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Problem of the Day—(+6pts Collect) **Take/Return Test and Short Notes** “If the midpoint of line CW is (7, 4) and point C is (9, -2), what is the coordinate of point W?” Section 4.0 “Triangles + Classifying Triangles” What is a Triangle? “A polygon with three angles (vertices) and three side (edges) that are line segments.” A D H Example #1: I. Name this Triangle? II. Name the 3 Vertices? III. Name the 3 Sides? Classifying Triangles **To Classify Triangles By (1) Angles and (2) Sides** By Angles: 1. Right—One Right, 90, Angle . 2. Equiangular—All Angles are Equal. 3. Acute—All Angles are Less than 90. 4. Obtuse—One Angle more than 90. By Sides: 5. Equilateral—All Sides are Equal, or Congruent 6. Isosceles—Two Sides are Equal, or Congruent. 7. Scalene—No Sides are Equal, or Congruent. Ex. #2: Classify each Triangle I. 5 I. Equiangular, Equilateral Triangle II. Obtuse, Isosceles Triangle 5 5 II. 12 12 104 III. Acute, Scalene Triangle III. All Angles Less then 90 and All Sides are not equal Worksheet “Classifying Triangles” Problem of the Day **Check Homework and Return Tests** Review SOL Problem: Write the Equation of the Circle whose center is (5, 7) and an outside point is (8, 11)? ( x h) ( y k ) r 2 2 2 Problem of the Day—6th Hour **Check Homework and Return Tests** 2 Review SOL Problems: 1. Write the Equation of the Circle whose center is (5, 7) and an outside point is (8, 11)? ( x h) ( y k ) r 2 2. 2 2 QR bisects <PQS. If <PQR = 3x + 4 and <SQR = 5x – 10. Find x and Find m<PQS? “Triangle Angle Sum” Investigation: Sum of a Triangle Worth: + 5Points 1. 2. 3. 4. 5. 6. Cut out the Triangle. Number the Angles Place the three angles adjacent, or next to, each other to form one angle. Glue Together as so. Answer Two Questions. Collect with Name. 2 1 3 2 3 1 Q#1: What angle is made by Angles 1, 2, and 3 after being cut out and taped together? Q#2: What is the sum, or what do the 3 angles, of a triangle add up to be? Triangle Angle-Sum Theorem “The sum of the three angle measures of a triangle add to be 180 Degrees.” m<A + m<B + m<C = 180 B A C How Many Degrees are Equiangular Triangles? 180 Degrees = 3 Angles 60 Degrees Ex. #1: Find m<B? B A Angles = 180 Degrees 117 <A + <B + <C = 180 117 + <B + 33 = 180 <B + 150 = 180 <B = 30 Degrees 33 C Example #2 Find <1, <2, and <3? 62 59 1 53 2 3 Ex. #3: Find x and y? y x 41 x: 180 = 90 – 41- x 49 = x y: “Vertical Angles Equal” If x = 49, then y = 49 Ex. #4: I. Find x? II. Find m<A; m<B; m<C? A x 2x + 11 B 2x + 4 C What are Exterior Angles? “Outside Angles formed by a side and an extension of an adjacent side. For the exterior there is two inside angles or remote interior angles.” m<1 = m<2 + m<3 1 Exterior Angle 2 3 Remote Interior Angles Ex. #5: Find m<1? 40 1 m<1 = m<2 + m<3 <1 = 40 + 30 <1 = 70 30 Ex. #6: Find m<2? 2 113 m<1 = m<2 + m<3 113 = <2 + 73 <2 = 40 73 Ex. #7: Find x and y? 31 y x 72 x = 77 and y = 103 Ex #8: Find x, y, and w? x: 180 = 65 – 39 – x 76 = x y: 180 – x = y 180 – 76 = y 104 = y 21 39 65 x y w w: 180 – 21 – y = w 180 – 21- 104 = w 55 = w Try Example #9 if needed Find x, y, and w? x = 70 y = 110 w = 30 40 30 80 x y w 1. Worksheets “Triangle Angle Sums” 2. Quiz (Next Class) --Name/Classifying Triangles --Solving Missing 180-degree Angles of Triangles Worth: +36pts Due: Next Tuesday, November 27th Problem of the Day **Check HW, then Quiz, then Short Notes** Try these SOL Triangle Problems: 1. I. B. 45 Degrees and II. C. 135 Degrees 2. C. 69 Degrees 3. B. 28 Degrees Then Short Notes “Congruent Polygons” Congruent Polygons “Two polygons that have matching corresponding Angle and Side Parts.” ‘Name in correct Angle Order’ Turn Turn Ex. #I: Tell the Congruent Triangles G GJH ______ J H M N T Ex. #II: Find Missing Parts of these Congruent Polygons Find x and y? Find x, y, and w? 56 x x 65 115 w y y 42 in 56 Ex. #III: Name the Corresponding, Side and Angle, Parts B D P X N A C R <X =<N <B =<R <A =<P <C =<D XB =RN AC =DP AX =PN BC =DR XBCA = NRDP Try Examples 1, 2, & 3: Name the Corresponding Parts Example #1 I. II. III. IV. HEV EH EV HV Example #2 I. A II. B III. V IV. T V. ABVT Example #3 x = 90 degrees y = 145 degrees w = 72 degrees 1. Worksheet—Congruent Figures 2. SOL Homework #4 --1st, 5th, and 7th Blocks Due Monday, November 26th --2nd and 6th Blocks Due Tuesday, November 27th Problem of the Day (11/26 1,5,7th Hours) **Check HW and Collect SOL HW #4 + Honors Activity (Tomorrow)** Do and Complete Problems 1 thru 3 Problem #1 I. Triangle RTH II. TR III. KQ IV. MK V. <H VI. <Q Problem #2 x = 90 degrees y = 156 degrees z = 114 degrees Problem #3 x = 54 degrees y = 54 degrees w = 41cm 4 Types of Triangle Congruence Example #1 Find x w/Congruent Triangles Given:If QRS = TUV and QS = 3x + 2 and TV = 7x – 6, Find x? Find QS and TV? Given:If QRS = TUV and QR = 5x + 2 and TU = 7x – 10, Find x? Find QR and TU? 1. SSS (Side-Side-Side) “If all three sides are equal, then triangles are congruent.” Shown: Reflexive Property “Triangles share middle line and equal to each other” A M D C AC = AC by “Reflexive Property” 2. SAS (Side-Angle-Side) “If two sides and the middle angle are equal, then triangles are congruent.” Shown: 3. ASA (Angle-Side-Angle) “If two angles and the middle side are equal, then triangles are congruent.” Shown: A S A A S A 4. AAS (Angle-Angle-Side) “If the two angles and the outside side are equal, then triangles are congruent.” Shown: A A A S A S Vertical Angles “Opposite Angles Equal” Try Example #2: Congruent by SSS, SAS, ASA, or AAS I. SSS II. ASA III. SAS IV. SAS V. ASA VI. SSS VII. ASA VIII.AAS IX. SAS X. AAS XI. SAS Example #3: Tell if SSS, SAS, ASA, or AAS w/Givens R B D X G H S Given: 1. SG = SH 2. RG = RH C Given: 1. BX = HX 2. <CBX = <DHX H 1. Worksheet—SSS, SAS, ASA, AAS 2. POP QUIZ 3. If not done, SOL HW #4 #1 Problem of the Day + POP QUIZ **Check HW** For #1-6 Pop Quiz, Tell if SSS, SAS, AAS, or ASA: 1. 2. 4. 5. 3. 6. Proving the 4 Types of Triangle Congruence Proofs: “Two Column” Column 1: Column 2: Statements 1. 2. 3. 4. Reasons 1. 2. 3. 4. Hints: Statement 1 Write Given Statement Statement 2 Write Given Statement Statement 3 You Provide Statement 4 Re-write the Prove Sentence from top Hints: Reason 1 Given Reason 2 Given Reason 3 You Provide Vertical Angles (<PTS = <CTS) or Reflexive Property ( ET = ET) Reason 4 SSS, SAS, ASA, or AAS Example #1 Reasons 1. 2. 3. 4. Statements 1. 2. 3. 4. Given 2. Given 1. AB = CB 2. AD = CD 1. 3. BD = BD 3. Reflexive Property 4. ABD = CBD 4. SSS Example #2 Statements 1. 2. 3. 4. Reasons 1. 2. 3. 4. 1. AC = EC 2. BC = DC 1. 3. <ACB = <ECD 3. Vertical Angles 4. ACB = Given 2. Given ECD 4. SAS Example #3 Statements 1. <Q = <S 2. <TRS = <RTQ 3. RT = RT 4. QRT = STR Reasons 1. Given 2. Given 3. Reflexive Property 4. AAS Example #4 Statements 1. LK = LM 2. LP = LJ 3. <MLP = <KLJ 4. JKL = PML Reasons 1. Given 2. Given 3. Vertical Angles 4. SAS Example #5 Statements 1. JK = MK 2. <GKJ = <GKM 3. GK = GK 4. GJK = GMK Reasons 1. Given 2. Given 3. Reflexive Property 4. SAS Problem of the Day, then Activity Given: 1. <A = <M 2. AX = MX A Y X Prove: AXT = MXY T Reasons Statements M 1. <A = <M 2. AX = MX 1. 3. <AXT = <MXY 3. Vertical Angles 4. AXT = Given 2. Given MXY 4. ASA Worth: +20 Points “Fill in the Statement, Reasons, and Proofs” Worksheet—Proving SSS, SAS, ASA, or AAS 2. Quiz (Next Tuesday December 4th for 1st, 5th, and 7th Blocks) (Next Wednesday December 5th for 2nd and 6th Blocks) 1. Solving for Congruent Polygons SSS, SAS, ASA, AAS Congruence Proofs of SSS, SAS, ASA, and AAS Proofs of CPCTC Problems HL Theorem Reflexive and Vertical Angle Properties Problem of the Day **Check Worksheet** Do and Complete Proofs 1 and 2 Statements and Reasons 1. 1. 2. 2. 3. 3. 4. 4. Statements and Reasons 1. 1. 2. 2. 3. 3. 4. 4. 1. HL Theorem 2. Proving w/CPCTC Right Triangle Parts Hypotenuse (H) Leg (L) Leg (L) Longest Side Opposite Right Angle = Hypotenuse Two Shorter Sides touching Right Angle = Legs Hypotenuse- Leg ‘HL’ Theorem “If 1 Hypotenuse, 1 Leg, and 1 Right Angle are Congruent, then the right triangles are congruent.” **Two Right <‘s, Two Equal Legs, Two Equal Hypotenuses** Ex #1: Name the Two Right Triangles equal by ‘HL’ O 3 N T 3 M 5 E 5 5 V B W 3 P PMN = TVW Ex #2: Congruent by ‘HL’? I. HL (yes/no)? II. HL (yes/no)? YES, by HL NO, by HL Ex #3a: What extra information do you need to prove these Triangles equal by ‘HL’ Q R B T D C <Q and <C are Right Angles Ex #3b: What extra information do you need to prove these Triangles equal by ‘HL’ B J H M Legs BJ = BH or Legs JM = HM Ex #3c: What extra information do you need to prove these Triangles equal by ‘HL’ B J H M Hypotenuse BM = BM Problem of the Day 1. Fill in the 2 Missing Blanks Prove: Triangles BAH = MAC B H th (7 Hour) 2. What extra information do you need to prove these Right Triangles equal by ‘HL’ Theorem? X A C M Proof: 1. BA = MA 1. Given 2. CA = HA 2. Given 3. __________ 3. Vertical Angles 4. BAH = MAC 4. ____________ E A J CPCTC “Corresponding Parts of Congruent Triangles are Congruent” ‘After proving Triangles Congruent by SSS SAS ASA or AAS, you prove the Remaining Sides or Angles by CPCTC.’ Statement #5: Side = Side or Angle = Angle Reason #5: “CPCTC” Example #4 Statements 1. 2. 3. 4. 5. Reasons 1. 2. 3. 4. 5. 1. AC = EC 2. BC = DC 1. 3. <ACB = <ECD 3. Vertical Angles ACB = ECD 5. AB = ED 4. SAS 4. Given 2. Given 5. CPCTC Example #5 Statements 1. 2. 3. 4. 5. Reasons 1. 2. 3. 4. 5. 1. <DCH = <MCH 2. <DHC = <MHC 1. 3. CH= CH 3. Reflexive Property DCH = MCH 5. <D = <M 4. ASA 4. Given 2. Given 5. CPCTC +5 Points Worksheet—HL Theorem and CPCTC 2. Quiz (Tuesday December 4th for 1st, 5th, and 7th Blocks) (Wednesday December 5th for 2nd and 6th Blocks) 1. Solving for Congruent Polygons SSS, SAS, ASA, AAS Congruence Proofs of SSS, SAS, ASA, and AAS Proofs of CPCTC Problems HL Theorem Reflexive and Vertical Angle Properties Problem of the Day + Collect Honors Activity **Check HW, Short Quiz, then Short Notes** 1. C. AC = EC 1. Given 2. Given 3. Vertical Angles 4. ASA 5. CPCTC 3. x = 5 and AC = 39 2. Then Short Notes Problem of the Day th (7 What is the Measurement of: I. m<MTV? II. m<TVB? III. m<NVB? T 63 M 65 V B N Hour) “Equilateral Triangles” Equilateral Triangles “If a triangle is Equilateral, then the triangle has (1) all equal 60 angles and (2) all equal sides.” XY = YW = WX = 12 <X = <Y = <W = 60 X 12 Y W Ex #1A. Two sides of an Equilateral Triangle have lengths of 2x + 4 and x + 8. I. Find x? II. Find the Length of each Side? Ex #1B. Two sides of an Equilateral Triangle have lengths of 3x + 12 and 7x - 8. I. Find x? II. Find the Length of each Side? Ex #2. Equilateral Triangle Find x and w? w 10x – 5 25 w w 25 Ex #3. Equilateral Triangle Find x? 8x + 4 Ex #4. Equilateral Triangle Find x, y, z, and w? 11w z x 6w + 30 x y Worksheet—Equilateral Triangles 2. Late/Missing Work 1. Problem of the Day **Check Homework and Return Quiz** In this Equilateral Triangle Problem, Find x, y, w, and z? x 2w + 10 8z – 12 5w - 2 x y “Isosceles Triangles” Investigation: Isosceles Triangles Worth: + 12 Points 1. 2. 3. 4. 5. 6. 7. 8. C Make a 5in Line. Open the compass up 4in. A B Make two arcs in the middle to create a connection point. Draw two lines to the connection point to create a triangle. Label Angles A, B, and C. DD Draw a Straight Line from Angle C down to make angle D. Answer Questions. Q#1: What are the Measures of Angles A, B, C, and D? Collect with Name. Q#2: What do you notice about the Measures of Angles A and B? Isosceles Triangles 1. Two equal leg sides. 2. Two equal base angles below the two equal sides. C A B Base Angles <A = <B AC = BC Thus, Isosceles Triangles 2 Equal Sides and 2 Equal Base Angles. Isosceles Triangle Word Problem #1 “If the legs of an Isosceles Triangle have lengths 2x + 4 and 1x + 8 and the base bottom side has a length 5x -2, What is x? What is the length of the base?” Isosceles Triangle Word Problem #2 “What is the measure of the top missing third vertex angle of an isosceles triangle if both base angles measure 42 Degrees and 42 Degrees?” Isosceles Triangle Examples #1 Find x and y for Both? I. II. 32 23 y 54 y x x 112 x x Isosceles Triangle Example #2 I. Find Angle F? II. Find Angle E? E 118 F G Isosceles Triangle Examples #3 Find x and w for Both? I. II. w w x x 47 x 130 Isosceles Triangle Examples #4 Find x, y, and w for Both? III. IV. w w 2x + 50 y x 53 7x y +9 Points 1. Worksheet on Isosceles Triangles 2. UNIT 4 TEST (Next Friday , Dec. 14th or Monday, Dec. 17th) Triangles Congruent Polygons Congruent Triangles SSS, SAS, ASA, AAS Problems SSS, SAS, ASA, AAS Proofs CPCTC and CPCTC Proofs HL Theorem Equilateral, Isosceles, and Right Triangle Problems and Solving for Missing Variables Problem of the Day **Check HW and Test Friday/Monday** 1. Slope = 0 2. (x + 1)^2 + (y – 10)^2 = 25 3. x = 32 and w = 45in 4. x = 10, y = 130, and w = 80 “Overlapping Congruent Triangle + O.C.T Proofs” Overlapping Triangles “Triangles that share a common side and/or common angle on top of each other.” Ex #1: Overlapping Triangles What common sides and common angles are shared between Triangles ACD and EDC? A E C D Ex #2: Overlapping Triangles What common sides and common angles are shared between Triangles ACD and ECB? A E B D C Ex #3: Reasons in this Overlapping Triangle Proof Statements 1. BA = DE 2. CA = CE Reasons C 3. <CAE = <CEA B D 4. AE = AE X 5. Triangle BAE = Triangle DEA A E Given: 1. BA = DE 2. CA = CE 6. <ABE = <EDA Prove: <ABE = <EDA Overlapping Proof Scramble #1 Statements 1. <ZXW = <YWX 2. <ZWX = <YXW 3. WX = WX 4. Triangle ZWX = Triangle YXW 5. ZW = YX Reasons 1. Given 2. Given 3. Reflexive Property 4. ASA 5. CPCTC Overlapping Proof Scramble #2 Statements 1. CA = CE 2. BA = DE 3. <CAE = <CEA 4. AE = AE 5. Triangle BAE = Triangle DEA Reasons 1. Given 2. Given 3. Base Angles of an Isosceles Triangle are Congruent. 4. Reflexive Property 5. SAS +8 Points 1. Worksheet on Overlapping Triangles 2. UNIT 4 TEST (Friday , Dec. 14th or Monday, Dec. 17th) Triangles Congruent Polygons Congruent Triangles SSS, SAS, ASA, AAS Problems SSS, SAS, ASA, AAS Proofs CPCTC and CPCTC Proofs HL Theorem Equilateral, Isosceles, and Right Triangle Problems and Solving for Missing Variables Problem of the Day **Check HW then Short Lesson and Test Review** 1. Which is Parallel to show the 2. y = 3/4x – 1 Ceiling Beam is Parallel to the Floor Beam? Which is Parallel to the Line above? Ceiling Beam A. y = 4/3x + 2 x z y Floor Beam y=w B. w = x C. w = z D. x = z A. w B. y = 3/4x – 5 C. y = -4/3x + 3 D. y = -3/4x + 6 Review Problem (+6pts) <ACB = 56 degrees x = 31 y=4 “Ratios and Proportions” Ratio—compares two quantities in a fraction form with one number over another number. a b Proportion—two equal ratios. a c b d To Solve Proportions “Cross-Multiply” a c b d ad = bc Multiply across from upper left to lower right and from upper right to lower left. x 5 y 8 What do you get? 8x = 5y Ex #1: Solve for x in these Proportions x 2 35 5 “Cross-Multiply” 5x= 70 x = 14 x x 1 2 3 “Cross-Multiply” 3x = 2(x + 1) 3x = 2x + 2 1x = 2 x=2 Ex #2: Solve for x in this Proportion x 7 x 5 2 5 “Cross Multiply” 5(x – 7) = 2(x + 5) 5x – 35 = 2x + 10 3x – 35 = 10 3x = 45 x = 15 Problem of the Day (7th Hour) Complete #1-2 1. Solve for x in this Proportional Ratio 2. Find x and y in this Isosceles Triangle Problem 2x 1 x 5 6 2 “Cross Multiply” 4x + 2 = 6x - 30 2 = 2x - 30 32 = 2x x = 16 y = 52 and x = 5 STUDY FOR UNIT 4 TEST (Friday , Dec. 14th or Monday, Dec. 17th) Triangles Congruent Polygons Congruent Triangles SSS, SAS, ASA, AAS Problems SSS, SAS, ASA, AAS Proofs CPCTC and CPCTC Proofs HL Theorem Equilateral, Isosceles, and Right Triangle Problems and Solving for Missing Variables Problem of the Day, then TEST Name the Angle Included by the Sides DE and DG? D E G Honors #37: Fill in the Missing Statements/Reasons Statements 1. HP = CK 2. GP = GK G 3. 4. Reasons 1. Given 2. Given 3. 4. Reflexive Property 5. Triangle HPK = H C X 6. Triangle CPK 5. 6. P K Given: 1. HP = CK 2. GP = GK Prove: HK = CP Extra Credit 2. SOL Homework #5 3. Honors—Notes on Tessellations 1.