ANGLE PAIRS In Two Lines Cut By A Transversal

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Name Per2.1 Angle Relationships in Parallel LinesVocabularyParallel linesSkew linesPerpendicular linesTransversalExample 1:1.2.Fill in the blank with parallel, perpendicular, or skewFill in the blank with parallel, perpendicular, or skew.(b) ̅ is to ̅̅̅̅̅. (c) ̅̅̅̅ is to ̅̅̅̅.(b) ̅̅̅̅ is to ̅̅̅̅. (c) ̅̅̅̅ is to ̅̅̅̅ .ANGLE PAIRS in two lines cut by a transversalCorresponding anglesConsecutive (same side) interior angles correspondingpositions. same side between the two linesAlternate interior anglesAlternate exterior angles alternate sides between the two lines alternate sides outside the two linesOther angle relationships that you will need to remember Vertical anglesLinear Pair opposite s with thesame vertex adjacent s that makea straight line1

Example 2: Classify the pair of numbered angles. Identify all pairs of the following angles.5. Identify the relationship between each pair of angles, ifany.b. Alternate interior angles2183 46 5c. Consecutive interior angles7d. Alternate exterior angles1) 1 and 74) 3 and 8e. Vertical Angles2) 4 and 65) 3 and 53) 8 and 76) 2 and 4a. Corresponding anglesf. Linear PairsWHEN LINES ARE PARALLEL! (magic happens HARRY POTTER!)Corresponding Angles PostulateIf two parallel lines are cut by atransversal, then pairs ofcorresponding anglesa .a1b2Statements1. 𝑎 𝑏Reasons1.2. 2.Statements1. 𝑎 𝑏1.2. 2.Alternate Interior Angles TheoremIf two parallel lines are cut by atransversal, then pairs ofalternate interior angles are.a3b4ReasonsAlternate Exterior Angles TheoremIf two parallel lines are cut by atransversal, then pairs of alternateexterior angles are .a5b6Statements1. 𝑎 𝑏1.Reasons2. 2.Consecutive Interior Angles TheoremIf two parallel lines are cut by atransversal, then pairs of consecutiveinterior angles are.a7b82Statements1. 𝑎 𝑏1.Reasons2. & are supp.3.2.3.

Example 3: Use the diagram below to find the angle measures. Explain your reasoning.1. If the the 4. If the 2. If the the what iswhat is the5. If the what is what is theExample 4: Finding all the angle measures.If and , find the measures of all the angles formed bythe parallel lines cut by the transversal.pq𝑚 𝑚 𝑚 𝑚 𝑚 𝑚 𝑚 𝑚 3. If the what is the6. If the the what isDO YOU NOTICE APATTERN? Describe it!ETHE HARRY POTTER SCAR! any angle with a dotFind its vertical and mark it with a dotCopy the same dot pattern on the other parallelConnect the dotsBDA If they both have a dot or are both blank (SAME) F If one has a dot and the other it blank (DIFFERENT) Example 5: If ̅̅̅̅ ̅̅̅̅, are the angles congruent or supplementary?1. and 2. and DBFHCGE3. and 4. AExample 6: Solve for x and explain your reasoning.1.2.3and 2. and 5. and

2.2 ConversesVocabularyConditional StatementEx: “If you have visited the statue of Liberty, then you have been to New York.”ConverseEx:Example 1: Write the converse of the given statement.1. If an animal has wings, then it can fly.2. If you are student, then you have a student I.D. card.3. All sharks have a boneless skeleton.4. All police officers eat donuts.Example 2: (a) Write the converse of the true statement. (b)Then decide whether the converse is trueor false. If false, provide a counterexample.1. If an animal is an owl, then it is also a bird.2. If two lines form right angles, then they areperpendicular.3. If an angle measures 130 , then it is obtuse.4. If two angles are adjacent, then they are congruent.Checkpoint1. Findbelow.E a counterexample to the statementEEEIf two angles are supplementary, then they are formed by two parallel lines cut by a transversal.a.b. Bc. Bd.BAA12F1A2D1DAD2FB12FF2. Write the converse of the statement below. Then determine whether each statement is true or false. If false, give acounterexample.Conditional Statement:If two angles are right angles, then they are congruent.Converse:4T or FT or F

2.3 Parallel & Perpendicular LinesExample 1: Solve for x and explain your reasoning. – 8)(3x 10)(11x 2y)Baby Proofs1. Given: ;Prove: M 1I2. Given: ;J Prove: aba3LJK2KM 4bLN3. Given: ;Prove: N 4. Given:Prove:5 ;

MORE PROOFS1. Given: ; Prove: ; Statements; 3. Given: ̅̅̅̅ ̅̅̅̅ Statements1. ̅̅̅̅ ̅̅̅̅ Reasons1. Prove: ;Statements1. 3sm2Reasons1. , 1rReasons1.nmProve: 1. 434. Given:1.Reasons n2Prove: ;m12. Given:Statements1.n6

PERPENDICULAR LINES So if two adjacent angles areTwo lines that form fourformed bym.lines, then theyare .nExample 2: Perpendicular Lines⃗⃗⃗⃗⃗ , complete the sentence using your new vocabulary.1. a. Given ⃗⃗⃗⃗⃗ b. IfVis a angle, because the definition of . and Tthe find the value of x. Explain.R2. If ̅̅̅̅S̅̅̅̅ , solve for x and y. Explain.PF4x-2 5x 11 Ey NQLET”S KEEP PRACTICING THOSE ANGLE NAMES!Name the angle pair. Then state if they are congruent or supplementary.̅̅̅̅ ̅̅̅̅a. e. b. f. c. g. d. h. 7GO

2.4 Perpendicular lines ProofsPERPENDICULAR LINES in proofGiven: 𝑚Statements1. 𝑚𝑛1 mn𝑛Reasons1. ANGLES CONGRUENCE THEOREMStatementsaAll right angles are .1. 𝑎4x-2 1 25x 11 by Example 2: Using Perpendicular lines in a proof.⃡⃗⃗⃗⃗ ,1. Given: ⃡⃗⃗⃗⃗Statements 1. ⃡⃗⃗⃗⃗ ⃡⃗⃗⃗⃗ , Prove: 2. KPAT3. Given: ̅̅̅̅ ̅̅̅̅ ;̅̅̅̅ ̅̅̅̅Prove: DBCReasons1. Angle Addition Postulate5. ̅̅̅̅Ais a right angle𝑏Statements̅̅̅̅ ; ̅̅̅̅Reasons̅̅̅̅1.PROVING LINES PARALLEL**REMEMBER: Magic happens only if the lines are parallel, so You can use angle measures to PROVE lines are parallel!When to use the CONVERSE!!!a b a b8

Example#1: Determine whether each set of lines are parallel or not. Explain!a)b)c)d)111 aOPROOFS1. Given: Prove: AIMR12mb13P4baQ2. Given: ; 12Prove: nN2PN2. Given: ; RAIProve: AJStatementsReasons1. ; 1.c13dP42QN3. Given : 10 9 ;AJ Prove:910,Statements1. 10 9 ; ,Statements1. and are supplementary, m Prove: m b5 68 1 910 3 a1 24 35 68 72. Given: and aresupplementary,aReasons249Reasons1.

Rememberthis?Transitive Property3. Given: , , Prove: If a b and b c, thenStatements1. , ,Reasons1. 35621PERPENDICULAR TRANSVERSAL THEOREMIfand,thenProofs4. Given:,1. 𝑚Statement1.2.4.k1. Given , 2.3.4. g5. Given:Prove:Reason3.6 57 8h,,Statement, 1.,,Reason1. gf76 83 12 45mn6. Given: m n;Prove: Statement1.34 m5Reason1.n10nt1. given2.,2 13 4sReasons2. Prove:Statements𝑡; 𝑛 𝑡m

2.5 Review Multiple Choice1. In the diagram line r is parallel to line s. Which of thefollowing statements must be true?A. m 3m 5B. m 5m 4C. m 2m 3 180D. m 2m 42. Given: line t line s and neither is perpendicular to lineg. Which of the following statements is false?stA. m 2 m 5 180g5 6317 8B. m 1 m 742C. m 3 m 5 18012r354s6D. m 23. In the diagram ⃡⃗⃗⃗ ⃡⃗⃗⃗⃗⃗ and m YRH 100 . Which ofthe following conclusions does not have to be true?A. m MHFB. m RHMC. SRT andD. SRY and4. Based on the diagram, which theorem or postulate wouldsupport the statement m RIP m SMY ?A. Alternate ExteriorAngles TheoremB. Alternate InteriorAngles TheoremC. Consecutive InteriorAngles TheoremD. Corresponding Postulate10080MHF are alternate exterior anglesRHV are alternate interior angles5. In the diagram below, . Which of the followingmust be true?rs1A.2 5 673B. 48tC. D. 6. Which type of angles are a counterexample to theconjecture below?“If two lines are parallel, then each pair of angles areAAsupplementary”.A. B. C. D. 7. In the diagram to the right, ⃡⃗⃗which angles are congruent?⃡⃗⃗⃗⃗ and ⃡⃗⃗m 3⃡⃗⃗⃗ thenCD348. In the diagram below, BBthe following does not have to be true?A. , A. B. , B. C. C. D. , D. . Which ofr12 5 6s4738t 10. In the diagram below, which pair of angles are alternateinterior angles?H9. Allison wanted to solve for , so she set up the equation. What would herreasoning be?J𝑥“If two parallel lines are intersected bya transversal, then 1 2𝑥A. and B. and SVRMTA. linear pairs are supplementary.”B. corresponding angles are supplementary.”C. alternate interior angles are congruent.”D. consecutive (same-side) interior angles are supplementary.”11C. and D. and KGXYL

11. Use the diagram to determine which of the pair ofangles is alternate exterior angles.A. 1 and 15B. 9 and 1512. To solve for x in the diagram below, Betty used theequation.ab1 23 48 7m11 12n6 5Betty can justify herequation by thefollowing statement:C. 4 and 119 1014 13D. 2 and 816“If two parallel lines are intersected by a transversal, then .15A.B.C.D.13. Use the diagram to determine which of the pair ofangles is corresponding angles.abA. 2 and 10B. 8 and 11C. 4 and 10D. 10 and 12EXTRA PRACTICE1. Given: ,Prove:m1l14 31 21 26 53 48 79 1014 1316alternate interior angles are congruent.alternate exterior angles are congruent.corresponding angles are congruent.consecutive interior angles are supplementary.14. Use the diagram to determine which of the pair ofangles is consecutive interior angles.A. 3 and 11B. 13 and 16C. 9 and 13D. 10 and 13mn11 1215,Statements1. , n1p12. Solve for x and y. Explain your reasoning for each equation you set up!(14x – 22) (2x 5y) (11x 14) 12b1 26 53 48 79 1014 1316Reasons,6 75 8a1.m11 1215n

2.5 Perimeter & AreaFormulas for Perimeter (P), Area (A), and Circumference (C)Rectangle or SquareTriangleP P A A b , h b , h Circle**NOTE** Height is always to the baseC Perimeter, Circumference:A r units (Ex: ) Area:Example 1: Find perimeter, circumference, and area1. Find the perimeter and area of the rectangle.2. Find the circumference and area of the circle. Leaveyour answers in terms of π.3. Find the perimeter and area of the figure.4. Find the perimeter and area of the figure.10 yd5. Find the area and circumference of the circle inscribed inthe square.6. Find the perimeter and area of the figure.15 yd12 yd13

AREA ON A COORDINATE PLANE (2 methods) Subtract values Count the units(4, 10)(4, 2)Example 2: Find the area of the figure shown.1.(2, 10)2.(10, 10)(13, 12)(2, 4)(2, 3)(13, 2)(13, 4)(10, 3)3.4.(4, 7)(9, 7)(4, -3)(9, -3)(7,7)(1, 2)(5, 2)Example 3: Find unknown length1. The base of a triangle is 12 feet. It’s area is 36square feet. Find the height of the triangle.3. The perimeter of a square is 128 inches.a. Find the length of one side of the square.2. The area of a rectangle is 243 square meters. The rectangle is threetimes its width. Find the length and width of the rectangle.4. The circumference of a circle is 14 centimeters. Findthe area of the circle.b. Then find the area of the square.14

SPIRAL REVIEWI. Points, Lines, Planes Collinear:Use the diagram below.a. Name a point that is collinear with C, S, and P.b. Name a point that is coplanar with A, C, andD.c. Circle the correct set of 3 collinear points.B, K, LK, M, LP, I, LG, S, C Coplanar:d. Circle the correct set of 4 coplanar points.G, O, J, PK, I, J, FL, C, I, OA, C, S, CII. Addition Postulates1. A is between H and T. If HA ,AT , and HT 35, solve for x and explain.2. If find EXTRA PROOF PRACTICE1. Given: ; Prove: and are complementaryProve: 2. Given:15, andand explain.

2.7 Composite AreaPARTIALLY SHADED FULLY SHADED – Arearegion Arearegion Areashaded – Areaunshaded Areashaded AreashadedExample 1: Fins the area of the shaded region.1. Find the area of the shaded region.2. Find the area of the shaded region. (Round to the tenths)18 in8m6 in8m4 in3. Find the area of the shaded region4. Find the area of the shaded region 2 cm6 yd11 cm6 yd5. Find the area of the figure below comprised of arectangle and a semicircle. 6. Find the area of the figure below comprised of a squareand a right triangle.16

Example 2: Composite figures1. Find the area and perimeter of the figure below if all linesegments meet at right angles. (Figure not drawn to scale)2. Find the area and perimeter of the figure below if all linesegments meet at right angles. (Figure not drawn to scale)SPIRAL REVIEWFor every 90 1 quadrant overRotationsSwitch #’s1. Ifis rotated 902. Ifis rotated 180counterclockwise about the origin,clockwise about the origin, thenthen what would be the coordinateswhat are the coordinates of itsof the new point?image?Translations1. Iftranslates totranslated to what point?thenisPRACTICE MAKES PERFECT!1. In the figure, ̅̅̅̅ and ̅ are intersected by ̅̅̅̅. and which of the following angles areknown as corresponding angles?KA. 𝐽𝑀𝑁1 2 B. 𝐽𝑀𝐿34ND. 𝐼𝑀𝐿2. Iftranslates totranslated to what point?Mthenis2. You are planting grass on a rectangular plot of land. You are alsobuilding a fence around the edge of the plot. The plot is 45 yardslong and 30 yards wide. How much area do you need to coverwith grass s eed? How many yards of fencing do you need?HGC. 𝑁𝑀𝐼3. Ifthen what are thecoordinates of its image after arotation 90 clockwise about theorigin?JIL3. Solve for x and explain your reasoning.17

K4.a. Write an equation that can be used to find the value of x and justify yourequation.H6x 8 7x-12 b. Find the value of x.3x-8 Hc. Find the measure of one of the acute angles.L5. Find the area of the triangle formed by thecoordinatesand.6. a) Write the converse of the statement.If two angles formed by parallel line cut by atransversal are corresponding angles, then they arecongruent.b) Is the converse true or false? If false, give acounterexample. and are complementaryProve: 2. Given:3. Given: ,Prove: a12b3 64 5r7 108 9s18


C. SRT and MHF are alternate exterior angles D. SRY and RHV are alternate interior angles 4. Based on the diagram, which theorem or postulate would support the statement RIPm SMY? A. Alternate Exterior Angles Theorem B. Alternate Interior Angles Theorem C. Consecutive

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