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Understanding the Properties of Equality and Congruence in Geometry

This presentation covers essential properties of equality and congruence in geometry, including addition, subtraction, multiplication, and division properties alongside reflexive, symmetric, and transitive properties. Key concepts explained include overlapping segments, vertical angles, and supplementary angles with illustrative examples and proofs. Interactive quizzes are included to reinforce understanding. By mastering these principles, students will build a solid foundation for solving geometric problems and proofs.

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Understanding the Properties of Equality and Congruence in Geometry

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Presentation Transcript

  1. Powerpoint Project 5 Adam and Nikki Birnbrey

  2. Properties of Equality to Know and Love • Addition Property- If a=b, then a+c= b+c • Subtraction Property- If a=b, then a-c=b-c • Multiplication Property- If a=b, then ac=bc • Division Property- If a=b and c doesn’t = 0 Then a/c= b/c • Substitution Property- If a=b you may replace a with b in any equation containing a in the result

  3. Equivalence Properties of Equality to Know and Love • Reflexive Property- For any real number a, a=a • Symmetric Property- For all real numbers a and b, if a=b, then b=a • Transitive Property- For all real numbers a, b, and c, if a=b and b=c, then a=c

  4. Equivalence Relation To Know And Love= any relation that satisfies these equivalence properties • Reflexive Property- Figure A=Figure A A • Symmetric Property- If Figure A=Figure B, then Figure B=Figure A A • Transitive Property- If Figure A=Figure B and Figure B=Figure C, Then Figure A=Figure C A B C

  5. Overlapping Theorem to Know and Love • Overlapping Segments- Given a segment with points A, B, C, & D arranged as shown the following statements are true: 1. If AB=CD then AC=BD 2. If AC=BD then AB=CD • Overlapping Angles- Given Angle AED with points B and C in its interior as shown, the following statements are true: 1. If Angle AEB=Angle CED then ? 2. If Angle AEC=Angle BED then ? A B C D A B C D E

  6. Vertical Angles Theorem to Know and Love • If two angles form a pair of vertical angles, then they are congruent. Given: angle 1 and 2 are vertical angles Prove: Angle 1 and 2 are congruent Statements Reasons 4 1 2 3 Given Linear pair property Substitution property of equality Subtraction property of equality Angle 1 and 2 are vertical angles 2. Angle 1 + angle 2= 180 Angle 2 + angle 3= 180 3. Angle 1+ angle 3= angle 2 + angle 3 4. Angle 1= angle 2 (1=2)

  7. Congruence Supplements Theorem to Know and Love • If two angles are supplements of congruent angles then the two angles are congruent Given: Angle 1=Angle 3, Angle 1 and angle 2 are supplementary, Angle 3 and 4 are supplementary Prove: Angle 2=Angle 4 Statements Reasons 3 4 1 2 1. <1+<2=180 <3+<4=180 2. <1+<2=<3+<4 3. <1=<3 4. <1+<2=<1+<4 5. <2=<4 Definition of Supplementary Angles Transitive Given Substitution Property Subtraction Property

  8. More Theorems to Know and Love • Theorem- Reflection across two parallel lines is equivalent to a translation of twice the distance between the lines and in a direction perpendicular to the lines • Theorem- Reflection across two intersecting lines is equivalent to a rotation about the point of intersection through twice the measure of the angle between the lines

  9. Vocabulary to Know and Love • Equivalence relation= Any relation that satisfies the Reflexive, Symmetric, and Transitive Properties • Inductive reasoning= The process of forming conjectures that are based on observations • Paragraph proof= A form of a proof in which one’s reasoning is explained in paragraph form, as opposed to a two-column proof • Theorem= A statement that has been proved deductively • Two column proof= A proof in which the statements are written in the left-hand column and the reasons are given in the right-hand column • Vertical angles= The opposite angles formed by two intersecting lines

  10. Practice to Know and Love(and a pooton of it) • Vertical Angles: Definition, illustrated examples, and an interactive practice quiz

  11. A real pooton of Practice to Know and Love (yeah math) • Given: 15x-5=10x+15 (use properties to prove) • Prove the Overlapping Segments Theorem Given: WX=YZ; Prove WY=XZ W X Y Z segment additon postulate ______property WX=YZ WX+XY= YZ+XY WX=XY=WY

  12. ….Extra poo • Identify the properties of equality that justify the conclusion. <B= <C; <C= <D <B= <D ________ AB=CD; CD=AB_______ AB+BC= BC+ CD; AC=BD_______

  13. Finding Values: Problemas (To Know and Love) Find x. 3X- 60 120-6X X+40 10X

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