7 A 8 o 25 Congruent Triangles C Why does AAS work? B D o o o 80 o 75 8 o 75 E F What do you know about angles C and F? So what other triangle congruence theorem does AAS come from?

8 Congruent Triangles However, ASS does work when the Hypotenuse and one Leg of a Right Triangle are congruent. Why? A C A C B B The Pythagorean Theorem forces the third pair of sides to be congruent, giving us SAS. This Triangle Congruence Postulate Is called Hypotenuse-Leg (HL).

9 Congruent Triangles What if the legs of two right triangles are congruent? Are the triangles congruent? A A B B This Triangle Congruence Postulate Is called Leg-Leg (LL). It is derived from SAS.

10 Congruent Triangles So, the Triangle Congruency Postulates are: Side-Angle-Side (SAS) Two congruent sides and their included angle. Side-Side-Side (SSS) All three sides are congruent. Angle-Side-Angle (ASA) Two angles and their included Side are congruent. Angle-Angle-Side (AAS) Two angles and a side that is NOT included by them. These postulates represent the minimum amount of information considered sufficient for two triangles to be congruent.

11 Congruent Triangles The Right Triangle Congruency Postulates are: Hypotenuse-Leg (HL) If the hypoteni and one leg of two right triangles are congruent, then the triangles are congruent. Leg-Leg (LL) If the legs of two right triangles are congruent, then the triangles are congruent. In order to use these postulates as reasons for triangle congruency in proofs, you would have to first demonstrate that the triangles are right triangles.

12 Definitions, Postulates, and Theorems Segment Addition Postulate (SAP) A B C AB BC AC Angle Addition Postulate (AAP) B A D C m BAD m DAC m BAC Definition of a Straight Angle D A B C m ABC 180 o m ABD m DBC 180 o ABD and DBC are supplementary

13 Definitions, Postulates, and Theorems A A B C Definition of a Segment Bisector B B D C D Median Altitude A C Definition of an Angle Bisector

14 Equally Wet

15 Equally Wet : Prove: C is the midpoint of AD BD AB,and AB DC Statements Reasons D C is the midpoint of AB AB CD AC CB CD CD ACD and BCD are right angles ACD BCD Def of a Midpoint Reflexive Property Definition of Right Angle Congruency Theorem A C B 7 ACD BCD SAS 8 AD BD CP

16 Equally Wet : Prove: C is the midpoint of AD BD AB,and AB DC D A C B Statements 1 C is the midpoint of AB 2 AB CD 3 AC CB 4 CD CD 5 ACD and BCD are right angles 6 ACD and BCD are right triangles 7 8 ACD BCD AD BD Reasons Def LL of Reflexive a Midpoint Property Definition of Definition of CP a right triangle

17 D A C B Theorem 4.9 : Prove: Statements ADB with AC CB and DC AD BD DC bisects D ADC BDC DC DC ADC BDC AC CB ACD BCD ACB is a straight angle ACD and BCD are supplementary ACD and BCD are right angles AD BD and DC the angle bisector of D AB Reasons Defintition of Reflexive SAS CP CP Definition of an angle Property a straight bisector angle Congruent Supplements Thm. DC AB Definition of

18 The Base Angles Theorem Isosceles triangles are symmetric. If we draw in the angle bisector of B, the triangle is symmetric about this axis of symmetry. Since the triangle is symmetric: ABD CBD What is the Converse of this Theorem? Is it true? B A C CP A C

19 The Base Angles Theorem How could we use this theorem to explain why all equilateral triangles are also equiangular? What is the Converse of this Theorem? B Is It True? A C

20 The Base Angles Theorem How could we use this theorem to explain why all equilateral triangles are also equiangular? What is the Converse of this Theorem? B Is It True? A C

21 Exterior Angle Theorem b a c d Which groups of angles equal 180? a b c 180 o c d 180 Can you use substitution to show that angle d is equal to the sum of angles a and b? a b c a b c d The measure of an exterior angle of a triangle is equal to the sum of the measures of the non adjacent interior angles (and consequently greater than the measures of either). d o

22 From Two Flowers to Three (4,8) (4,2) (14,2)

23 The Perpendicular Bisector Theorem Converse : Prove: AD DB, DCis drawn such that DC DCis the bisector of AB toab D 1 AD DB 2 DC AB 3 ACD and BCD are right 's Definition of A C B DC DC ACD and BCD are right ADC BDC AC CB s Reflexive Definition of HL CP Property right DC is the Bisector of AB Definition of a Bisector

24 The Perpendicular Bisector Theorem Converse : Prove: AD DB, DCis drawn such that DC DCis the bisector of AB toab D A C B AD DB DC AB ACD and BCD are right ACD BCD ABDis Isosceles CAD CBD CAD CBD AC CB 's Definition of Right Def CP DC is the Bisector of AB Definition of a Bisector of Theorem Base Angles AAS Isosceles Triangles Theorem

25 The Perpendicular Bisector Theorem Converse : AD DB Prove: DCis DCis drawn such that C is the midpoint of the bisector of AB AB D 1 AD DB 2 AC CB Definition of a midpoint 3 DC DC Reflexive Property 4 ADC BDC SSS A C B 5 ADC BDC CP 6 DC bisects ADB Def of bisector 7 ADB is Isosceles Def of Isosceles Triangles 8 DC is the Bisector of AB Theorem 4.9

26 D A C B The Perpendicular Bisector Theorem Converse : Prove: AD DB DCis drawn such that C is the midpoint of DCis the bisector of AB AD DB AC CB DC DC ADC BDC ACD BCD ACB is a straight ACD and BCD are supp. Definition of Reflexive SSS CP Def of AB a midpoint Property a Straight Angle 8 ACD and BCD are right s Congruent Supplements 9 10 DC AB Definition DC is the Bisector of AB Definition of a Bisector