Chapter 2 Summary. Carnegie Learning
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1 Chaper Summary Key Terms inducion (.1) deducion (.1) counerexample (.1) condiional saemen (.1) proposiional form (.1) proposiional variables (.1) hypohesis (.1) conclusion (.1) ruh value (.1) ruh able (.1) supplemenary angles (.) complemenary angles (.) adjacen angles (.) linear pair (.) verical angles (.) posulae (.) heorem (.) Euclidean geomery (.) Addiion Propery of Equaliy (.3) Subracion Propery of Equaliy (.3) Reflexive Propery (.3) Subsiuion Propery (.3) Transiive Propery (.3) flow char proof (.3) wo-column proof (.3) paragraph proof (.3) consrucion proof (.3) conjecure (.4) converse (.5) Posulaes and Theorems Linear Pair Posulae (.) Segmen Addiion Posulae (.) Angle Addiion Posulae (.) Righ Angle Congruence Theorem (.3) Congruen Supplemen Theorem (.3) Congruen Complemen Theorem (.3) Verical Angle Theorem (.3) Corresponding Angle Posulae (.4) Alernae Inerior Angle Theorem (.4) Alernae Exerior Angle Theorem (.4) Same-Side Inerior Angle Theorem (.4) Same-Side Exerior Angle Theorem (.4) Corresponding Angle Converse Posulae (.5) Alernae Inerior Angle Converse Theorem (.5) Alernae Exerior Angle Converse Theorem (.5) Same-Side Inerior Angle Converse Theorem (.5) Same-Side Exerior Angle Converse Theorem (.5) 195
2 .1 Idenifying and Comparing Inducion and Deducion Inducion uses specific examples o make a conclusion. Inducion, also known as inducive reasoning, is used when observing daa, recognizing paerns, making generalizaions abou he observaions or paerns, and reapplying hose generalizaions o unfamiliar siuaions. Deducion, also known as deducive reasoning, uses a general rule or premise o make a conclusion. I is he process of showing ha cerain saemens follow logically from some proven facs or acceped rules. Kyra sees coins a he boom of a founain. She concludes ha if she hrows a coin ino he founain, i oo will sink. Tyler undersands he physical laws of graviy and mass and decides a coin he hrows ino he founain will sink. The specific informaion is he coins Kyra and Tyler observed a he boom of he founain. The general informaion is he physical laws of graviy and mass. Kyra s conclusion ha her coin will sink when hrown ino he founain is inducion. Tyler s conclusion ha his coin will sink when hrown ino he founain is deducion..1 Idenifying False Conclusions I is imporan ha all conclusions are racked back o given ruhs. There are wo reasons why a conclusion may be false. Eiher he assumed informaion is false or he argumen is no valid. Erin noiced ha every ime she missed he bus, i rained. So, she concludes ha nex ime she misses he bus i will rain. Erin s conclusion is false because missing he bus is no relaed o wha makes i rain..1 Wriing a Condiional Saemen A condiional saemen is a saemen ha can be wrien in he form If p, hen q. The porion of he saemen represened by p is he hypohesis. The porion of he saemen represened by q is he conclusion. If I plan an acorn, hen an oak ree will grow. A solid line is drawn under he hypohesis, and a doed line is drawn under he conclusion. 196 Chaper Inroducion o Proof
3 .1 Using a Truh Table o Explore he Truh Value of a Condiional Saemen The ruh value of a condiional saemen is wheher he saemen is rue or false. If a condiional saemen could be rue, hen is ruh value is considered rue. The firs wo columns of a ruh able represen he possible ruh values for p (he hypohesis) and q (he conclusion). The las column represens he ruh value of he condiional saemen ( p q). Noice ha he ruh value of a condiional saemen is eiher rue or false, bu no boh. Consider he condiional saemen, If I ea oo much, hen I will ge a somach ache. p q p q T T T T F F F T T F F T When p is rue, I ae oo much. When q is rue, I will ge a somach ache. I is rue ha when I ea oo much, I will ge a somach ache. So, he ruh value of he condiional saemen is rue. When p is rue, I ae oo much. When q is false, I will no ge a somach ache. I is false ha when I ea oo much, I will no ge a somach ache. So, he ruh value of he condiional saemen is false. When p is false, I did no ea oo much. When q is rue, I will ge a somach ache. I could be rue ha when I did no ea oo much, I will ge a somach ache for a differen reason. So, he ruh value of he condiional saemen in his case is rue. When p is false, I did no ea oo much. When q is false, I will no ge a somach ache. I could be rue ha when I did no ea oo much, I will no ge a somach ache. So, he ruh value of he condiional saemen in his case is rue..1 Rewriing Condiional Saemens A condiional saemen is a saemen ha can be wrien in he form If p, hen q. The hypohesis of a condiional saemen is he variable p. The conclusion of a condiional saemen is he variable q. Consider he following saemen: If wo angles form a linear pair, hen he sum of he measures of he angles is 180 degrees. The saemen is a condiional saemen. The hypohesis is wo angles form a linear pair, and he conclusion is he sum of he measures of he angles is 180 degrees. The condiional saemen can be rewrien wih he hypohesis as he Given saemen and he conclusion as he Prove saemen. Given: Two angles form a linear pair. Prove: The sum of he measures of he angles is 180 degrees. Chaper Summary 197
4 . Idenifying Complemenary and Supplemenary Angles Two angles are supplemenary if he sum of heir measures is 180 degrees. Two angles are complemenary if he sum of heir measures is 90 degrees. Y Z V W X In he diagram above, angles YWZ and ZWX are complemenary angles. In he diagram above, angles VWY and XWY are supplemenary angles. Also, angles VWZ and XWZ are supplemenary angles.. Idenifying Adjacen Angles, Linear Pairs, and Verical Angles Adjacen angles are angles ha share a common verex and a common side. A linear pair of angles consiss of wo adjacen angles ha have noncommon sides ha form a line. Verical angles are nonadjacen angles formed by wo inersecing lines n m Angles and 3 are adjacen angles. Angles 1 and form a linear pair. Angles and 3 form a linear pair. Angles 3 and 4 form a linear pair. Angles 4 and 1 form a linear pair. Angles 1 and 3 are verical angles. Angles and 4 are verical angles. 198 Chaper Inroducion o Proof
5 . Deermining he Difference Beween Euclidean and Non-Euclidean Geomery Euclidean geomery is a sysem of geomery developed by he Greek mahemaician Euclid ha included he following five posulaes. 1. A sraigh line segmen can be drawn joining any wo poins.. Any sraigh line segmen can be exended indefiniely in a sraigh line. 3. Given any sraigh line segmen, a circle can be drawn ha has he segmen as is radius and one poin as he cener. 4. All righ angles are congruen. 5. If wo lines are drawn ha inersec a hird line in such a way ha he sum of he inner angles on one side is less han wo righ angles, hen he wo lines ineviably mus inersec each oher on ha side if exended far enough. Euclidean geomery: Non-Euclidean geomery:. Using he Linear Pair Posulae The Linear Pair Posulae saes: If wo angles form a linear pair, hen he angles are supplemenary. R 38 P Q S m PQR 1 m SQR 5 180º 38º 1 m SQR 5 180º m SQR 5 180º 38º m SQR 5 14º Chaper Summary 199
6 . Using he Segmen Addiion Posulae The Segmen Addiion Posulae saes: If poin B is on segmen AC and beween poins A and C, hen AB 1 BC 5 AC. A B C 4 m 10 m AB 1 BC 5 AC 4 m 1 10 m 5 AC AC 5 14 m. Using he Angle Addiion Posulae The Angle Addiion Posulae saes: If poin D lies in he inerior of angle ABC, hen m ABD 1 m DBC 5 m ABC. C D A 4 39 B m ABD 1 m DBC 5 m ABC 4º 1 39º 5 m ABC m ABC 5 63º 00 Chaper Inroducion o Proof
7 .3 Using Properies of Real Numbers in Geomery The Addiion Propery of Equaliy saes: If a, b, and c are real numbers and a 5 b, hen a 1 c 5 b 1 c. The Subracion Propery of Equaliy saes: If a, b, and c are real numbers and a 5 b, hen a c 5 b c. The Reflexive Propery saes: If a is a real number, hen a 5 a. The Subsiuion Propery saes: If a and b are real numbers and a 5 b, hen a can be subsiued for b. The Transiive Propery saes: If a, b, and c are real numbers and a 5 b and b 5 c, hen a 5 c. Addiion Propery of Equaliy applied o angle measures: If m 1 5 m, hen m 1 1 m 3 5 m 1 m 3. Subracion Propery of Equaliy applied o segmen measures: If m AB 5 m CD, hen m AB m EF 5 m CD m EF. Reflexive Propery applied o disances: AB 5 AB Subsiuion Propery applied o angle measures: If m 1 5 0º and m 5 0º, hen m 1 5 m. Transiive Propery applied o segmen measures: If m AB 5 m CD and m CD 5 m EF, hen m AB 5 m EF..3 Using he Righ Angle Congruence Theorem The Righ Angle Congruence Theorem saes: All righ angles are congruen. F H J K FJH GJK G Chaper Summary 01
8 .3 Using he Congruen Supplemen Theorem The Congruen Supplemen Theorem saes: If wo angles are supplemens of he same angle or of congruen angles, hen he angles are congruen. V Y Z X W VWZ XWY.3 Using he Congruen Complemen Theorem The Congruen Complemen Theorem saes: If wo angles are complemens of he same angle or of congruen angles, hen he angles are congruen Using he Verical Angle Theorem The Verical Angle Theorem saes: Verical angles are congruen s 1 3 and 4 0 Chaper Inroducion o Proof
9 .4 Using he Corresponding Angle Posulae The Corresponding Angle Posulae saes: If wo parallel lines are inerseced by a ransversal, hen corresponding angles are congruen. r s The angle ha measures 50 and 1 are corresponding angles. So, m º. The angle ha measures 130 and are corresponding angles. So, m 5 130º..4 Using he Alernae Inerior Angle Theorem The Alernae Inerior Angle Theorem saes: If wo parallel lines are inerseced by a ransversal, hen alernae inerior angles are congruen m 1 n The angle ha measures 63 and 1 are alernae inerior angles. So, m º. The angle ha measures 117 and are alernae inerior angles. So, m 5 117º. Chaper Summary 03
10 .4 Using he Alernae Exerior Angle Theorem The Alernae Exerior Angle Theorem saes: If wo parallel lines are inerseced by a ransversal, hen alernae exerior angles are congruen. d c The angle ha measures 11 and 1 are alernae exerior angles. So, m º. The angle ha measures 59 and are alernae exerior angles. So, m 5 59º..4 Using he Same-Side Inerior Angle Theorem The Same-Side Inerior Angle Theorem saes: If wo parallel lines are inerseced by a ransversal, hen same-side inerior angles are supplemenary p 1 q The angle ha measures 81 and 1 are same-side inerior angles. So, m º 81º 5 99º. The angle ha measures 99 and are same-side inerior angles. So, m 5 180º 99º 5 81º. 04 Chaper Inroducion o Proof
11 .4 Using he Same-Side Exerior Angle Theorem The Same-Side Exerior Angle Theorem saes: If wo parallel lines are inerseced by a ransversal, hen same-side exerior angles are supplemenary g 1 h The angle ha measures 105 and 1 are same-side exerior angles. So, m º 105º 5 75º. The angle ha measures 75 and are same-side exerior angles. So, m 5 180º 75º 5 105º..5 Using he Corresponding Angle Converse Posulae The Corresponding Angle Converse Posulae saes: If wo lines inerseced by a ransversal form congruen corresponding angles, hen he lines are parallel. j 80 k 80 Corresponding angles have he same measure. So, j k. Chaper Summary 05
12 .5 Using he Alernae Inerior Angle Converse Theorem The Alernae Inerior Angle Converse Theorem saes: If wo lines inerseced by a ransversal form congruen alernae inerior angles, hen he lines are parallel m Alernae inerior angles have he same measure. So, m..5 Using he Alernae Exerior Angle Converse Theorem The Alernae Exerior Angle Converse Theorem saes: If wo lines inerseced by a ransversal form congruen alernae exerior angles, hen he lines are parallel. x y Alernae exerior angles have he same measure. So, x y..5 Using he Same-Side Inerior Angle Converse Theorem The Same-Side Inerior Angle Converse Theorem saes: If wo lines inerseced by a ransversal form supplemenary same-side inerior angles, hen he lines are parallel v w Same-side inerior angles are supplemenary: So, v w. 06 Chaper Inroducion o Proof
13 .5 Using he Same-Side Exerior Angle Converse Theorem The Same-Side Exerior Angle Converse Theorem saes: If wo lines inerseced by a ransversal form supplemenary same-side exerior angles, hen he lines are parallel. b c Same-side exerior angles are supplemenary: So, b c. Chaper Summary 07
14 08 Chaper Inroducion o Proof
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