Geometry Final Review. Chapter 1. Name: Per: Vocab. Example Problems


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1 Geometry Final Review Name: Per: Vocab Word Acute angle Adjacent angles Angle bisector Collinear Line Linear pair Midpoint Obtuse angle Plane Pythagorean theorem Ray Right angle Supplementary angles Complementary angles Vertical angles Perpendicular lines Straight angle Segment bisector Segment addition postulate Distance formula Midpoint formula Angle addition postulate Congruent Chapter 1 Definition Example Problems Find the distance between F( 1, 4 ) and G( 6, 2 ) Find the midpoint of the segment between H( 3, 2 ) and K( 8, 1 ) Find x, leave in simplified radical form x
2 Given that Q R S U T Multiple Choice Practice If two angles form a linear pair, they are. A. Congruent B. Supplementary C. Complementary D. vertical Find the coordinates of the midpoint on for L( 10, 8 ) and M( 2, 6 ) A. ( 6, 7 ) B. ( 8, 2 ) C. ( 12, 14 ) D. ( 5, 2 ) The measure of the supplement of an angle is 30 less than four times the measure of the complement of the angle. Find the measure of the angle. A. 130 B. 60 C. 40 D. 50 Find m C if C D, m C = 3x 5, and D = 2x + 5 A. 65 B. 10 C. 30 D. 25 Vocab Word Chapter 2 Definition Conditional statement Hypothesis Conclusion Conjecture Counterexample Deductive reasoning Inductive reasoning Ifthen statement Inverse Converse Contrapositive Law of detachment Law of syllogism 2
3 Important Theorems Theorem What does it say 22: Supplement Theorem Example Problems Determine if each conjecture is true or false, explain your answer Given: X, Y, and Z are collinear and XY = YZ. Conjecture: Y is the midpoint of Given: 1 and 2 are supplementary Conjecture: 1 2 Write each conditional in ifthen form Every cloud has a silver lining A rectangle has four right angles Write the converse, inverse and contrapositive If a rectangle has four congruent sides, then it is a square. Converse: Inverse: Contrapositive: Determine if statement (3) follows from statement (2) and (1) by the law of detachment or the law of syllogism. If it does state which law was used. If it does not, write invalid. (1) All pilots must pass a physical examination. (2) Kris Thomas must pass a physical examination (3) Kris Thomas is a pilot (1) If a student is enrolled in Nipomo High, then the student has an ID number. (2) Jenny Jones is enrolled at Nipomo High. (3) Jenny Jones has an ID number. 3
4 Multiple Choice Practice Which of the following best describes deductive reasoning? A. using logic to draw conclusions based on accepted statements B. accepting the meaning of a term without definition C. defining mathematical terms to correspond with physical objects D. inferring a general truth by examining a number of specific examples Find the value of x. A. 7 B. 14 C D. 26 3x + 2 4x 5 Which statement follows from statements (1) and (2) by the Law of Syllogism? (1) If an object is a square, then it is a rhombus. (2) If an object is a rhombus, then it is an equilateral. A. An object is a rhombus. B. If an object is an equilateral, then it is a square. C. If an object is a square, then it is an equilateral. D. An object is a square. "Two lines in a plane always intersect in exactly one point." Which of the following best describes a counterexample to the assertion above? A. coplanar lines B. parallel lines C. perpendicular lines D. intersecting lines Identify the hypothesis of the following ifthen statement. If Ralph does his math homework, then he will get a good grade on the quiz. A. He will not get a good grade on the quiz. B. Ralph may do his math homework. C. He will get a good grade on the quiz. D. Ralph does his math homework. Which statement is the inverse of the statement angles with the same measure are congruent? A. If two angles do not have the same measure, then they are not congruent. B. If two angles are not congruent, then they do not have the same measure. C. If two angles have the same measure, then they are congruent. D. If two angles are congruent, then they have the same measure. Vocab Parallel Lines Word Chapter 3 Definition Alternate interior angles Alternate exterior angles Consecutive angles Corresponding angles 4
5 Skew lines transversal Slope of a line Important Theorems Theorem What does it say Corresponding angles postulate Alternate interior angle theorem Consecutive interior angle theorem Alternate exterior angles theorem Perpendicular transversal theorem Converse of Corresponding angles postulate Converse of Alternate interior angle theorem Converse of Consecutive interior angle theorem Converse of Alternate exterior angles theorem Converse of Perpendicular transversal theorem 5
6 Postulate 32 Postulate 33 Example Problems In the figure, S X Y Z 4 = 7 = 3 = 6 = 5 = 8 = T 6 Find the value of x, y, and z. 42 3x ( y + 7 ) z Find the slope of the line parallel to the line through ( 3, 0 ) and ( 4, 5 ). Find the slope of the line perpendicular to the line through ( 2, 9 ) and ( 1, 4 ). Find the value of x so that l ( 5x  8 ) ( 3x + 20 ) m Determine whether each statement is true or false. Explain your reasoning l n and 11 are alternate interior angles 2. 4 and 9 are alternate exterior angles 3. 7 and 11 are corresponding angles k m Multiple Choice Practice Find x What is the m 2? A. 7 B. 7 C. 91 D. 132 A. 15 B. 75 C. 90 D
7 Complete the following proof. 1 b Given: Prove: b w w 2 Statements Reasons Problem # Problem # Problem #7 4. b w 4. Problem #8 5. A) Given B) Prove C) Alternate Exterior Angles Theorem D) Alternate Exterior Angles Converse 6. A) Given B) Corresponding Angles Theorem C) Alternate Exterior Angles Converse D) Vertical Angles are congruent 7. A) Transitive Property B) Reflexive Property C) Corresponding Angles Theorem D) Corresponding Angles Converse 8. A) Prove B) Corresponding Angles Converse C) Corresponding Angles Theorem D) Lines are always parallel Vocab Word Acute triangle Equilateral triangle Chapter 4 Definition Obtuse triangle Equiangular triangle Isosceles triangle Scalene triangle Right triangle Exterior angle CPCTC 7
8 Important Theorems Theorem What does it say Angle sum theorem Third angle theorem Exterior angle theorem SAS SSS ASA AAS Isosceles Triangle theorem Thm: 47 Converse of the isosceles triangle theorem Example Problems Use the distance formula to classify the triangle by the measures of its sides. ABC with vertices A( 6, 4 ), B( 2, 4 ), and C( 2, 7 ) Find the value of x 20 x 15 Find x 45 (3x + 16) 112 8
9 If angle find the measure of each P Q S 6 R Complete the congruence statement ABX B A X D C Determine which theorem or postulate can be used to prove the triangles are congruent, if it is not possible state that. 1 = 2 = 3 = 4 = 5 = 6 = Multiple Choice Practice Use the proof to answer the question below. Given AB BC; D is the midpoint of AC A Prove: ABD CBD B Statement Reason 1. AB BC 1. Given 2. D is the midpoint of AC 2. Given 3. AD CD 3. Def of midpoint 4. BD BD 4. Reflexive Prop 5. ABD CBD 5.? What reason can be used to prove that the triangles are congruent? A. AAS B. ASA C. SAS D. SSS Find the values of x and y: D C x In the figure below, AC DF and A D. D A F C Which additional information would be enough to prove that ABC DEF? A. AB DE B. AB BC C. BC EF D. BC DE Given H L and HJ JL. Which of the following is true? I E B K J 106 y y H L A. x = 32, y = 74 B. x = 74, y = 54 C. x = 74, y = 106 D. x = 32, y = 106 A. HIJ JKL by SAS B. HIJ KLJ by ASA C. HIJ KLJ by SAS D. HIJ LKJ by ASA 9
10 triangle? A. Isosceles B. Obtuse C. Acute D. Equilateral, which term does not describe the 3x  2 A C 7 x + 4 B Which of the following is not the way to prove two triangles congruent? 1. SSS 2. SAS 3. CPCTC 4. SSA 5. SAA 6. AAA A. 1, 2, and 5 B. 3, 4, and 6 C. 3 only D. 4 only Vocab Word Altitude of a triangle Angle bisector of a triangle Chapter 5 Definition Median of a triangle Perpendicular bisector of a triangle Important Theorems Theorem What does it say HL HA LL LA Triangle inequality Theorem 10
11 (6y 2) Example Problems State the additional information needed to prove the triangles are congruent by the given theorem LL HA HL A B R Q Z U V V W X Y C D S T Find the value of x so that MNP ONP by LL P Find the values of x and y so that ABC DEF by HA B C D In ABC identify the following: C F G D M 2x  4 N 8 O A F E A H Median: Altitude: Angle bisector: Perpendicular bisector: E B Multiple Choice Practice Which of the following sets of numbers could represent the lengths of the sides of a triangle? A. 2, 2, 5 B. 3, 3, 5 C. 4, 4, 8 D. 5, 5, 15 Two sides of a triangle have lengths of 7 and 13. The third side has a length that is? A. 6 < 3rd side < 13 B. 6 < 3rd side < 20 C. 13 < 3rd side < 20 D. 6.5 < 3rd side < 19.5 Which may not contain a vertex of a triangle? A. Perpendicular bisector B. Altitude C. Median D. Angle bisector Multiply a number by 5, subtract 6, multiply the result by 3, then add 8. If the final result is 80, what number did you start with? A. 6 B. 8 C. 30 D
12 Vocab trapezoid Word Chapter 6 Definition parallelogram quadrilateral Rhombus Square Rectangle Median of a trapezoid Isosceles trapezoid diagonal Important Theorems Theorem What does it say
13 Name: Name: Name: Name: Name: Properties: Properties: Properties: Properties: Properties: Parallelograms Directions: 1. Consider each of the six polygons above. Write the name and properties of each in the spaces provided. 2. Label the figures to show all congruent sides, diagonals, angles, and segments. 3. Label congruent angles, and show any special relationships such as supplementary angles or parallel sides. 13
14 Example Problems In the parallelogram find the values of x, y and z Determine if the quadrilateral is a parallelogram Find the values of x and y that makes the quadrilateral a parallelogram 120 y x + 22 y 35 x z 12 8 x 5x + 4y Use the rhombus below to find x PQRS is an isosceles trapezoid with bases use the following to solve If TV = x + 7 and PS + QR = 5x + 2, find x True or false. Explain your answer. If the diagonals of a quadrilateral bisect each other, then it is a rectangle. m 5 = 2( x + 1 ) m 3 = 4( x + 1 ) P T Q S V R Multiple Choice Practice What values of a and b make MNOP a parallelogram? If ABCD is a parallelogram, then what is the length of BD? A 10 B 11 C 12 D 14 14
15 Find the perimeter of the rectangle: There is always a pair of noncongruent sides in a A. 95 in B. 60 in (2x + 5) in 30 in A. Parallelogram B. Trapezoid C. Rhombus D. quadrilateral C. 25 in (3x 15) in D. 150 in Vocab Word Geometric mean Similar polygons Chapter 7 Definition Important Theorems Theorem What does it say AA similarity SSS Similarity SAS Similarity Thm: 74 Triangle proportionality Thm: 75 Thm: 76 Thm: 77 Thm:
16 Thm: 79 Thm: 710 Thm: 711 Example Problems Determine if the triangles are similar, if they are, tell how you know Solve for x 6 5 x 10 5 x  3 x 16
17 . If BP = 8, AP = 6, DF = 2x + 1 and EY = 2x 4, find DY. If STV~ PQM, the perimeter of PQM is 28, Find the perimeter of STV. A B P C E P 8 Q M S T 10 x  4 V x 1.5 D Y F Multiple Choice practice problems Find x x + 6 x 8 16 A. 2 B. 12 C. 12 D. 6 In. If EI = 8, IF = 4, and EH = 5, find HG. E? R F I A. 1 B. 2 C. 2.5 D. 10 H G A. B. C. D. P S Q N L O M 17
18 Vocab Sine Word Chapter 8 Definition Cosine Tangent Law of sines Law of cosines Pythagorean triple Geometric mean Important Theorems Theorem What does it say Converse of the Pythagorean therorem Thm: 86 Thm:
19 Trigonometry Flow Chart: START: How many side lengths do you already have? Draw and Label the Triangle : : : : EXAMPLE: 5 in. 25 o x Visit the Chief! 19
20 Trigonometry Flow Chart: START: How many angles do you already have? Draw and Label the Triangle A Example: 13 B 12 C C Example: Chosen Angle 20
21 Example problems What is the area in square inches of the triangle below? In the figure below, sin A = 0.7 What is the length of? A right triangle s hypotenuse has a length of 5. If one leg has length 2, what is the length of the other leg? (Hint: Draw the triangle.) What is the value for x in the triangle below? A new road is being constructed to relieve traffic congestion in a residential neighborhood. The plan for the old road and the new road is shown below. How many fewer miles will the commuters travel on the new road? An 8foot ladder is leaning against a wall. Approximately how far up the wall does the ladder reach? Multiple choice practice problems Which equation should be used to find the length of? A 13 ft ladder is leaning against a wall. The top of the ladder touches the wall 12 ft above the ground. The bottom of the ladder is 5 ft from the bottom of the wall. What is the sine of the angle formed by the ground and the base of the ladder? 21
22 If RSTW is a rhombus, what is the area of ΔWXT? What is the approximate height, in feet, of the tree in the figure below? A 36 B 36 3 C 48 D 18 3 What is the approximate value of x in the triangle below? A 3.4 units B 4.2 units C 4.9 units D 7.3 units 22
23 Vocab Minor arc Word Chapter 9 Definition Major arc Central angle chord Tangent of a circle Inscribed angle Intercepted arc Secant Semicircle Radius diameter Important Theorems Theorem What does it say Thm:91 Thm: 92 Thm: 93 Thm: 94 Thm: 95 Thm: 96 Thm:
24 Thm: 910 Thm: 911 Thm: 912 Thm: 913 Thm: 914 Thm: 915 Thm: 916 Vocabulary Using Correct Symbols, give an example of: 1. Minor Arc: 2. Major Arc: 3. SemiCircle: 4. Diameter: 5. Radius: 6. Chord: 7. Inscribed Angle: 8. Central Angle: 24
25 Arcs and Angles Segments Theorems ON CrissCross Two Tangents From the Same Point x y ON Outside x Whole A Diameter Perpendicular to a Chord IN Outside 2 A Tangent and a Radius OUT Equation of a Circle Equation: Center: Radius: Quadrilateral Inscribed in a Circle 8y x 3x 84 o Example problems 42 x x x 25
26 x x x Multiple Choice Practice Problems Refer to circles B and D in the figure below. If BC = 5 and CD = 5, find AE. What is the measure of the angle formed by the hands of a clock at 4 o'clock? A. 20 B. 15 C. 10 D. 25 A B D C E A. 60 B. 120 C. 30 D. 90 In the figure below, is a diameter of the circle. If US = 9, find SV. A. 4.5 B. 27 C. 18 D. 9 Find the value of y. A. 19 B. 11 C. D. Find the value of y to the nearest tenth A. 7.2 B. 7.6 C. 7.5 D. 8.0 Find an equation of the circle that has a diameter with endpoints at (6, 10) and (2, 4). Hint: find the center of the circle by finding the center f the diameter. A. (x  2) 2 + (y  7) 2 = 25 B. (x + 2) 2 + (y  4) 2 = 100 C. (x  4) 2 + (y  3) 2 = 53 D. (x  6) 2 + (y  10) 2 =
27 Vocab apothem Word Chapter 10 Definition Concave polygon Convex polygon Regular polygon Important Theorems Theorem What does it say Interior Angle Sum Theorem Exterior Angle Sum Theorem Example Problems Find the measure of one interior angle of a regular 24gon Find the measure of one exterior angle of a regular 16gon Find the interior angle sum of a regular dodecagon Is there a regular polygon with an interior angle sum of 9000? If so, what is it? Find the area of the following octagon apothem = 14.1 side = 11.7 Find the area of the shaded region
28 Multiple Choice Practice Problems A trapezoid has an area of sq. in., and its two bases are 7 and 12 inches long. Find the height of the trapezoid A. 7 in. B in. C. 8.5 in. D. 9.2 in. Find the area of a regular octagon with an apothem of 8.5. E. 239 F. 296 G. 182 H. 340 Find the area of this figure. Find the area of the shaded region between the circle of diameter 4 feet and the equilateral triangle. Round to the nearest tenth. A. 104 sq. units B. 98 sq. units C. 200 sq. units D. 168 sq. units A. 9.4 ft 2 B ft 2 C. 8.7 ft 2 D ft 2 Vocab cylinder Word Chapter 11 Definition sphere cone Prism Pyramid 28
29 Important Formulas Solid Lateral Area Surface Area Volume Prism Cylinder Pyramid Cone Sphere Example Problems Find surface area and volume of each solid A sphere with a diameter of 6.2 in Sketch the net of the solid 29
30 Multiple Choice Practice Problems Find the surface area of a cylinder with a radius of 2 meters and a height of 6 meters. A m 2 B m 2 C m 2 D m 2 Find the volume of the right prism. A. 144 cm 3 B. 192 cm 3 C. 264 cm 3 D. 216 cm 3 Find the volume to the nearest tenth of a sphere with a radius of 4.2 cm. Find the surface area of the solid. Round to the nearest tenth. A cm 3 B cm 3 C cm 3 D cm 3 A in 2 B in 2. C in 2 D in 2 30
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