Geometry Midterm Review (Chapters: 1, 2, 3, 4, 5, 6)

Transcription

1 Geometry Midterm Review (Chapters: 1, 2, 3, 4, 5, 6) Algebra Review Systems of equations Simplifying radicals Equations of Lines: slope-intercept, point-slope Writing and solving equations: linear and quadratic Parallel lines Chapter 1 Topics 1.1 Undefined Terms point, line, plane Collinear, Coplanar Segment Endpoint Ray Opposite Rays Postulates Points, Lines, and Planes Intersections 1.2 Coordinate Ruler Postulate Distance Congruent Segments Segment Addition Postulate Midpoint Segment Bisector 1.3 Angle Interior of an Angle/Exterior of an Angle Measure of an angle/degree Protractor Postulate Types of Angles Congruent Angles Angle Addition Postulate Angle Bisector 1.4 Adjacent Angles Linear Pair Complementary Angles Supplementary Angles Vertical Angles 1.5 Perimeter (P) and Area (A) of: Rectangle, Triangle, Square Parallelogram Circumference (C) and Area (A) of: Circles 1.6 Midpoint Formula Parts of a right triangle Finding distance using: Distance Formula Pythagorean Theorem Chapter 2 Topics 2.1 Inductive Reasoning Finding and describing a pattern Writing conjecture Counterexamples 2.2 Conditional Statement Hypothesis Conclusion Writing Conditional Statements Truth Value Negation Related Conditionals -Converse -Inverse -Contrapositive Logically Equivalent Statements 2.3 Deductive Reasoning Law of Detachment Law of Syllogism Making Conclusions 2.4 Biconditional Statement Truth Value of a Biconditional Statement Definition Polygon Triangle Quadrilaterals Definition Biconditional 2.5 Properties of Equality Distributive Property Algebraic Proof D=rt Properties of Congruence Difference between Congruence and Equivalencies 2.6 Geometric Proofs (2-column) Theorem Linear Pair Theorem Congruent Supplements Theorem Right Angle Congruence Theorem Congruent Complements Theorem Vertical Angle Theorem 2.7 Common Segments Theorem If two congruent angles are supplementary, then each angle is a right angle.

2 Chapter 3 Topics 3.1 Parallel Lines/Planes Perpendicular Lines Skew Lines Transversal Corresponding Angles Alternate Interior Angles Same-Side Interior Angles Alternate Exterior Angles 3.2 Corresponding Angles Postulate Alternate Interior Angles Theorem Alternate Exterior Angles Theorem Same-Side Interior Angles Theorem If transversal is perpendicular to parallel lines, then all angles are right angles. 3.3 Converse of the Corresponding Angles Postulate Converse of the Alternate Interior Angles Theorem Converse of the Alternate Exterior Angles Theorem Converse of the Same-Side Interior Angles Theorem Parallel Postulate 3.4 Perpendicular Bisector Distance from a point to a line If 2 intersecting lines form a linear pair of congruent angles, then the lines are perpendicular. Perpendicular Transversal Theorem If 2 coplanar lines are perpendicular to the same line, then the 2 lines are parallel to each other. 3.5 Slope from a graph/coordinates Positive/Negative/Zero/Undefined Slopes Parallel Lines Theorem Perpendicular Lines Theorem 3.6 Point-Slope Form Slope-Intercept Form Vertical Lines Horizontal Line Transform between both equations Graphing Lines Pairs of Lines: -Parallel Lines -Intersecting Lines/Perpendicular Lines -Coinciding Lines Chapter 4 Topics 4.1 Classifying Triangles by Angles and Sides Using Triangle Classification 4.2 Triangle Sum Theorem Auxiliary Line Corollary The acute angles of a right triangle are complementary. The measure of each equiangular triangle is 60 degrees. Interior/Exterior Interior Angles/Exterior Angles Remote Interior Angles Exterior Angle Theorem 3 rd Angles Theorem 4.3 Congruent Corresponding Angles and Corresponding Sides 2 polygons are congruent iff their corresponding sides and angles are congruent CPCT-Corresponding Parts of Congruent Triangles Proving Triangles Congruent 4.4 Triangle Rigidity SSS Included Angle SAS AAS Verifying Triangle Congruence 4.5 Included Side ASA HL/HA/LA 4.6 CPCTC Corresponding Parts of Congruent Triangles are Congruent Remember: SSS, SAS, ASA, AAS, HL, HA, LA use corresponding parts to prove triangles congruent CPCTC uses congruent triangles to prove corresponding parts are congruent 4.8 Isosceles Triangle Legs, Vertex Angle, Base, Base Angles Isosceles Triangle Theorem (ITT) Converse of Isosceles Triangle Theorem If a triangle is equilateral, then it is equiangular.

3 Chapter 5 Topics 5.1 Equidistant Perpendicular Bisector Theorem Converse of Perpendicular Bisector Theorem Angle Bisector Theorem Converse of Angle Bisector Theorem Applying Angle Bisector Theorem 5.2 Concurrent Circumcenter Theorem Incenter Incenter Theorem Inscribed 5.3 Median of a Triangle Centroid of a Triangle Centroid Theorem Altitude of a Triangle Orthocenter of a triangle 5.4 Midsegment of a Triangle Triangle Midsegment Theorem 5.5 Indirect Proof In a triangle, the longer side is opposite the larger angle In a triangle, the larger angle is opposite the longer side Inequality Properties: -Addition Property of Inequality -Subtraction Property of Inequality -Multiplication Property of Inequality -Division Property of Inequality -Transitive Property of Inequality -Comparison Property Triangle Inequality Theorem Chapter 6 Topics 6.1 Side of a Polygon, Vertex of a Polygon, Diagonal of a Polygon Names of Polygons Definition of Polygon Regular Polygon Concave Convex Polygon Angle-Sum Theorem Polygon Exterior Angle Sum Theorem 6.2 Properties of a parallelogram If quad is a parallelogram, then its opposite sides are congruent If a quad is a parallelogram, then its opposite angles are congruent If a quad is a parallelogram, then its consecutive angles are supplementary. If a quad is a parallelogram, then its diagonals bisect each other. 6.3 Conditions for Parallelograms Quad with 1 pair of opposite sides parallel and congruent is a parallelogram. Quad with opposite sides congruent is a parallelogram Quad with opposite angles congruent is a parallelogram Quad with angles supp. to consecutive angles is a parallelogram Quad with diagonals bisecting each other is a parallelogram 5.6 Hinge Theorem Converse of Hinge Theorem Simplifying Radicals 5.7 Pythagorean Triples Pythagorean Inequality Theorems Triangle Theorem Triangle Theorem

4 Algebra Review Simplify completely. 1.) ) 24 2x 3x 2.) ) ) ) 8x ( 3 3x 2 + 5x) 4.) ) 8x ( 3 3x 2 5x) 5.) ) ( x 2 3) 2

5 Solve the following equations. 11.) x 2 = ( x + 3) 2 14.) 2x 2 7x = 5 12.) 4x 2 = ) 3y 2 + 4y = 2y 2 2y 9 13.) x 2 3x = ) ( 4x 9) 2 = 0

6 17.) Write an equation for the line that goes through the points ( 8,9) and ( 6,5). ( ) and N( 6,8) 18.) What is the equation of the perpendicular bisect of MN if M 3,5? 19.) What is the equation for a line parallel to y = 1 through point 2 x 8 ( 8, 5 )?

7 Review Chapter 1 Use the following diagram for problems ) Two opposite rays.!!" 21.) A point on XF. 22.) The intersection of the two planes. 23.) A plane containing X, V, and P. 24.) The intersection of two planes is a(n). 25.) M is between N and R. MR=15 m. NR=17.6 m. Write an equation and solve for MN. Draw a diagram. 26.) LH bisects GK at M. GM=(2x+6) m and GK = 24 m. Write an equation and solve for x. Draw a diagram. Use the following diagram for problems !VTS is a right angle. 27.) Name two acute angles. 29.) Name two angles that form a linear pair. 28.) Name two obtuse angles. 30.) Name a pair of angles that are supplementary and congruent.

8 31.)!!! " BD bisects #ABC, m#abd = 1 2 y2 + 10, and m#dbc = ( 4y + 4). What is m#abc? Use the following diagram for problem ) AB = 6x + 4, BC = x + 8, and AC = x Solve for x. If m!a = (4x 30) and m!b = 54.3 find the measure of the following. Write an equation to solve. 33.) Complement of A 34.) Supplement of B Find the perimeter and area in problems 35 and 36. Draw a diagram. All units are in inches. 35.) Rectangle with length = x + 4 and width = x. 36.) Triangle: with a=3x, b=10, c=x+6, height=2x where b is the base.

9 37.) Find the circumference of a circle with radius 4cm. Leave circumference in terms of pi. 38.) Find the area of a circle with diameter 12 ft. Leave area in terms of pi. 39.) Find the area of a circle with circumference ( 37π)ft. Leave area in terms of pi. ( ) and N( 8,0) 40.) Find the coordinates of the midpoint of MN with endpoints M 2,6. 41.) K is the midpoint of HL. H has coordinates ( 1, 7) and K has corrdinates ( 9,3). Solve for the coordinates of L.

10 Review Chapter 2 42.) Find the next two items in the pattern, then write a conjecture about the pattern: 0.7, 0.07, 0.007, 43.) Find the next two items in the pattern, then write a conjecture about the pattern: 3,6,9,15,24,39, ) Show that the conjecture is false by providing a counterexample: For every integer n, n 5 is positive. 45.) Show that the conjecture is false by providing a counterexample: Two complementary angles are not congruent. 46.) Write the inverse and determine the truth value for the inverse: All even numbers are divisible by ) Write the converse and determine the truth value for the converse: A triangle with one right angle is a right triangle. 48.) Write the contrapositive and determine the truth value for the contrapositive: If n 2 = 144, then n = 12. Determine if each conjecture in 49 and 50 is valid and by which law of deductive reasoning. 49.) If n is a natural number, then n is an integer. n is a rational number, if n is an integer. If n is is a rational number, then n is a natural number.

11 50.) If you do your homework, then your grade will be at least a C. You have a C-. 51.) For the conditional, If an angle is a right, then its measure is 90 degrees, write the converse and a biconditional statement. Converse: Biconditional: 52.) Write the definition as a biconditional: An acute triangle is a triangle with three acute angles. 53.) Write, solve, and justify each step of an equation. J is a point on segment GH. Draw a diagram. All lengths are in meters. GJ = 2x, JH = 3x 9, GH = 4x 4.

12 Review Chapter 3 For problems 54-61, identify the following using the diagram on the right. 54.) One pair of parallel segments 55.) One pair of skew segments 56.) One pair of perpendicular segments 57.) One pair of parallel planes 58.) One pair of alternate interior angles 59.) One pair of corresponding angles 60.) One pair of alternate exterior angles 61.) One pair of same-side interior angles 62.) Use diagram below to solve for x and y.

13 Use the diagram to the right to answer questions In problems 63-66, state the theorem/postulate that is related to the measures of the angles in each pair. Then, write an solve an equation to find the unknown angle measure ) m!1 = 120 ; m!5 = ( 60x) l m 64.) m!8 = ( 75x 30) ; m!3 = ( 30x + 60) 65.) m!4 = ( 50x + 20) ; m!6 = ( 100x 80) 66.) m!3 = ( 3x 2 30) ; m!7 = ( 4x 10)

14 In problems 67-70, name the theorem/postulate that proves l " m. 67.)!2!6 68.)!5! )!1!7 70.)!2 and!5 are supplementary 71.) Write and solve an inequality for x. l m 72.) Solve to find x and y: 73.) Determine if XY &&& and &&&& AB are parallel, perpendicular, or neither. X( 0, 2); Y( 1,2); A( 2,5); B( 3,1)

15 ( ) and ( 3, 5) 74.) Write the equation of the line through 1,3 in slope-intercept form. Review Chapter 4 Use the following diagram for problems to classify each triangle by its angles and sides. 75.) MNQ: 76.) NQP: 77.) MNP: 78.) Write and solve an equation to find the side lengths of the triangle. 79.) Solve for m!abd.

16 80.) Write and solve an equation to find m!n and m!p. 81.) Given ΔABC ΔJKL, AB = ( 2x + 12)m, JK = ( 4x 50)m, write and solve an equation to find x and AB. Draw a diagram. 82.) Write and solve an equation to find y. All lengths are in centimeters.

17 83.) Write and solve an equation to find x. Review Chapter 5 Use the diagram to the right for problems 84 and ) Given that m!abd = 16, find the m!abc. 85.) Given that m!abd = ( 2x + 12), and m!cbd = ( 6x 18), find m!abc. Use the diagram to the right for problems 86 and ) Given that FH is the perpendicular bisector of EG, EF = ( 4y 3)m, and FG = ( 6y 37)m, write and solve an equation for FG. 87.) Given that EF=10.6 ft, EH=4.3 ft, and FG=10.6 ft. Write and solve an equation for EG.

18 88.) Write an equation for the perpendicular bisector of the segment with endpoints X( 7,9) and Y( 3,5) 89.) ED, FD, and GD are the perpendicular bisectors of triangle ABC. ED=8 ft, DC=17 ft. What is BD? 90.) JP, KP, and HP are angle bisectors of ΔHJK. PL=3 in, LK=4 in, and JP=5 in. Find the distance from P to HK. Use the figure to the right for items In ΔABC, AE=12 ft, DG=7 ft, and BG=9 ft. Find each length. 91.) AG 92.) GC 93.) BF For items 94-96, use ΔMNP with vertices M( 4, 2),N( 6, 2),P ( 2,10 ). Find the coordinates of each point. 94.) The centroid

19 95.) The orthocenter 96.) The circumcenter Use the diagram to the right for items Find each measure. All lengths are in miles. 97.) ED 98.) AB 99.) m!bfe 100.) Write and solve an equation for n. All lengths are in meters.

20 101.) ΔXYZ is the midsegment triangle of ΔWUV. What is the perimeter of ΔXYZ? All lengths are in feet. 102.) Draw ΔABC, then write the angles in order from smallest to largest. AB=7 m, BC=9 m, and AC=8 m. 103.) Draw ΔDEF, then write the sides in order from shortest to longest m!e = 61 and m!f = ) The lengths of two sides of a triangle are 17cm and 12cm. Find the range of possible lengths for the third side. 105.) Can a triangle have sides with lengths 2.7 m, 3.5 m, and 5.8 m? Would the triangle be acute, obtuse, or acute? Show your work.

21 106.) Ray wants to place a chair 10 feet from his television set. Can the distance from the chair to his fireplace be 6 ft and the distance between the fireplace and the television is 8 feet? Draw a diagram. Show your work or explain your answer. 107.) Compare m!bac and m!dac below. 108.) Compare EF and GF below. 109.) Find the range of values for k. 111.) Compare m!abc and m!def below. 110.) Find the range of values for z. 112.) Compare PS and QR below.

22 113.) Find the value of x. All lengths are in yards. If necessary, write your answer as a simplified radical. 114.) An entertainment center is 52 inches wide and 40 inches high. Will a TV with a 60 inch diagonal fit in it? 115.) Find the missing side length, if a=5 cm and b=12 cm. Do the side lengths form a Pythagorean triple? Explain. 116.) Can the sides of a triangle measure 7 m, 11 m, and 15 m? If so, classify the triangle as acute, obtuse, or right. Find the values of the variables. Give your answers in simplest radical form. All sides lengths are in inches. 117.) 119.) 118.) 120.)

23 Find the perimeter and area of each figure. Give your answers in simplest radical form. 121.) A square with a diagonal length of 20 cm. 122.) An equilateral triangle with height 24 in. 123.) In a right triangle, the hypotenuse has length 18 3z and one leg has length z + 4. Draw a diagram and write an inequality to show all possible values for z. All lengths are in feet. Review Chapter ) Find the sum of the interior angle measures of a convex 11-gon. 125.) Find the measure of each interior angle of a regular 18-gon. 126.) Find the measure of each exterior angle of a regular 15-gon.

24 In parallelogram PNWL, PW and NL are diagonals that intersect at point M. NW=12 m, PM=9 m, and m WLP = 144. Draw the diagram and find the following measures. 127.) PW 128.) m!pnw QRST is a parallelogram. Find each measure. All lengths are in feet. 129.) TQ 130.) m!t 131.) Three vertices of parallelogram ABCD are A( 2, 6),B( 1,2), and C( 5,3). Find the coordinates of vertex D. Show your reasoning.

25 132.) Determine if QWRT must be a parallelogram. Justify your answer. 133.) Show that the quadrilateral with vertices E( 1,5),F ( 2,4 ),G( 0, 3) and H( 3, 2) is a parallelogram.