Lesson 13.1 The Premises of Geometry

Transcription

1 Lesson 13.1 The Premises of Geometry Name Period ate 1. Provide the missing property of equality or arithmetic as a reason for each step to solve the equation. Solve for x: 5(x 4) 2x 17 Solution: 5(x 4) 2x 17 a. 5x 20 2x 17 3x x 37 x b. c. d. e. In Exercises 2 4, identify each statement as true or false. If the statement is true, tell which definition, property, or postulate supports your answer. If the statement is false, give a counterexample. 2. If M M, then M is the midpoint of. 3. If P is on and is not, then mp mp If PQ ST and PQ KL, then ST KL. 5. omplete the flowchart proof. :, P Q, P Q Show: P Q P Q P Q Postulate 84 HPTER 13 iscovering Geometry Practice Your Skills

2 Lesson 13.2 Planning a Geometry Proof Name Period ate For these exercises, you may use theorems added to your theorem list through the end of Lesson In Exercises 1 3, write a paragraph proof or a flowchart proof for each situation. 1. :, P Q P Show: P Q Q 2. : PQ ST, QPR STU Show: PR UT Q R T P U S 3. : Noncongruent, nonparallel segments,, and Show: x y z 180 x a b y c z iscovering Geometry Practice Your Skills HPTER 13 85

3 Lesson 13.3 Triangle Proofs Name Period ate Write a proof for each situation. You may use theorems added to your theorem list through the end of Lesson : XY ZY, XZ WY W 2. :,, Show: WXY WZY Show: X M Z Y 3. : MN QM, NO QM, 4. :, E, P is the midpoint of MO R Show: QMN RON Show: E O E N P M Q 86 HPTER 13 iscovering Geometry Practice Your Skills

4 Lesson 13.4 Quadrilateral Proofs Name Period ate In Exercises 1 6, write a proof of each conjecture on a separate piece of paper. You may use theorems added to your theorem list through the end of Lesson The diagonals of a bisect each other. (Parallelogram iagonals ) 2. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a. (onverse of the Parallelogram iagonals ) 3. The diagonals of a rhombus bisect each other and are perpendicular. (Rhombus iagonals ) 4. If the diagonals of a quadrilateral bisect each other and are perpendicular, then the quadrilateral is a rhombus. (onverse of the Rhombus iagonals ) 5. If the base angles on one base of a trapezoid are congruent, then the trapezoid is isosceles. (onverse of the Isosceles Trapezoid ) 6. If the diagonals of a trapezoid are congruent, then the trapezoid is isosceles. (onverse of the Isosceles Trapezoid iagonals ) In Exercises 7 9, decide if the statement is true or false. If it is true, prove it. If it is false, give a counterexample. 7. quadrilateral with one pair of parallel sides and one pair of congruent angles is a. 8. quadrilateral with one pair of congruent opposite sides and one pair of parallel sides is a. 9. quadrilateral with one pair of parallel sides and one pair of congruent opposite angles is a. iscovering Geometry Practice Your Skills HPTER 13 87

5 Lesson 13.5 Indirect Proof Name Period ate 1. omplete the indirect proof of the conjecture: In a triangle the side opposite the larger of two angles has a greater measure. : Show: Proof: with m m ssume ase 1: If, then is by. y,, which contradicts. So,. ase 2: If, then it is possible to construct point on such that, by the Segment uplication Postulate. onstruct, by the Line Postulate. is. omplete the proof In Exercises 2 5, write an indirect proof of each conjecture. 2. :, Show: 3. If two sides of a triangle are not congruent, then the angles opposite them are not congruent. 4. If two lines are parallel and a third line in the same plane intersects one of them, then it also intersects the other. 88 HPTER 13 iscovering Geometry Practice Your Skills

6 Lesson 13.6 ircle Proofs Name Period ate Write a proof for each conjecture or situation. You may use theorems added to your theorem list through the end of Lesson If two chords in a circle are congruent, then their arcs are congruent. 2. : Regular pentagon E inscribed in circle O, with diagonals and Show: and trisect E E O 3. : Two circles externally tangent at R, common external tangent segment TS T S Show: TRS is a right angle R 4. : Two circles internally tangent at T with chords T and T of the larger circle intersecting the smaller circle at and Show: T iscovering Geometry Practice Your Skills HPTER 13 89

7 Lesson 13.7 Similarity Proofs Name Period ate Write a proof for each situation. You may use theorems added to your theorem list through the end of Lesson : with Show: 2 2. The diagonals of a trapezoid divide each other into segments with lengths in the same ratio as the lengths of the bases. 3. In a right triangle the product of the lengths of the two legs equals the product of the lengths of the hypotenuse and the altitude to the hypotenuse. 4. If a quadrilateral has one pair of opposite right angles and one pair of opposite congruent sides, then the quadrilateral is a rectangle. 90 HPTER 13 iscovering Geometry Practice Your Skills

8 7. RS 22.5 cm, E 20 cm 8. x 20 cm; y 7.2 cm 9. p cm; q cm LESSON 12.1 Trigonometric Ratios 1. sin P p r 2. cos P q r 3. tan P p q 4. sin Q q r 5. sin T cos T tan T sin R x x x m m m 30 w 15. sin 40 ; 2 8 w 18.0 cm x 16. sin 28 ; 1 4 x 7.4 cm 17. cos 17 7 y; 3 y 76.3 cm 18. a t z 76 LESSON 12.2 Problem Solving with Right Triangles 1. rea 2 cm 2 2. rea 325 ft 2 3. rea 109 in 2 4. x y a 7.6 in. 7. iameter 20.5 cm bout 2.0 m 11. bout ft 12. bout 22.6 ft LESSON 12.3 The Law of Sines 1. rea 46 cm 2 2. rea 24 m 2 3. rea 45 ft 2 4. m 14 cm 5. p 17 cm 6. q 13 cm 7. m 66, m mp 37, mq mk 81, mm Second line: about 153 ft, between tethers: about 135 ft LESSON 12.4 The Law of osines 1. t 13 cm 2. b 67 cm 3. w 34 cm 4. m 76, m 45, m m 77, mp 66, ms ms 46, mu 85, mv bout bout 43.0 cm 9. bout 34.7 in. LESSON 12.5 Problem Solving with Trigonometry 1. bout 2.85 mi/h; about m 50.64, m 59.70, m bout 8.0 km from Tower 1, 5.1 km from Tower 2 4. bout 853 miles 5. bout 530 ft of fencing; about 11,656 ft 2 LESSON 13.1 The Premises of Geometry 1. a. b. istributive property c. Subtraction property d. ddition property e. ivision property 2. False 3. False 4. True; transitive property of congruence and definition of congruence 5. P Q LESSON 13.2 Planning a Geometry Proof Proofs may vary. 1. P Q M P P Q Postulate P Q P Q Postulate P Q Postulate PQ Q Postulate P Q S Postulate PT P Q Third ngle iscovering Geometry Practice Your Skills NSWERS 113

9 2. PQ ST 3. a b c 180 Triangle Sum PQR TSU I QRP TUS Third ngle PR UT onverse of E a x V x b c 180 Substitution x y c 180 Substitution x y z 180 Substitution QPR STU b y V c z V 2. Proof: Statement Reason is bisector 4. onverse of ngle of isector efinition of angle bisector 6. is a right 6. efinition of angle perpendicular 7. is a right 7. efinition of angle perpendicular Right ngles re ongruent S 3. MN QM MN NO Transitivity NO QM LESSON 13.3 Triangle Proofs Proofs may vary. 1. XY ZY XZ WY WY WY Reflexive property WXY WZY SS XZY is isosceles efinition of isosceles triangle YM is the altitude from vertex Y efinition of altitude and vertex angle YM is angle bisector of XYZ Isosceles Triangle Vertex ngle XYM ZYM efinition of angle bisector QMN and NMO are supplementary Linear Pair Postulate 4. Proof: Statement MNO is isosceles efinition of isosceles triangle NMO NOP IT QMN RON Supplements of ongruent ngles Reason RON and NOP are supplementary Linear Pair Postulate 2. is isosceles 2. efinition of isosceles triangle IT 4. E NSWERS iscovering Geometry Practice Your Skills

10 5. E 5. Transitivity 6. E 6. onverse of Postulate 7. E 7. Postulate is a right 9. efinition of angle perpendicular 10. E is a right 10. efinition of right angle angle, transitivity 11. E 11. efinition of perpendicular LESSON 13.4 Quadrilateral Proofs Proofs may vary. 1. : is a Show: and bisect each other at M I M M PT is a efinition of I M M S Postulate 2. : M M, M M M Show: is a Proof: Statement Reason 1. M M M M 2. efinition of congruence and bisect each other at M efinition of bisect, definition of congruence M M M PT Opposite Sides 3. M M M M 4. efinition of congruence 5. M M 5. V 6. M M 6. SS Postulate PT onverse of I 9. M M 9. V 10. M M 10. SS Postulate PT onverse of I 13. is a 13. efinition of 3. : is a rhombus Show: and bisect each other at M and is a efinition of rhombus and bisect each other Parallelogram iagonals M and M are supplementary Linear Pair Postulate is a rhombus M M Rhombus ngles M M SS Postulate M M PT M is a right angle ongruent and Supplementary efinition of perpendicular M efinition of rhombus M M Reflexive property iscovering Geometry Practice Your Skills NSWERS 115

11 4. : and bisect each other at M and Show: is a rhombus (See flowchart at bottom of page.) 5. : is a trapezoid with and Show: is isosceles Proof: Statement Reason E 1. is a trapezoid 1. with 2. onstruct E 2. Parallel Postulate 3. E is a 3. efinition of 4. E 4. Opposite Sides ongruent 5. E 5. Postulate E 7. Transitivity 8. E is isosceles 8. onverse of IT 9. E 9. efinition of isosceles triangle Transitivity 11. is isosceles 11. efinition of isosceles trapezoid M 6. : is a trapezoid with and Show: is isosceles Proof: Statement Reason 1. is a trapezoid 1. with 2. onstruct E 2. Parallel Postulate 3. and E intersect 3. Line Intersection at F Postulate 4. F is a 4. efinition of 5. F 5. Opposite Sides ongruent F 7. Transitivity 8. F is isosceles 8. efinition of isosceles triangle 9. F F 9. IT 10. F 10. Opposite ngles 11. F 11. I Transitivity Reflexive property SS Postulate PT 16. is isosceles 16. efinition of isosceles trapezoid F E Lesson 13.4, Exercise 4 is a onverse of the Parallelogram iagonals Opposite Sides ll 4 sides are congruent Transitivity is a rhombus efinition of rhombus and bisect each other at M M M efinition of bisect, definition of congruence Opposite Sides M M M M SS Postulate PT Reflexive property M and M are right angles efinition of perpendicular M M Right ngles ongruent 116 NSWERS iscovering Geometry Practice Your Skills

12 7. False 8. False Therefore the assumption,, is false, so. 2. Paragraph Proof: ssume 9. True : with and Show: is a and are supplementary Interior Supplements Supplements of ongruent ngles is a and are supplementary Interior Supplements It is given that. y the reflexive property. So by SS,. Then by PT. ut this contradicts the given that. So. 3. : with Show: Paragraph Proof: ssume If, then by the onverse of the IT, is isosceles and. ut this contradicts the given that. Therefore,. 4. : oplanar lines k,, and m, k, and m intersecting k Show: m intersects Paragraph Proof: ssume m does not intersect If m does not intersect, then by the definition of parallel, m. ut because k, by the Parallel Transitivity, k m. This contradicts the given that m intersects k. Therefore, m intersects. m k onverse of Opposite ngles LESSON 13.6 ircle Proofs LESSON 13.5 Indirect Proof Proofs may vary. 1. ssume ase 1: If, then is isosceles, by the definition of isosceles. y the IT,, which contradicts the given that m m. So,. ase 2: is isosceles. 1. : ircle O with Show: onstruct O, O, O, O Line Postulate O y the Exterior ngle, m1 m2 m4, so m1 m4. y the ngle Sum Postulate, m2 m3 m, so m3 m. ut is isosceles, so m4 m3 by the IT. So, by transitivity, m1 m4 m3 m, or m1 m, which contradicts the given that m m. So,. O O efinition of circle, definition of radii O O SSS Postulate O O PT efinition of congruence, definition of arc measure, transitivity O O efinition of circle, definition of radii iscovering Geometry Practice Your Skills NSWERS 117

13 2. Paragraph Proof: hords,, and E are congruent because the pentagon is regular. y the proof in Exercise 1, the arcs,, and E are congruent and therefore have the same measure. me 1 2 me by the Inscribed ngles Intercepting rcs. Similarly, m 1 2 m and m 1 2 m. y transitivity and algebra, the three angles have the same measure. So, by the definition of trisect, the diagonals trisect E. 3. Paragraph Proof: onstruct the common internal tangent RU (Line Postulate, definition of tangent). Label the intersection of the tangent and TS as U. T U S LESSON 13.7 Similarity Proofs 1. Similarity Postulate efinition of similar triangle 2 Reflexive property R Multiplication property TU RU SU by the Tangent Segments. TUR is isosceles by definition because TU RU. So, by the IT, T TRU. all this angle measure x. SUR is isosceles because RU SU, and by the IT, S URS. all this angle measure y. The angle measures of TRS are then x, y, and (x y). y the Triangle Sum, x y (x y) 180. y algebra (combining like terms and dividing by 2), x y 90. ut mtrs x y, so by transitivity and the definition of right angle, TRS is a right angle. 2. : Trapezoid with, and and intersecting at E Show: E E E E I I E 4. Paragraph Proof: onstruct tangent TP (Line Postulate, definition of tangent). PT and T both have the same intercepted arc, T. Similarly, PT and T have the same intercepted arc, T. So, by transitivity, the Inscribed ngles Intercepting rcs, and algebra, T and T are congruent. Therefore, by the onverse of the Postulate,. E E Similarity Postulate E E E E efinition of similarity P T 118 NSWERS iscovering Geometry Practice Your Skills

14 3. : with right, Show: is right efinition of perpendicular Right ngles re ongruent 4. : with right angles and, Show: is a rectangle Similarity Postulate efinition of similarity Multiplication property is right Reflexive property Proof: Statement Reason 1. onstruct 1. Line Postulate 2. and are 2. right angles Right ngles re ongruent Reflexive property HL ongruence PT 8. m m 8. efinition of congruence 9. m m 9. Triangle Sum m m efinition of right angle 11. m m 11. Subtraction property m m 12. Substitution m m 13. ngle ddition m Postulate 14. m Transitivity 15. m efinition of right angle 16. m m 16. Quadrilateral Sum m m m Substitution property and subtraction property efinition of congruence 19. is a rectangle 19. Four ongruent ngles Rectangle iscovering Geometry Practice Your Skills NSWERS 119

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