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Name 59Chapter 4 – Triangle Congruence4.1Scalene triangle - A triangle with all three sides havingdifferent lengths.Terms, Postulates and Theorems4.2SSS Congruence Postulate (Side-Side-Side)If the sides of one triangle are congruent to the sides of asecond triangle, then the triangles are congruent.Equilateral triangle - All sides of a triangle are congruent.Isosceles triangle - A triangle with at least two sidescongruent. Legs of an isosceles triangle - The congruent sidesin an isosceles triangle. Vertex angle - The angle formed by the legs in anisosceles triangle. Base - The side opposite the vertex angle. Base angles - The angles formed by the base.SAS Congruence Postulate (Side-Angle-Side)If two sides and the included angle of one triangle arecongruent to two sides and an included angle of anothertriangle, then the triangles are congruent.Median: a segment in a triangle that connects a vertex tothe midpoint of the opposite side.Altitude: a segment in a triangle that connects a vertex tothe side opposite forming a perpendicular.Isosceles Triangle TheoremIf two sides of a triangle are congruent, then the anglesopposite those sides are congruent.Angle Bisector: a segment that bisects an angle in atriangle and connects a vertex to the opposite side.Corollary 4-1 - A triangle is equilateral if and only if it isequiangular.Theorem 4.1 – If a median is drawn from the vertex angleof an isosceles triangle, then the median is also an anglebisector and an altitude.Acute triangle - A triangle with all acute angles.Equiangular triangle - A triangle with all angles congruent.Obtuse triangle - A triangle with one obtuse angle.Right triangle - A triangle with one right angle. Hypotenuse - The side opposite the right angle ina right triangle. Legs of a right triangle - The two sides that formthe 90 .Converse to the Isosceles Triangle TheoremIf two angles of a triangle are congruent, then the sidesopposite those angles are congruent.4.3ASA Congruence Postulate (Angle-Side-Angle)If two angles and the included side of one triangle arecongruent to two angles and the included side of anothertriangle, the triangles are congruent.AAS Congruence Postulate (Angle-Angle-Side)If two angles and a nonincluded side of one triangle arecongruent to the corresponding two angles and side of asecond triangle, the two triangles are congruent.Corollary 4-2 - Each angle of an equilateral trianglemeasures 60 .Definition of Congruent Triangles (CPCTC) - Two trianglesare congruent iff their corresponding parts are congruent.4.4HL Congruence Theorem (HL) – If the hypotenuse and legof one right triangle are congruent to the hypotenuse andleg of another right triangle, then the triangles arecongruent.Geometry 59
Geometry 60Geometry 60
Name 61TrianglesNotes Section 4.1Classify by SidesClassify by AnglesScalene triangle - A triangle with all three sides havingdifferent lengths.Acute triangle - A triangle with all acute angles. Acute angle - An angle greater than 0 and lessthan 90 .Equilateral triangle - All sides of a triangle are congruent.Equiangular triangle - A triangle with all angles congruent.Isosceles triangle - A triangle with at least two sidescongruent. Legs of an isosceles triangle - The congruent sidesin an isosceles triangle. Vertex angle - The angle formed by the legs in anisosceles triangle. Base - The side opposite the vertex angle. Base angles - The angles formed by the base.Obtuse triangle - A triangle with one obtuse angle. Obtuse angle - An angle more than 90 and lessthan 180 .Right triangle - A triangle with one right angle. Right angle - An angle that is 90 . Hypotenuse - The side opposite the right angle ina right triangle. Legs of a right triangle - The two sides that formthe 90 .Isosceles Triangle TheoremIf two sides of a triangle are congruent, then the anglesopposite those sides are congruent.Converse to the Isosceles Triangle TheoremIf two angles of a triangle are congruent, then the sidesopposite those angles are congruent.Corollary 4-1 - A triangle is equilateral if and only if it isequiangular.Corollary 4-2 - Each angle of an equilateral trianglemeasures 60 .Geometry 61
Geometry 62Definition of Congruent Triangles (CPCTC) - Two trianglesare congruent iff their corresponding parts are congruent.HName congruent figures.5.NEOFX6.Find the value of x.1.2.7.3.8.List pairs of corresponding parts.4.Geometry 62
Name 63SSS and SASNotes Section 4.2SSS Congruence Postulate (Side-Side-Side)State if the two triangles are congruent. If they are, stateIf the sides of one triangle are congruent to the sides of awhy.second triangle, then the triangles are congruent.1.SAS Congruence Postulate (Side-Angle-Side)If two sides and the included angle of one triangle arecongruent to two sides and an included angle of anothertriangle, then the triangles are congruent.2.3.Median: a segment in a triangle that connects a vertex tothe midpoint of the opposite side.Altitude: a segment in a triangle that connects a vertex tothe side opposite forming a perpendicular.4.5.Angle Bisector: a segment that bisects an angle in atriangle and connects a vertex to the opposite side.Theorem 4.1 – If a median is drawn from the vertex angleof an isosceles triangle, then the median is also an anglebisector and an altitude.6.Geometry 63
Geometry 64Geometry 64
Name 65AAS and ASANotes Section 4.3ASA Congruence Postulate (Angle-Side-Angle)4.If two angles and the included side of one triangle arecongruent to two angles and the included side of anothertriangle, the triangles are congruent.5.AAS Congruence Postulate (Angle-Angle-Side)If two angles and a nonincluded side of one triangle arecongruent to the corresponding two angles and side of asecond triangle, the two triangles are congruent.6.7.State if the two triangles are congruent. If they are, statewhy.1.8.2.9.3.Geometry 65
Geometry 66Geometry 66
Name 67HLNotes Section 4.4HL Congruence Theorem (HL) – If the hypotenuse and legof one right triangle are congruent to the hypotenuse andleg of another right triangle, then the triangles arecongruent.Geometry 67
Geometry 68Geometry 68
Name 69Chapter 4 Summary1. Summarize the main idea of the chapter2. Terms (Include name and definition). Also include key example or picture for each term
Geometry 703. Theorems and Postulates. Also include key example for each theorem or postulate4. Key examples of the most unique or most difficult problems from notes, homework or application.Geometry 70
Name 71Bisectors, Medians and AltitudesMedian: a segment in a triangle that connects a vertex tothe midpoint of the opposite side.Altitude: a segment in a triangle that connects a vertex tothe side opposite forming a perpendicular.Notes Section 5.1Draw and label a figure to illustrate each situation.#1) ̅̅̅̅𝑃𝑇 and ̅̅̅̅𝑅𝑆 are medians of triangle PQR and intersectat V.̅̅̅̅ is a median and an altitude of ABC.#2) 𝐴𝐷#3) DEF is a right triangle with right angle at F. ̅̅̅̅𝐹𝐺 is amedian of DEF and ̅̅̅̅𝐺𝐻 is the perpendicular bisector of̅̅̅̅𝐷𝐸 .Angle Bisector: a segment that bisects an angle in atriangle and connects a vertex to the opposite side.State whether each sentence is always, sometimes, ornever true.Perpendicular Bisector: a segment in a triangle thatpasses through the midpoint of a side and isperpendicular to that side.Theorem 5-1.2: A point is on the perpendicular bisectorIFF it is equidistant from the endpoints of the segment.#4) Three medians of a triangle intersect at a point insidethe triangle.#5) The three angle bisectors of a triangle intersect at apoint outside the triangle.B#6) The three altitudes of a triangle intersect at a vertex ofthe triangle.
Geometry 72#7) ̅̅̅̅𝐴𝐷 is an altitude of ABC. Find BD.#10) Find m ABC if ̅̅̅̅𝐵𝐷 is an angle bisector of ABC.CB𝑥 7m ABC 13x 4A6x 3ºD2𝑥 15ADCB̅̅̅̅is a median of ABC. Find 𝑚 𝐴𝐷𝐶#8) 𝐴𝐷̅̅̅̅ is a perpendicular bisector of 𝐵𝐶̅̅̅̅ . Find x and y.#11) 𝐴𝐷CCDA3𝑥 7𝑥 7A𝑥 152𝑥 15DBB#9) Find the midpoint of A(2, 4) and B(-5, 8)Geometry 72
Name 73Chapter 6 – QuadrilateralsTerms, Theorems & PostulatesSection 6.2Parallelogram: a quadrilateral with both pairs of opposite sidesparallel.Section 6.5Rhombus: A quadrilateral with four congruent sides. (Also couldbe defined as a parallelogram with four congruent sides.)Theorem 6-1: Opposite sides of a parallelogram are congruent.Theorem 6-13.14: A parallelogram is a rhombus IFF its diagonalsare perpendicular.Theorem 6-2: Opposite angles of a parallelogram are congruent.Theorem 6-3: Consecutive angles in a parallelogram aresupplementary.Theorem 6-4: If a parallelogram has one right angle then it hasfour right angles.Theorem 6-5: The diagonals of a parallelogram bisect eachother.Theorem 6-6: Each diagonal of a parallelogram separates theparallelogram into two congruent triangles.Theorem 6-15: Each diagonal of a rhombus bisects a pair ofopposite angles.Square:(a rectangular rhombus; a rhombicular rectangle.) Aquadrilateral that is both a rhombus and a rectangle.Section 6.6Trapezoid: a quadrilateral with exactly one pair of parallel sides.Bases: the parallel sides of a trapezoid.Legs:the nonparallel sides of a trapezoid.Pair of base angles: two angles in a trapezoid that share acommon base.Section 6.3Theorem 6-7: If both pairs of opposite sides of a quadrilateralare congruent, then the quadrilateral is a parallelogram.Theorem 6-8: If one pair of opposite sides of a quadrilateral isboth parallel and congruent, then the quadrilateral is aparallelogram.Theorem 6-9:If the diagonals of a quadrilateral bisect eachother, then the quadrilateral is a parallelogram.Theorem 6-10: If both pairs of opposite angles in a quadrilateralare congruent, then the quadrilateral is a parallelogram.Section 6.4Rectangle: a quadrilateral with four right angles. (Also coulddefine as a parallelogram with one right angle.)Isosceles trapezoid: a trapezoid with congruent legs.Theorem 6-16:Both pairs of base angles of an isosceles trapezoid arecongruent.Theorem 6-17:The diagonals of an isosceles trapezoid are congruent.Median of a Trapezoid:a segment that connects the midpoints of the legs.Theorem 6-18:The median of a trapezoid is parallel to the bases and itsmeasure is one half the sum of the measures of the bases.Theorem 6-11.12: A parallelogram is a rectangle IFF itsdiagonals are congruent.The quiz will consist of one matching section and one multiple choice section. The matching section will contain all terms and the theoremsthat have names. The multiple choice section will contain all theorems, postulates, and corollaries that have no names. I will remove aword from the sentence and give you three or four choices to complete the sentence.Geometry 73
Geometry 74Geometry 74
Name 75Solving Systems of EquationsNotes Section 6.1Solve each system of equations by substitution orelimination. If the system does not have exactly onesolution, state whether it has no solution or infinitelymany solutions.#1)x 7Substitution5y x 12PRO TIPS#4)If possibleyou maywant totransformone or bothof your EQs.Such asgetting rid offractions ordecimals.-3x – 4y 52x 2y 0#1) In one ofthe EQ, solvefor avariable.#2)5x 4y -92x – 4y -40#2) Thensubstitutefor thevariable intothe otherEQ.#5)4 2𝑥 𝑦 232y -2x 207#3) Solve theEQ.#4) Thensubstitutethe value ofthe variableinto one ofthe EQ andsolve.#3)y 3x – 23x – y 7If at anypoint whilesolving anEQ you get atruestatementsuch as, 9 9, then theanswer isinfinitelymanysolutions. Ifat any pointyou get afalsestatement,such as 3 7, then theanswer is nosolution.#6)x 2y -2.75x .15y 2.55Geometry 75
Geometry 76Geometry 76
Name 77ParallelogramsNotes Section 6.2Parallelogram: a quadrilateral with both pairs of oppositesides parallel.Theorem 6-1: Opposite sides of a parallelogram arecongruent.Theorem 6-2: Opposite angles of a parallelogram arecongruent.Theorem 6-3: Consecutive angles in a parallelogram aresupplementary.Theorem 6-4: If a parallelogram has one right angle thenit has four right angles.Theorem 6-5: The diagonals of a parallelogram bisecteach other.Theorem 6-6: Each diagonal of a parallelogram separatesthe parallelogram into two congruent triangles.Geometry 77
Geometry 78Is each quadrilateral a parallelogram? Justify youranswer.#1)20 With the given information, answer each question.#5) Given parallelogram PQRS with m P 2y andm Q 4y 30, find the m R and m S.160 #2)50 50 50 If each quadrilateral is a parallelogram, find the value of x,y, and z.#3)x80 yz#4)xzy15 Geometry 7870 #6) If NCTM is a parallelogram, m N 12x 10y 5,m C 9x, and m T 6x 15y, find m M.
Name 79Tests for ParallelogramsTheorem 6-7:If both pairs of opposite sides of a quadrilateral arecongruent, then the quadrilateral is a parallelogram.Notes Section 6.3Determine if each quadrilateral must be a parallelogram.Justify your answer.#1)AC55SB#2)72 Theorem 6-8:If one pair of opposite sides of a quadrilateral is bothparallel and congruent, then the quadrilateral is aparallelogram.72 AC#3)SBTheorem 6-9:If the diagonals of a quadrilateral bisect each other, thenthe quadrilateral is a parallelogram.Use parallelogram ABCD and the given information to findeach value.#4) m ABC 50 . Find m BCDACABTSBDTheorem 6-10:If both pairs of opposite angles in a quadrilateral arecongruent, then the quadrilateral is a parallelogram.C#5) AB 11, BC 2, m ADC 84 . Find DC.ABACTSBDCGeometry 79
Geometry 80#6) What values must x and y be in order for quadrilateralto be a parallelogram? ST x 3y, TA 6, PT 4x 2y andTN 14SN#7) The coordinates of the vertices of quadrilateral ABCDare A(-1, 3), B(2, 1), C(9, 2), and D(6, 4). Determine if thequadrilateral ABCD is a parallelogram.Option 1: Use the distance formula to find the length of all four sides.*If opposite lengths are the same, then the quad is a parallelogram.Option 2: Use the slope formula to find the slope of all four sides.*If opposite slopes are the same, then the quad is a parallelogram.TPOption 3: Find the slopes and lengths of one pair of opposite sides.*If the pair of opposite sides have the same slope and length, then thequad is a parallelogram.AOption 4: Find the midpoints of the diagonals.*If the midpoints of the diagonals are the same, then the quad is aparallelogram.A(-1, 3), B(2, 1), C(9, 2), and D(6, 4).Geometry 80
Name 81RectanglesNotes Section 6.4Rectangle: a quadrilateral with four right angles. (Also#2) m 1 40 . Find m 2could define as a parallelogram with one right angle.)MCT31P2HRA4TETheorem 6-11.12: A parallelogram is a rectangle IFF itsdiagonals are congruent.RETCUse rectangle MATH and given information to solve eachproblem.#3) If a quadrilateral has one pair of congruent sides, it isa rectangle.#1) HP 10. Find MT.MA1HDraw a counterexample to show that each statementbelow is false.23P4T#4) If a quadrilateral has two pairs of congruent sides, it isa rectangle.Geometry 81
Geometry 82Find the values of x and y in rectangle PQRS.#5) TR 3x – 12y, TQ -2x 9y 4, ST 3PQTSRDetermine whether ABCD is a rectangle. Explain.#6) A(1, 2), B(3, 6), C(9, 3), D(7, -1)Option 1: Use the distance formula to find the length of all four sides.Use the slope formula to find the slopes of two consecutive sides.*If opposite lengths are the same, and consecutive slopes are perpendicular,then the quad is a rectangle.Option 2: Use the slope formula to find the slope of all four sides.*If opposite slopes are parallel and consecutive slopes are perpendicular,then the quad is a rectangle.Option 3: Find the midpoints of the diagonals.Find the lengths of the diagonals.*If the midpoints of the diagonals are the same and the diagonals are thesame length, then the quad is a rectangle.Geometry 82
Name 83Squares and RhombiNotes Section 6.5Rhombus:Name all the quadrilaterals – parallelogram, rectangle,A quadrilateral with four congruent sides. (Also could berhombus, or square – that have each property.defined as a parallelogram with four congruent sides.)#1) The opposite sides are parallel.RH#2) The opposite sides are congruent.MO#3) All sides are congruent.Theorem 6-13.14:A parallelogram is a rhombus IFF its diagonals areperpendicular.RH#4) It is equiangular and equilateral.Use rhombus BEAC with BA 10 to determine whethereach statement is true or false. Justify your answer.#5) CE 10MOBTheorem 6-15:Each diagonal of a rhombus bisects a pair of oppositeangles.REHCAH̅̅̅̅ ̅̅̅̅#6) 𝐶𝐸𝐴𝐵MBOEHSquare:(a rectangular rhombus; a rhombicular rectangle.) Aquadrilateral that is both a rhombus and a rectangle.SRCAQAGeometry 83
Geometry 84Use rhombus IJKL and the given information to solve eachproblem.#7) If m 3 4(x 1) and m 5 2(x 1), find x.I421To determine if a quad is a rhombus.The midpoints of the diagonals must be the sameand the diagonals must be perpendicular3To determine if a quad is a square.The quad must be a rectangle and a rhombus.5KGeometry 84To determine if a quad is a parallelogram.The diagonals must have the same midpoint.To determine if a quad is a rectangle.The midpoints of the diagonals must be the sameand the diagonals must have the same length.6LJDetermine whether EFGH is a parallelogram, rectangle,rhombus, or square. List all that apply.#8) E(6, 5), F(2, 3), G(-2, 5), H(2, 7)
Name 85TrapezoidsNotes Section 6.6Trapezoid: a quadrilateral with exactly one pair of parallelTheorem 6-17:sides.The diagonals of an isosceles trapezoid are congruent.Bases: the parallel sides of a trapezoid.Legs:the nonparallel sides of a trapezoid.Median of a Trapezoid:a segment that connects the midpoints of the legs.Pair of base angles: two angles in a trapezoid that share acommon base.Isosceles trapezoid: a trapezoid with congruent legs.Theorem 6-18:The median of a trapezoid is parallel to the bases and itsmeasure is one half the sum of the measures of the bases.Theorem 6-16:Both pairs of base angles of an isosceles trapezoid arecongruent.Geometry 85
Geometry 86If possible, draw a trapezoid that has the followingcharacteristics. If the trapezoid cannot be drawn,explain why.#1)Four congruent sides.#2)One right angle.#3)One pair of opposite angles congruent.#4)Congruent diagonals.#6) If the measure of the median of an isosceles trapezoidis 7.5, what are the possible integral measures for thebases?̅̅̅̅ is the median of a trapezoid with bases ̅̅̅̅#7) 𝑈𝑅𝑂𝑁̅̅̅. If the coordinates of the points are U(2,and ̅𝑇𝑆2), R(6, 2), O(6, -2), N(0, -2), find the coordinates ofT and S.PQRS is an isosceles trapezoid with bases ̅̅̅̅𝑃𝑆 and̅̅̅̅𝑄𝑅 . Use the figure and the given information tosolve each problem.#5)If TV 2x 5 and PS QR 5x 3, find x.PSTVQGeometry 86R
Name 87Chapter 6 Summary1. Summarize the main idea of the chapter2. Terms (Include name and definition). Also include key example or picture for each termGeometry 87
Geometry 883. Theorems and Postulates. Also include key example for each theorem or postulate4. Key examples of the most unique or most difficult problems from notes, homework or application.Geometry 88
Name 89Chapter 7 – SimilaritySection 7.1Ratio: a comparison of two quantities.Proportion: an equation stating that two ratios are equal.Section 7.2Rate: a ratio of two measurements that may havedifferent types of units.Similar Polygons: Two polygons are similar IFF theircorresponding angles are congruent and the measures oftheir corresponding sides are proportional.Scale Factor: The ratio of the lengths of twocorresponding sides of two similar polygonsSection 7.3AA Similarity: If two angles of one triangle are congruentto two angles of another triangle, then the triangles aresimilar.SSS Similarity: If the measures of the corresponding sidesof two triangles are proportional, then the triangles aresimilar.SAS Similarity: If the measures of two sides of a triangleare proportional to the measures of two correspondingsides of another triangle and the included angles arecongruent, then the triangles are similar.Theorem 7-3: Similarity of triangles is reflexive,symmetric, and transitive.Terms, Theorems & PostulatesSection 7.4Triangle Proportionality: A line, that intersects two sidesof a triangle in two distinct points, is parallel to the thirdside IFF it separates these sides into segments ofproportional lengths.Theorem 7-6: a segment whose endpoints are themidpoints of two sides of a triangle is parallel to the thirdside of the triangle and its length is one-half the length ofthe third side.Corollary 7-1: If three or more parallel lines intersect twotransversals, then they cut off the transversalsproportionally.Corollary 7-2: If three or more parallel lines cut offcongruent segments on one transversal then they cut offcongruent segments on every transversal.Section 7.5Proportional Perimeter: If two triangles are similar, thenthe perimeters are proportional to the measures ofcorresponding sides.Proportional Altitudes Theorem: If two triangles aresimilar, then the measures of the corresponding altitudesare proportional to the measures of the correspondingsides.Proportional Angle Bisectors Theorem: If two triangles aresimilar, then the measures of the corresponding anglebisectors are proportional to the measures of thecorresponding sides.Proportional Medians Theorem If two triangles aresimilar, then the measures of the corresponding mediansare proportional to the measures of the correspondingsides.Angle Bisector Theorem: An angle bisector in a triangleseparates the opposite side into segments that have thesame ratio as the other two sides.Geometry 89
Geometry 90The quiz will consist of one matching section and one multiple choice section. The matching section will contain all terms and the theoremsthat have names.The multiple choice section will contain all theorems, postulates, and corollaries that have no names. I will remove a word from thesentence and give you three or four choices to complete the sentence.Geometry 90
Name 91Properties of ProportionsNotes Section 7.1Ratio: a comparison of two quantities.#2)109 30𝑥 2What is the ratio of female students to male students inthis class?What is the ratio to Twinkie riders to car riders in thisclass?Proportion: an equation stating that two ratios are equal.#3)𝑥 610 2𝑥 53Example:Solve each proportion.𝑥#1) 12830#4)7 𝑥9 26Geometry 91
Geometry 92#5) On a bike, the ratio of the number of rear sprocketteeth to the number of front sprocket teeth is equivalentto the number of rear sprocket wheel revolutions to thenumber of pedal revolutions. If there are 8 rear sprocketteeth and 18 front sprocket teeth, how many revolutionsof the rear sprocket wheel will occur for 5 revolutions ofthe pedal?#7) The ratio of the measures of the angles of a triangle is3:5:7. What is the measure of each angle in the triangle?#6) One way to determine the strength of a bank is tocalculate its capital-to-assets ratio as a percent. A weakbank has a ratio of less than 4%. The Gnaden NationalBank has a capital of 177,000 and assets of 4,450,000.Is it a weak bank? Explain.#8) On a map of Ohio, three fourths of an inch represents15 miles. If it is approximately 10 inches from Sanduskyto Cambridge on the map, what is the actual distance inmiles?Geometry 92
Name 93Similar PolygonsNotes Section 7.2Similar Polygons: Two polygons are similar IFF theirDetermine whether each pair of figures is similar. Justifycorresponding angles are congruent and the measures ofyour answer.their corresponding sides are proportional.#1)DA60 2cmC60 50 1.6cm70 B1.8cm#2)2.4cm3cm50 F70 E2.7cmDA60 7cmScale Factor: The ratio of the lengths of twocorresponding sides of two similar polygonsC41 90 6cm9.2cm49 5cm4cmBF50 53 3cmEDraw and label a pair of polygons for each. If it isimpossible to draw two such figures, write “Mission:Impossible.”#3) two pentagons that are similar#4) two squares that are not similarGeometry 93
Geometry 94Given two similar polygons find the value of x and y.#5)Make a scale drawing using the given scale.#8) A basketball court is 84 feet by 50 feet.1Scale: inch 2 ft.88103y6x#6)51880 x12y80 IF quadrilateral PQRS is similar to ABCD, find the scalefactor of quadrilateral PQRS to quadrilateral ABCD.#7)BQ11R33C110 110 155541810 A10 P7Geometry 9421SD
Name 95Similar TrianglesNotes Section 7.3AA Similarity: If two angles of one triangle are congruentTheorem 7-3: Similarity of triangles is reflexive,to two angles of another triangle, then the triangles aresymmetric, and transitive.similar.ReflexiveSSS Similarity: If the measures of the corresponding sidesof two triangles are proportional, then the triangles aresimilar.SymmetricTransitiveSAS Similarity: If the measures of two sides of a triangleare proportional to the measures of two correspondingsides of another triangle and the included angles arecongruent, then the triangles are similar.Geometry 95
Geometry 96#1) Determine if each pair of triangles is similar. If similar,state the reason and find the missing measure.#3) If TS 6, QP 4, RS x 1, and QR 3x – 4, find thevalue of xT94RQ2S63x#2) In the figure, ST // PR , QS 3, SP 1, and TR 1.2.Find QT.P#4) Identify the similar triangles in each figure. Explainyour answer.AQSTDP90 RCGeometry 96B
Name 97Parallel Lines & Proportional PartsTriangle Proportionality: A line, that intersects two sidesof a triangle in two distinct points, is parallel to the thirdside IFF it separates these sides into segments ofproportional lengths.Notes Section 7.4#1) Find the value of x.710Midsegment: A segment in a triangle with endpoints thatare the midpoints of two sides of the triangle.12xTheorem 7-6: A midsegment is parallel to the third side ofthe triangle and its length is one-half the length of thethird side.Corollary 7-1: If three or more parallel lines intersect twotransversals, then they cut off the transversalsproportionally.#2) Determine if BD // AE .CA 15, AB 3, CD 8, CE 10CBADECorollary 7-2: If three or more parallel lines cut offcongruent segments on one transversal then they cut offcongruent segments on every transversal.Geometry 97
Geometry 98#3) Find the value of x.2x#5) Find the value of x and y.4 x36268xy#6) Find the value of x.#4) Find the value of x.13x9x 416x131214Geometry 989
Name 99Parts of Similar TrianglesProportional Perimeter Theorem: If two triangles aresimilar, then the perimeters are proportional to themeasures of corresponding sides.Notes Section 7.5Angle Bisector Theorem: An angle bisector in a triangleseparates the opposite side into segments that have thesame ratio as the other two sides.Proportional Altitudes Theorem: If two triangles aresimilar, then the measures of the corresponding altitudesare proportional to the measures of the correspondingsides.#1) Find the value of x.9x8Proportional Angle Bisectors Theorem: If two triangles aresimilar, then the measures of the corresponding anglebisectors are proportional to the measures of thecorresponding sides.5#2) Find the value of x.xx 346Proportional Medians Theorem: If two triangles aresimilar, then the measures of the corresponding mediansare proportional to the measures of the correspondingsides.Geometry 99
Geometry 100#3) ABC is similar to XYZ. Segments ̅̅̅̅𝐴𝐾 and ̅̅̅̅𝑄𝑋 aremedians of the triangles.AK 4, BK 3, YZ x 2, QX 2x – 5. Find QZ. ABC is similar to XYZ. Determine if each proportion istrue or false.XZAYQBKCYAZXB#4)#6)Geometry 100Q𝐴𝐵𝑋𝑌𝐵𝐶𝑌𝑍 ��𝐶𝐴𝐵𝐴𝐾 𝑋𝑄𝑌𝑍𝑋𝑌𝑋𝑄
Name 101Chapter 7 Summary1. Summarize the main idea of the chapter2. Terms (Include name and definition). Also include key example or picture for each termGeometry 101
Geometry 1023. Theorems and Postulates. Also include key example for each theorem or postulate4. Key examples of the most unique or most difficult problems from notes, homework or application.Geometry 102
Name 103Transformations – IsometriesNotes T.1 (G.CO.A.2)An ISOMETRIC TRANSFORMATION (RIGID MOTION) is aA NON-ISOMETRIC TRANSFORMATION (NON-RIGIDtransformation thatMOTION) is a transformationSynonym for isometryIsometric TransformationsNon-Isometric TransformationsRotations, Translations, & ReflectionsDilations and StretchesThis is aThis is CD'D'DF'BB'This is aThis is aFEIJHJ'F'G'E'H'This is also aThis is aL'LMKGI'M'LMKNL'M'K'N'K'Geometry 103
Geometry 1041.Circle which of the following are isometrictransformations? (there may be more than 1 answer)And determine which transformation took place bywriting reflection, translation, rotation, dilation,stretch or other under each image.3.Determine the coordinates of the image, plot theimage and determine if it is an isometrictransformation.B4Pre-Image2AC-55-2-4Image AImage BImage Ca) Pre-Image Points2.Circle which of the following are isometrictransformations? (there may be more than 1 answer)And determine which transformation took place bywriting reflection, translation, rotation, dilation,stretch or other under each image.A (-1,1)B (0,4)C (4,1)Isometry? Yes or NoTransformationCoordinate Rule(x,y) (-y, x)Image PointsA’ ( , )B’ ( , )Transformation Type:C’ ( , )Pre-Image4.Determine the coordinates of the image, plot theimage and determine if it is an isometrictransformation.4Image AImage BImage CB2-5A5C-2-4a) Pre-Image PointsA (0,0)B (1,3)C (5,0)Isometry? Yes or NoTransformationCoordinate Rule(x,y) (x, -2y)Image PointsA’ ( , )B’ ( , )Transformation Type:Geometry 104C’ ( , )
Name 105Transformations – SymmetryWhat does it mean to carry a shape onto its
AAS and ASA Notes Section 4.3 ASA Congruence Postulate (Angle-Side-Angle) If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent. AAS Congruence Postulate (Angle-Angle-Side) If two angles and a
