Geometry Summary Sheet 3.0.pdf - Geometry Summary Sheet...

This preview shows page 1 out of 48 pages.

We have textbook solutions for you!
The document you are viewing contains questions related to this textbook.
Precalculus: Mathematics for Calculus
The document you are viewing contains questions related to this textbook.
Chapter 1 / Exercise 1
Precalculus: Mathematics for Calculus
Redlin/Stewart
Expert Verified

Unformatted text preview: Geometry​ ​Summary​ ​Sheet Welcome.​ ​This​ ​is​ ​the​ ​third​ ​iteration​ ​of​ ​a​ ​summary​ ​sheet​ ​started​ ​by​ ​Brian​ ​Kwong​.​ ​Special​ ​thanks to​ ​Shadman​ ​Khandaker​,​ ​Daniel​ ​Roz​,​ ​as​ ​well​ ​as​ ​my​ ​old​ ​peers​ ​who​ ​served​ ​as​ ​brilliant​ ​editors from​ ​the​ ​start.​ ​Finally,​ ​I​ ​personally​ ​hope​ ​that​ ​y’all​ ​will​ ​find​ ​this​ ​summary​ ​sheet​ ​useful,​ ​and​ ​as always,​ ​cheers.​ ​The​ ​express​ ​version​ ​can​ ​be​ ​found​ ​here​.​ ​Also,​ ​we​ ​are​ ​now​ ​accepting​ ​new editors!​ ​The​ ​application​ ​form​ ​can​ ​be​ ​found​ ​here​. Table​ ​of​ ​Contents: 1. Logic 2. Fundamentals 3. Congruence​ ​&​ ​Equality 4. Triangles 5. Parallel​ ​Lines​ ​&​ ​Quadrilaterals 6. Similarity​ ​&​ ​Proportions 7. Trigonometry 8. Right​ ​Triangles 9. Area 10. Circles 11. Parabolas 12. The​ ​Coordinate​ ​Plane 13. Applied​ ​Algebra 14. Locus,​ ​Constructions​ ​&​ ​Concurrence​ ​Theorems 15. Transformations 16. Solid​ ​Geometry 17. Planar​ ​Geometry Logic: Statements​​ ​can​ ​be​ ​written​ ​in​ ​two​ ​forms,​ ​either​ ​as​ ​a​ ​declarative​,​ ​or​ ​a​ ​conditional​. 1. A​ ​declarative​​ ​statement​ ​directly​ ​states​ ​a​ ​fact​ ​as​ ​true,​ ​such​ ​as​ ​“Two​ ​straight​ ​∡​ ​are​ ​≅”. 2. A​ ​conditional​​ ​statement​ ​has​ ​two​ ​parts,​ ​an​ ​antecedent​​ ​and​ ​a​ ​consequent​.​ ​The antecedent​ ​justifies​ ​the​ ​statement​ ​stated​ ​true​ ​in​ ​the​ ​consequent.​ ​An​ ​example​ ​is​ ​“If​ ​two angles​ ​are​ ​straight​ ​angles,​ ​then​ ​they​ ​are​ ​congruent”.​ ​Also​ ​known​ ​as​ ​“If…​ ​then”​ ​form. The​ ​negation​​ ​of​ ​a​ ​statement​ ​is​ ​its​ ​opposite​.​ ​In​ ​notation,​ ​it​ ​is​ ​represented​ ​as​ ​~P​ ​(not​ ​P). The​ ​law​ ​of​ ​the​ ​contrapositive​​ ​states​ ​that​ ​given​ ​a​ ​statement,​ ​P​ ​→​ ​Q,​ ​its​ ​contrapositive,​ ​~Q​ ​→​ ​~P is​ ​always​ ​true​ ​as​ ​well. The​ ​chain​ ​rule​​ ​states​ ​that​ ​if​ ​P​ ​→​ ​Q,​ ​and​ ​Q​ ​→​ ​R,​ ​then​ ​P​ ​→​ ​R,​ ​acting​ ​transitive-like​ ​in​ ​its properties. This​ ​is​ ​a​ ​truth​ ​table​,​ ​showing​ ​what​ ​each​ ​statement’s​ ​truth​ ​value​ ​is,​ ​depending​ ​on​ ​what​ ​the truth​ ​values​ ​of​ ​statements​ ​P​ ​and​ ​Q​ ​are. ​ ​p ​ ​ ​q p​ ​Λ​ ​q​ ​(​ ​If​ ​p​ ​AND​ ​q​ ​is true,​ ​p​ ​^​ ​q​ ​is​ ​true) p​ ​→​ ​q​ ​(if​ ​p,​ ​then​ ​q) Conditional p​ ​←→​ ​q​ ​(p​ ​if​ ​and​ ​only if​ ​q)​ ​Biconditional ​ ​T ​ ​ ​T ​ ​ ​ ​ ​T ​ ​ ​ ​ ​T ​ ​ ​ ​ ​ ​ ​T ​ ​T ​ ​ ​F ​ ​ ​ ​ ​F ​ ​ ​ ​ ​F ​ ​ ​ ​ ​ ​ ​F ​ ​F ​ ​ ​T ​ ​ ​ ​ ​F ​ ​ ​ ​ ​T ​ ​ ​ ​ ​ ​ ​F ​ ​F ​ ​ ​F ​ ​ ​ ​ ​F ​ ​ ​ ​ ​T ​ ​ ​ ​ ​ ​ ​T Types​ ​of​ ​Logic​ ​Statements: - Original:​​ ​ ​ ​p​ ​→​ ​q - Converse:​​ ​q​ ​→​ ​p - Inverse:​​ ​~p​ ​→​ ​~q - Contrapositive:​​ ​ ​~q​ ​→ ​~p Back​ ​to​ ​Top Fundamentals: The​ ​Structure​ ​of​ ​Geometry: A​ ​deductive​ ​structure​​ ​is​ ​where​ ​new​ ​conclusions​ ​are​ ​justified​ ​through​ ​proven​ ​conclusions​ ​and assumptions. 1. Undefined​ ​Terms: a. Point: b. Line: c. Plane: d. Set: e. Between: 2. Postulate:​​ ​An​ ​unproved​ ​assumption​ ​(Not​ ​always​ ​reversible) 3. Definitions:​​ ​States​ ​the​ ​meaning​ ​of​ ​a​ ​term​ ​/​ ​idea.​ ​(Always​ ​reversible) 4. Theorems​ ​/​ ​Conclusions:​​ ​A​ ​statement​ ​that​ ​can​ ​be​ ​proven.​ ​(Not​ ​always​ ​reversible) Proving​ ​Statements: Alright,​ ​it’s​ ​great​ ​that​ ​we​ ​can​ ​prove​ ​statements,​ ​but​ ​how​ ​do​ ​we​ ​go​ ​about​ ​proving​ ​them​ ​in​ ​the first​ ​place? Well,​​ ​proofs​​ ​are​ ​lists​ ​of​ ​steps​ ​leading​ ​up​ ​to​ ​the​ ​statement​ ​one​ ​is​ ​trying​ ​to​ ​prove​ ​(theorem). There​ ​are​ ​two​ ​main​ ​ways​ ​of​ ​proving​ ​a​ ​statement,​ ​either​ ​directly​​ ​or​ ​indirectly​. 1. A​ ​direct​ ​proof​​ ​is​ ​a​ ​proof​ ​where​ ​one​ ​tries​ ​to​ ​prove​ ​a​ ​statement​ ​using​ ​the​ ​given information,​ ​assumptions​ ​(postulates),​ ​and​ ​proven​ ​theorems​ ​to​ ​reach​ ​a​ ​conclusion. 2. An​ ​indirect​ ​proof​​ ​is​ ​a​ ​proof​ ​where​ ​one​ ​tries​ ​to​ ​prove​ ​a​ ​statement​ ​by​ ​disproving​ ​all​ ​of​ ​the other​ ​possibilities,​ ​therefore,​ ​making​ ​the​ ​desired​ ​conclusion​ ​true. Proofs​ ​can​ ​also​ ​be​ ​written​ ​in​ ​two​ ​ways,​ ​either​ ​the​ ​two-column​​ ​proof,​ ​or​ ​in​ ​paragraph​​ ​form. 1. The​ ​two-column​​ ​proof​ ​is​ ​a​ ​type​ ​of​ ​proof​ ​in​ ​which​ ​one​ ​column,​ ​labeled​ ​statements​,​ ​is​ ​a specific​ ​list​ ​of​ ​steps​ ​relating​ ​to​ ​only​ ​the​ ​diagram​ ​at​ ​hand.​ ​The​ ​other​ ​column,​ ​reasons​,​ ​is​ ​a more​ ​generalized​​ ​list​ ​that​ ​justifies​ ​the​ ​conclusions​ ​made​ ​in​ ​the​ ​statement​ ​column. These​ ​are​ ​definitions,​ ​postulates,​ ​and​ ​theorems​ ​that​ ​can​ ​be​ ​applied​ ​in​ ​numerous​ ​proofs. The​ ​reasons​ ​given​ ​usually​ ​have​ ​two​ ​parts,​ ​the​ ​antecedent​​ ​and​ ​consequent​.​ ​The consequent​ ​corresponds​ ​to​ ​the​ ​statement,​ ​while​ ​the​ ​antecedent​ ​acts​ ​as​ ​a​ ​justification​ ​to concluding​ ​that​ ​statement. 2. The​ ​paragraph​ ​proof​​ ​is​ ​a​ ​type​ ​of​ ​proof​ ​where​ ​only​ ​the​ ​statements​​ ​are​ ​given.​ ​These​ ​are also​ ​written​ ​in​ ​If…​ ​Then​​ ​form,​ ​or​ ​antecedent​ ​then​ ​consequent​ ​form.​ ​These​ ​are​ ​more commonly​ ​used​ ​in​ ​more​ ​advance​ ​courses. Theorems​ ​for​ ​Indirect​ ​Proofs: 1) Law​ ​of​ ​the​ ​excluded​ ​middle:​​ ​Either​ ​p​ ​OR​​ ​~p​ ​is​ ​true.​ ​No​ ​other​ ​possibility​ ​exists. 2) Law​ ​of​ ​Contradiction:​​ ​Both​ ​q​ ​AND​​ ​~q​ ​cannot​ ​be​ ​true​ ​at​ ​the​ ​same​ ​time. 3) Postulate​ ​of​ ​Elimination:​​ ​If​​ ​all​ ​other​ ​possibilities​ ​are​ ​false,​ ​then​​ ​the​ ​only​ ​remaining​ ​one must​ ​be​ ​true. Definitions: 1. A​ ​midpoint​​ ​is​ ​a​ ​point​ ​that​ ​divides​ ​a​ ​segment​ ​into​ ​two​ ​congruent​ ​segments. 2. A​ ​segment​ ​bisector​​ ​is​ ​a​ ​segment,​ ​ray,​ ​or​ ​line,​ ​that​ ​intersects​ ​another​ ​segment​ ​at​ ​its midpoint. 3. An​ ​angle​​ ​is​ ​the​ ​union​ ​of​ ​two​ ​rays​ ​with​ ​a​ ​common​ ​endpoint​ ​and​ ​no​ ​other​ ​points​ ​in common. 4. An​ ​angle​ ​bisector​​ ​is​ ​a​ ​segment,​ ​ray,​ ​or​ ​line,​ ​that​ ​divides​ ​an​ ​angle​ ​into​ ​two​ ​congruent angles. 5. Auxiliary​ ​Lines​​ ​are​ ​lines,​ ​rays,​ ​or​ ​segments​ ​that​ ​do​ ​not​ ​appear​ ​in​ ​a​ ​diagram,​ ​but​ ​their existence​ ​is​ ​guaranteed​ ​by​ ​a​ ​postulate. a. Two​ ​points​ ​determine​ ​exactly​ ​one​ ​line. b. Every​ ​∡​ ​has​ ​a​ ​bisector. 6. If​​ ​two​ ​lines​ ​intersect​ ​to​ ​form​ ​right​ ​∡s​ ​and​ ​≅​ ​adjacent​ ​∡s,​ ​then​​ ​they​ ​are​ ​⏊. 7. If​​ ​two​ ​∡s​ ​are​ ​suppl.​ ​&​ ​≅,​ ​then​​ ​they​ ​are​ ​right​ ​∡s. Theorems: 1) Theorems​ ​of​ ​Equidistance: - If​​ ​2​ ​points​ ​are​ ​equidistant​ ​from​ ​the​ ​endpoints​ ​of​ ​a​ ​segment,​ ​then​​ ​the​ ​line​ ​joining​ ​them​ ​is the​ ​⏊​ ​bisector​ ​of​ ​that​ ​segment. - If​​ ​a​ ​point​ ​lies​ ​on​ ​the​ ​⏊​ ​bisector​ ​of​ ​a​ ​segment,​ ​then​​ ​the​ ​point​ ​is​ ​equidistant​ ​from​ ​the endpoints​ ​of​ ​the​ ​segment. - If​​ ​a​ ​point​ ​is​ ​equidistant​ ​from​ ​the​ ​endpoints​ ​of​ ​a​ ​segment,​ ​then​​ ​it​ ​lies​ ​on​ ​the​ ​⏊​ ​bisector of​ ​that​ ​segment. 2) Theorems​ ​of​ ​Inequality: - If​​ ​2​ ​∡s​ ​of​ ​a​ ​△​ ​are​ ​ =/ ,​ ​then​​ ​the​ ​sides​ ​opposite​ ​are​ ​ =/ AND​​ ​the​ ​side​ ​opposite​ ​the​ ​larger​ ​∡ is​ ​the​ ​larger​ ​side. - If​​ ​2​ ​sides​ ​of​ ​a​ ​△​ ​are​ ​ =/ ,​ ​then​​ ​the​ ​∡s​ ​opposite​ ​are​ ​ =/ AND​​ ​the​ ​∡​ ​opposite​ ​the​ ​larger​ ​side is​ ​the​ ​larger​ ​∡. - The​ ​measure​ ​of​ ​an​ ​exterior​ ​∡​ ​of​ ​a​ ​△​ ​is​ ​greater​ ​than​​ ​either​ ​remote​ ​interior​ ​∡. 3) Theorems​ ​of​ ​Uniqueness: - At​ ​a​ ​point​ ​on​ ​a​ ​given​ ​line,​ ​one​ ​and​ ​only​​ ​⏊​ ​can​ ​exist​ ​on​ ​the​ ​given​ ​line. - At​ ​a​ ​point​ ​not​ ​on​ ​a​ ​given​ ​line,​ ​one​ ​and​ ​only​​ ​⏊​ ​can​ ​exist​ ​on​ ​the​ ​given​ ​line. - A​ ​line​ ​segment​ ​has​ ​one​ ​and​ ​only​​ ​1​ ​midpoint. Back​ ​to​ ​Top Congruence​ ​&​ ​Equality: Theorems​ ​to​ ​Know: - If​​ ​2​ ​∡s​ ​are​ ​(right/straight)​ ​∡s,​ ​then​​ ​they​ ​are​ ​≅. - If​​ ​2​ ​∡s​ ​are​ ​(suppl./comp.)​ ​to​ ​the​ ​same​ ​∡,​ ​then​​ ​they​ ​are​ ​≅. - If​​ ​2​ ​∡s​ ​are​ ​(suppl./comp.)​ ​to​ ​≅​ ​∡s,​ ​then​​ ​they​ ​are​ ​≅. - If​​ ​a​ ​(segment/∡s)​ ​is​ ​(added/subtracted)​ ​(to/from)​ ​≅​ ​(segments/∡s),​ ​then​​ ​their (sums/differences)​ ​are​ ​≅. - If​​ ​≅​ ​(segments/∡s)​ ​are​ ​(added/subtracted)​ ​(to/from)​ ​≅​ ​(segments/∡s),​ ​then​​ ​their (sums/differences)​ ​are​ ​≅. - If​​ ​(segments/∡s)​ ​are​ ​≅,​ ​then​​ ​their​ ​like​ ​(multiples/divisions)​ ​are​ ​≅. - If​​ ​(segments/∡s)​ ​are​ ​≅​ ​to​ ​the​ ​same​ ​(segment/∡s),​ ​then​​ ​they​ ​are​ ​≅​ ​to​ ​each​ ​other. - If​​ ​(segments/∡s)​ ​are​ ​≅​ ​to​ ​≅​ ​(segments/∡s),​ ​then​​ ​they​ ​are​ ​≅​ ​to​ ​each​ ​other. - If​​ ​2​ ​∡s​ ​are​ ​vertical,​ ​then​​ ​they​ ​are​ ​≅. - If​​ ​2​ ​sides​ ​of​ ​a​ ​△​ ​are​ ​≅,​ ​then​​ ​the​ ​∡s​ ​opposite​ ​those​ ​sides​ ​are​ ​≅. - If​​ ​2​ ​∡s​ ​of​ ​a​ ​△​ ​are​ ​≅,​ ​then​​ ​the​ ​sides​ ​opposite​ ​those​ ​∡s​ ​are​ ​≅. Postulates​ ​of​ ​Equality: - Reflexive:​​ ​Anything​ ​is​ ​equal​ ​to​ ​itself.​ ​Ex)​ ​a​ ​=​ ​a - Symmetric:​​ ​If​​ ​a​ ​=​ ​b,​ ​then​​ ​b​ ​=​ ​a. - Transitive:​​ ​If​​ ​a​ ​=​ ​b​ ​&​ ​b​ ​=​ ​c,​ ​then​​ ​a​ ​=​ ​c. - Substitution:​​ ​If​ ​a​ ​=​ ​b,​ ​then​ ​b​ ​can​ ​substitute​ ​for​ ​a​ ​&​ ​a​ ​can​ ​substitute​ ​for​ ​b. Postulates​ ​of​ ​Operations: - Addition:​​ ​If​​ ​a​ ​=​ ​b​ ​&​ ​c​ ​=​ ​d,​ ​then​​ ​a​ ​+​ ​c​ ​=​ ​b​ ​+​ ​d. - Subtraction:​​ ​If​​ ​a​ ​=​ ​b​ ​&​ ​c​ ​=​ ​d,​ ​then​​ ​a​ ​-​ ​c​ ​=​ ​b​ ​-​ ​d. - Multiplication:​​ ​If​​ ​a​ ​=​ ​b​ ​&​ ​c​ ​=​ ​d,​ ​then​​ ​a​ ​*​ ​c​ ​=​ ​b​ ​*​ ​d. a. If​​ ​equals​ ​are​ ​multiplied​ ​by​ ​equals,​ ​then​​ ​the​ ​products​ ​are​ ​equal. - Division:​​ ​If​​ ​a​ ​=​ ​b​ ​&​ ​c​ ​=​ ​d​ ​&​ ​c​ ​ =/ 0​ ​and​ ​d​ ​ =/ 0,​ ​then​​ ​a​ ​/​ ​c​ ​=​ ​b​ ​/​ ​d. a. If​​ ​equals​ ​are​ ​divided​ ​by​ ​equals,​ ​then​​ ​the​ ​quotients​ ​are​ ​equal. Back​ ​to​ ​Top Triangles: Proving​ ​Triangles​ ​Congruent: 1) SSS​ ​Theorem​ ​of​ ​≅​ ​△s:​​ ​Needs​ ​all​ ​3​ ​sides​ ​of​ ​a​ ​△​ ​≅​ ​to​ ​the​ ​corresponding​ ​sides​ ​of​ ​a​ ​2nd​ ​△. 2) SAS​ ​Postulate​ ​of​ ​≅​ ​△s:​​ ​Needs​ ​2​ ​sides​ ​and​ ​the​ ​included​ ​∡​ ​of​ ​a​ ​△​ ​≅​ ​to​ ​the​ ​corresponding parts​ ​of​ ​a​ ​second​ ​△. 3) ASA​ ​Postulate​ ​of​ ​≅​ ​△s:​​ ​Needs​ ​2​ ​∡s​ ​and​ ​the​ ​included​ ​side​ ​of​ ​a​ ​△​ ​≅​ ​to​ ​the​ ​corresponding parts​ ​of​ ​a​ ​second​ ​△. 4) AAS​ ​Theorem​ ​of​ ​≅​ ​△s:​​ ​Needs​ ​2​ ​∡s​ ​and​ ​a​ ​non-included​ ​side​ ​of​ ​a​ ​△​ ​≅​ ​to​ ​the corresponding​ ​parts​ ​of​ ​a​ ​second​ ​△. Triangle​ ​Theorems: - If​​ ​a​ ​△​ ​is​ ​equilateral,​ ​then​​ ​it​ ​is​ ​equiangular. - If​​ ​a​ ​△​ ​is​ ​equiangular,​ ​then​​ it​ ​is​ ​equilateral. - If​​ ​a​ ​line​ ​bisects​ ​1​ ​side​ ​of​ ​a​ ​△​ ​and​​ ​is​ ​parallel​ ​to​ ​a​ ​2nd​ ​side,​ ​then​​ ​it​ ​bisects​ ​the​ ​3rd​ ​side. Triangle​ ​Definitions: - If​​ ​a​ ​line​ ​segment​ ​is​ ​drawn​ ​from​ ​any​ ​vertex​ ​to​ ​the​ ​midpoint​ ​of​ ​the​ ​opposite​ ​side,​ ​then​​ ​it is​ ​a​ ​median​ ​of​ ​a​ ​△. - If​​ ​a​ ​line​ ​segment​ ​is​ ​drawn​ ​from​ ​any​ ​vertex​ ​to​ ​the​ ​opposite​ ​side,​ ​&​ ​is​ ​⏊​ ​to​ ​that​ ​side,​ ​then it​ ​is​ ​an​ ​altitude​ ​of​ ​a​ ​△. Measures​ ​and​ ​Angles: - The​ ​sum​ ​of​ ​the​ ​measures​ ​of​ ​three​ ​∡s​ ​in​ ​a​ ​triangle​ ​is​​ ​180. - If​ ​2​ ​∡s​ ​of​ ​a​ ​△​ ​are​ ​≅​ ​to​ ​2​ ​∡s​ ​of​ ​another​ ​△,​ ​then​ ​the​ ​3rd​ ​∡s​ ​are​ ​≅. - The​ ​acute​ ​∡s​ ​of​ ​a​ ​rt.​ ​△​ ​are​​ ​complementary. Back​ ​to​ ​Top Parallel​ ​Lines​ ​&​ ​Quadrilaterals: Proving​ ​Lines​ ​Parallel: 1) If​​ ​2​ ​lines​ ​are​ ​cut​ ​by​ ​a​ ​transversal​ ​such​ ​that​ ​the​ ​alternate​ ​interior​ ​∡s​ ​are​ ​≅,​ ​then​​ ​the​ ​lines are​ ​parallel. 2) If​​ ​2​ ​lines​ ​are​ ​cut​ ​by​ ​a​ ​transversal​ ​such​ ​that​ ​the​ ​alternate​ ​exterior​ ​∡s​ ​are​ ​≅,​ ​then​​ ​the​ ​lines are​ ​parallel. 3) If​​ ​2​ ​lines​ ​are​ ​cut​ ​by​ ​a​ ​transversal​ ​such​ ​that​ ​the​ ​corresponding​ ​∡s​ ​are​ ​≅,​ ​then​​ ​the​ ​lines​ ​are parallel. 4) If​​ ​2​ ​lines​ ​are​ ​⏊​ ​to​ ​the​ ​same​ ​line,​ ​then​​ ​they​ ​are​ ​parallel​ ​to​ ​each​ ​other. 5) If​​ ​2​ ​lines​ ​are​ ​parallel​ ​to​ ​the​ ​same​ ​line,​ ​then​​ ​those​ ​2​ ​lines​ ​are​ ​parallel. Angles​ ​from​ ​Parallel​ ​Lines: 1) If​​ ​2​ ​parallel​ ​lines​ ​are​ ​cut​ ​by​ ​a​ ​transversal,​ ​then​​ ​the​ ​alternate​ ​interior​ ​∡s​ ​are​ ​≅. 2) If​​ ​2​ ​parallel​ ​lines​ ​are​ ​cut​ ​by​ ​a​ ​transversal,​ ​then​​ ​the​ ​alternate​ ​exterior​ ​∡s​ ​are​ ​≅. 3) If​​ ​2​ ​parallel​ ​lines​ ​are​ ​cut​ ​by​ ​a​ ​transversal,​ ​then​​ ​the​ ​corresponding​ ​∡s​ ​are​ ​≅. Definitions: 1) Polygon:​​ ​The​ ​union​ ​of​ ​a​ ​set​ ​of​ ​points​ ​P​1​,P​2​,P​3​,...,P​n-1​,P​n​​ ​with​ ​the​ ​line​ ​segments P​1​P​2​,P​2​P​3​,...,P​n-1​P​n​,P​n​P​1​ such​ ​that​ ​if​ ​any​ ​segments​ ​intersect,​ ​they​ ​only​ ​intersect​ ​at ​ P​1​,P​2​,...,P​n​. 2) Quadrilaterals: - Quadrilateral:​​ ​A​ ​polygon​ ​with​ ​four​ ​sides. - Parallelogram:​​ ​A​ ​quadrilateral​ ​in​ ​which​ ​both​ ​pairs​ ​of​ ​opposite​ ​sides​ ​are​ ​parallel. Properties​ ​are:​ ​opposite​ ​sides​ ​are​ ​≅​ ​and​ ​parallel,​ ​opposite​ ​angles​ ​ ​are​ ​≅,​ ​consecutive angles​ ​are​ ​supp,​ ​and​ ​diagonal​ ​bisect​ ​each​ ​other. - Rectangle:​​ ​A​ ​parallelogram​ ​in​ ​which​ ​at​ ​least​ ​one​ ​angle​ ​is​ ​a​ ​right​ ​∡. - Rhombus:​​ ​A​ ​parallelogram​ ​in​ ​which​ ​at​ ​least​ ​one​ ​pair​ ​of​ ​consecutive​ ​sides​ ​are​ ​≅. - Square:​​ ​A​ ​parallelogram​ ​that​ ​is​ ​both​ ​a​ ​rectangle​ ​and​ ​a​ ​rhombus. - Trapezoid:​​ ​A​ ​quadrilateral​ ​that​ ​has​ ​at​ ​least​ ​one​ ​pair​ ​of​ ​opposite​ ​sides​ ​parallel. - Isosceles​ ​Trapezoid:​​ ​A​ ​trapezoid​ ​in​ ​which​ ​the​ ​non-parallel​ ​sides​ ​are​ ​≅. 3) Supplementary​ ​Definitions: - Regular​ ​Polygon:​​ ​A​ ​polygon​ ​that​ ​is​ ​both​ ​equilateral​ ​and​ ​equiangular. - Convex​ ​Polygon:​​ ​A​ ​polygon​ ​in​ ​which​ ​each​ ​interior​ ​angle​ ​has​ ​a​ ​measure​ ​less​ ​than​ ​180. - Exterior​ ​Angle​ ​of​ ​a​ ​Polygon:​​ ​An​ ​∡​ ​that​ ​is​ ​adjacent​ ​and​ ​supplementary​ ​to​ ​an​ ​interior​ ​∡​ ​of the​ ​polygon. Parallelogram: 1) Proving​ ​a​ ​Quadrilateral​ ​a​ ​Parallelogram: - If​​ ​two​ ​pairs​ ​of​ ​opposite​ ​sides​ ​of​ ​a​ ​quadrilateral​ ​are​ ​≅,​ ​then​​ ​it’s​ ​a​ ​parallelogram. - If​​ ​two​ ​pairs​ ​of​ ​opposite​ ​sides​ ​of​ ​a​ ​quadrilateral​ ​are​ ​parallel,​ ​then​​ ​it’s​ ​a​ ​parallelogram. [Def] - If​​ ​one​ ​pair​ ​of​ ​opposite​ ​sides​ ​of​ ​a​ ​quadrilateral​ ​are​ ​both​ ​≅​ ​and​ ​parallel,​ ​then​​ ​it’s​ ​a parallelogram. - If​​ ​the​ ​diagonals​ ​of​ ​a​ ​quadrilateral​ ​bisect​ ​each​ ​other,​ ​then​​ ​it’s​ ​a​ ​parallelogram. 2) Parts​ ​Proven​ ​given​ ​a​ ​Parallelogram: - Opposite​ ​sides​ ​of​ ​a​ ​parallelogram​ ​are​ ​≅. - Opposite​ ​sides​ ​of​ ​a​ ​parallelogram​ ​are​ ​parallel.​ ​[Def] - Opposite​​ ​∡s​ ​of​ ​a​ ​parallelogram​ ​are​ ​≅. - Diagonals​ ​of​ ​a​ ​parallelogram​ ​bisect​ ​each​ ​other. - Consecutive​ ​∡s​ ​of​ ​a​ ​parallelogram​ ​are​ ​supplementary. Rectangle: 1) Proving​ ​a​ ​Quadrilateral​ ​a​ ​Rectangle: - If​​ ​a​ ​parallelogram​ ​has​ ​at​ ​least​ ​one​ ​right​ ​∡,​ ​then​​ ​it​ ​is​ ​a​ ​rectangle.​ ​[Def] - If​ ​the​ ​diagonals​ ​of​ ​a​ ​parallelogram​ ​are​ ​≅,​ ​then​​ ​it​ ​is​ ​a​ ​rectangle. Rhombus: 1) Proving​ ​a​ ​Quadrilateral​ ​a​ ​Rhombus: - If​​ ​either​ ​diagonal​ ​of​ ​a​ ​parallelogram​ ​bisect​ ​2​ ​∡s​ ​of​ ​a​ ​Parallelogram,​ ​then​​ ​it​ ​is​ ​a​ ​Rhombus. - If​​ ​a​ ​pair​ ​of​ ​consecutive​ ​sides​ ​of​ ​a​ ​parallelogram​ ​are​ ​≅,​ ​then​​ ​it​ ​is​ ​a​ ​Rhombus.​ ​[Def] Isosceles​ ​Trapezoid: 1) Parts​ ​Proven​ ​given​ ​an​ ​Isosceles​ ​Trapezoid: - Diagonals​ ​of​ ​an​ ​Isosceles​ ​Trapezoid​ ​are​ ​≅. - Upper/Lower​ ​Base​ ​∡s​ ​of​ ​an​ ​Isosceles​ ​Trapezoid​ ​are​ ​≅. - Non-Parallel​ ​sides​ ​of​ ​an​ ​Isosceles​ ​Trapezoid​ ​are​ ​≅.​ ​[Def] Back​ ​to​ ​Top Similarity​ ​&​ ​Proportions: Definitions: - Proportion:​​ ​An​ ​equation​ ​that​ ​states​ ​2​ ​or​ ​more​ ​ratios. - Ratio:​​ ​The​ ​quotient​ ​of​ ​2​ ​#s. - Rate:​​ ​The​ ​ratio ​of​ ​2​ ​measures. - Similar​ ​Polygons:​​ ​Polygons​ ​such​ ​that... 1. The​ ​ratio​ ​of​ ​corresponding​ ​sides​ ​are​ ​equal. 2. Corresponding​ ​∡s​ ​are​ ​≅. - Golden​ ​mean​:​ ​The​ ​shorter​ ​part​ ​is​ ​to​ ​the​ ​longer​ ​part​ ​as​ ​the​ ​longer​ ​part​ ​is​ ​to​ ​the​ ​whole. - M​ ​is​ ​the​ ​mean​ ​proportional​​ ​between​ ​a​ ​and​ ​b​ ​or​ ​m​ ​is​ ​the​ ​geometric​ ​mean​​ ​of​ ​a​ ​and​ ​b,​ ​if m^2​ ​=​ ​(square​ ​root​ ​of)​ ​a​ ​X​ ​b. - M​ ​is​ ​the​ ​arithmetic​ ​mean​​ ​of​ ​a​ ​and​ ​b​ ​if​ ​and​ ​only​ ​if​ ​m​ ​=​ ​(a​ ​+​ ​b)/2. Proportion​ ​Theorems: 1) If​ ​2​ ​△s​ ​are​ ​similar,​ ​then​​ ​the​ ​ratio​ ​of​ ​their​ ​corresponding​ ​sides​ ​are​ ​equal. 2) If​​ ​a​ ​line​ ​is​ ​parallel​ ​to​ ​1​ ​side​ ​of​ ​a​ ​△​ ​&​ ​intersects​ ​the​ ​other​ ​2​ ​sides,​ ​then​​ ​it​ ​divides​ ​those​ ​2 sides​ ​proportionally. 3) If​​ ​3​ ​or​ ​more​ ​parallel​ ​lines​ ​are​ ​intersected​ ​by​ ​2​ ​transversals,​ ​then​​ ​the​ ​parallel​ ​lines​ ​divide the​ ​transversals​ ​proportionally. 4) If​​ ​a​ ​ray​ ​bisects​ ​an​ ​∡​ ​of​ ​a​ ​△,​ ​then​​ ​it​ ​divides​ ​the​ ​opposite​ ​into​ ​segments​ ​that​ ​are proportional​ ​to​ ​the​ ​adjacent​ ​sides. 5) The​ ​ratio​ ​of​ ​the​ ​areas​ ​of​ ​2​ ​~​ ​△s​ ​are​ ​equal​ ​to​​ ​the​ ​square​ ​of​ ​the​ ​ratio​ ​of​ ​any​​ ​pair​ ​of corresponding​ ​sides. 6) The​ ​ratio​ ​of​ ​the​ ​(perimeter/altitude)​ ​of​ ​2​ ​~​ ​△s​ ​are​ ​equal​ ​to​​ ​the​ ​ratio​ ​of​ ​any​​ ​pair​ ​of corresponding​ ​sides. 7) In​​ ​a​ ​proportion,​ ​the​ ​product​ ​of​ ​the​ ​means​ ​is​ ​equal​ ​to​​ ​the​ ​product​ ​of​ ​the​ ​extremes. 8) If​​ ​the​ ​product​ ​of​ ​2​ ​#s​ ​is​ ​equal​ ​to​ ​the​ ​product​ ​of​ ​2​ ​other​ ​#s,​ ​then​​ ​either​ ​pair​ ​may​ ​be​ ​the extremes​ ​while​ ​the​ ​other​ ​pair​ ​is​ ​the​ ​means. Proving​ ​Triangles​ ​Similar: - AAA​ ​Postulate​ ​of​ ​~​ ​△s:​​ ​Needs​ ​all​ ​3​ ​∡s​ ​of​ ​a​ ​△​ ​≅​ ​to​ ​the​ ​corresponding​ ​∡s​ ​of​ ​a​ ​second​ ​△. - AA​ ​Theorem​ ​of​ ​~​ ​△s:​​ ​Needs​ ​2​ ​∡s​ ​of​ ​a​ ​△​ ​≅​ ​to​ ​the​ ​corresponding​ ​∡s​ ​of​ ​a​ ​second​ ​△. - SSS​ ​Postulate​ ​of​ ​~​ ​△s:​​ ​Needs​ ​all​ ​the​ ​ratios​ ​of​ ​corresponding​ ​sides​ ​to​ ​be​ ​equal​ ​to​ ​each other. - SAS​ ​Postulate​ ​of​ ​~​ ​△s:​​ ​Needs​ ​the​ ​ratios​ ​of​ ​2​ ​corresponding​ ​sides​ ​to​ ​be​ ​equal​ ​to​ ​each other,​ ​as​ ​well​ ​as​ ​the​ ​included​ ​∡s​ ​of​ ​the​ ​2​ ​pairs​ ​of​ ​corresponding​ ​sides​ ​≅. Ratios​ ​based​ ​on​ ​~​ ​△s: - If​​ ​2​ ​△s​ ​are​ ​~,​ ​then​​ ​the​ ​ratio​ ​of​ ​their​ ​areas​ ​is​ ​equal​ ​to​ ​the​ ​square​ ​of​ ​any​ ​ratio​ ​of​ ​any​ ​pair of​ ​corresponding​ ​sides. - If​​ ​2​ ​△s​ ​are​ ​~,​ ​then​​ ​the​ ​ratio​ ​of​ ​any​ ​pair​ ​of​ ​corresponding​ ​altitudes​ ​are​ ​equal​ ​to​ ​the​ ​ratio of​ ​any​ ​pair​ ​of​ ​corresponding​ ​sides. Back​ ​to​ ​Top Trigonometry Note:​ ​SohCahToa​ ​works​ ​only​ ​if​ ​the​ ​triangle​ ​is​ ​right.​ ​They​ ​can​ ​only​ ​be​ ​applied​ ​to​ ​either​ ​acute​ ​∡. For​ ​example,​ ​it​ ​is​ ​not​ ​possible​ ​to​ ​have​ ​Sin90,​ ​Cos90,​ ​or​ ​Tan90. All​ ​you​ ​need​ ​to​ ​know​ ​is​ ​SohCahToa.​ ​The​ ​Sine​ ​of​ ​one​ ​of​ ​the​ ​acute​ ​∡​ ​is​ ​equal​ ​to​ ​the​ ​Cosine of​ ​the​ ​other​ ​acute​ ​∡.​ ​Same​ ​applies​ ​for​ ​the​ ​reverse.​ ​The​ ​Tangent​ ​of​ ​one​ ​of​ ​the​ ​acute​ ​angles​ ​is equal​ ​to​ ​the​ ​reciprocal​ ​of​ ​the​ ​Tangent​ ​of​ ​the​ ​other​ ​acute​ ​angle. The​ ​three​ ​trigonometric​ ​functions​ ​are: 1) Sine​​ ​=​ ​Length​ ​of​ ​Side​ ​Opposite​ ​∡​ ​/​ ​Length​ ​of​ ​Hypotenuse 2) Cosine​​ ​=​ ​Length​ ​of​ ​Side​ ​Adjacent​ ​∡​ ​/​ ​Length​ ​of​ ​Hypotenuse 3) Tangent​ ​=​ ​Length​ ​of​ ​Side​ ​Opposite​ ​∡​ ​/​ ​Length​ ​of​ ​Side​ ​Adjacent​ ​∡ Note:​ ​To​ ​find​ ​sides,​ ​use​ ​Sine,​ ​Cosine,​ ​and​ ​Tangent.​ ​To​ ​find​ ​angles,​ ​use​ ​Sin-1​ ​,​ ​Cos-1​ ​,​ ​and​ ​Tan-1​ Note:​ ​Angle​ ​of​ ​elevation​ ​goes​ ​upwards,​ ​such​ ​as​ ​looking​ ​up​ ​to​ ​a​ ​point​ ​while​ ​on​ ​the​ ​ground,​ ​while the​ ​angle​ ​of​ ​depres...
View Full Document

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture