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Math 1: Semester 2 3-3 (Angle Addition) 0-5:00 Attendance QD. A pair shoes costs $50.95 and they are on sale for 25% off. What is the final cost? $38.21 5:00-25:00 Lesson Objective- To measure angles using a protractor and determining the bisector of an angle NotesAngle Addition Postulate- For any angle, PQR, if a point, B, is the interior then, two angles can be formed, which equal the original angle, so PQB + BQR = PQR P B Q Angle Bisector- R separates the angle into two, equal angles Q W P R is the angle bisector of P&m QPW = m WPR Examples #1) Angles addition Find the following anglesa) m KNM M L 110° 25° K N 1 b) m 2 D C 75° 2 A c) m B JKL L (2x-10)° J 4x° K M #2 Angles bisector If bisects F CFE and m C 1 2 CFE = 70; find m CFD and m DFE D E 35° ActivityHave students draw an angle (encourage large angles to make the measuring easier), then have them find the angle bisector by using a protractor. 25:00-50:00 Homework: (108-109) & angle bisector construction (using compass) Nearing Proficient: Proficient: Adv. Proficient: 11-22 (odds) 11-22 (all) 11-22 (all); 23-25 2 Math 1: Semester 2 3-5 (Complementary & Supplementary Angle) 0-5:00 Attendance 5:00-25:00 Lesson Objective- To identify the properties of complimentary and supplementary angles NotesComplementary Angles- Two angles sums equal 90 degrees Supplementary angles- Two angles sums equal 180 degrees Supplementary Postulate- If two angles form a linear pair, then they are supplementary angles Examples #1) Determine if the following are complementary AGF & BGC AGB & BGC FGE & CGD FGE & FGA Yes No Yes No A B F 55° 80° 90° G C 10° 90° E D #2 Name a pair of supplementary angles, which are not adjacent angles? AGB & DGE #3 Find the measure of an angle that is angle that is supplementary to 145 BGC? 25:00-50:00 Activity: Group work Have students work with the two angles learned today by creating a table- one rewording the mathematical definition and example through illustration 3 Homework: Worksheet 3-4 & 3-5 Math 1: Semester 2 3-4 (Adjacent & Linear Pair Angles) 0-5:00 Attendance 5:00-25:00 Lesson Objective- To understand the relationship of angles, which share a common side- forming adjacent angles and linear pairs NotesAdjacent Angles- Angles that share a common side & have the same vertex Common side 1 2 Same vertex Linear Pair- Adjacent angles, which their angles added up equal 180 degrees 1 2 Examples #1) Adjacent angles Determine whether a) No 1 1& 2 2 are adjacent angles b) Yes 1 2 c) No 4 1 2 #2 Linear Pair Using the figure below construct as many linear pairs as possible a) MCA ACE b) HCE MCH c) TCM TCE d) ACH not possible T H E C A M 25:00-50:00 Activity Have students choose the one of the following: #1 VerbalAsk the student to help a friend who confuses adjacent and linear angles by writing a paragraph explaining the difference between these two types of angles. They may draw sketches to aid in their writing. #2 KineticAsk students to look for angles at home. They should notice that many angles are right angles. Have them note which angles are not right angles and where they appear. Encourage students to use a protractor to verify their observations. Also, have them note where they see adjacent angles and linear pairs. Homework: (113-114) Nearing Proficient: Proficient: Adv. Proficient: 8-19 (odds) 8-19 (all) 8-19 (all); 22-26 5 Math 1: Semester 2 3-6 (Congruent Angles) 0-5:00 QD Attendance If a dinner will cost you $65 but you have a coupon for 25% off. How much is the final bill? 5:00-25:00 Lesson Objective- Students will identify congruent and vertical angles and learn their uses. ReviewDo the 5-minute check NotesCongruent Angles- Two angles that have the same measure Vertical anglesFormed by two intersecting lines; of the four angles formed, they are only the nonadjacent angles A D B C Thm Vertical angle Vertical angles are congruent Thm 3-2 & 3-3 If two angles are congruent, then there compliments or supplements are congruent Thm 3-4 & 3-5 If two angles are complementary or supplementary to the same angle, then they are congruent Thm. 3-6 If two angles are congruent and supplementary, then each angle is right Thm. 3-7 All right angles are congruent 6 Examples 1) Find the values of x a) b) 100° 75° x° (x-18)° x = 100 2) Suppose A B. IF B = 47, find the supplementary angle to A 1st A = 47 2nd – The supplement would be 180 – 47 = 133° 3) Given: 1 is supplementary to 3 is supplementary to 2 = 105 degrees Find m 1 & m 3 2 2 Since both angles (1 & 3) are supplementary to 130-105 = 75 degrees 2, they both have the same measure- 25:00-50:00 Activity: Logic Have students write a short paragraph explaining why vertical angles are congruent. They cannot use examples but must use the facts (theorems) learned in this chapter. After students are finished, pair them and have them explain their method to each other. Homework: (126-127) Nearing Proficient: Proficient: Adv. Proficient: 9-14 (odds) 9-14 (odds); 15-20 (odds) 9-14 (all); 15-20 (all) Math 1: Semester 2 3-7 (Perpendicular Lines) 0-5:00 Attendance 7 QD If a big screen TV costs $ 1200 but it’s discounted by 15% off. How much is the TV? 5:00-25:00 Lesson Objective- Students will identify, construct, and use properties of perpendicular lines. NotesPerpendicular lines- Lines that intersect at a 90 degree angle, use symbol What statements can be said of the figure below? 1 2 4 3 a) 1 is a right or 90 degrees, so l m b) 1 & 3 are vertical angles, so 1 3 and m 3 = 90 c) 1 & 4 form a linear pair, so m 1 + m 4 = 180; 1 & supplementary; and m 4 = 90 d) 2 & 4 are vertical angles, so 2 4 and m 2 = 90 4 are Thm. If two lines are perpendicular, then all four angles are right angles Examples M 7 P Q 1 2 R 5 6 8 S O 3 4 N 1) Answer the questions by referring to the figure abovea) PRN is an acute angle False b) 4 8 True c) QS OP False 8 d) 2) 7 is an obtuse angle True Answer the questions by referring to the figure below Find m 1 and m 2. Given: AC BD; m 1 = 8x-2; m 2 = 16x –4 A B 2 F E 1 D C Since, AC BD, we know that Find m So, 8x – 2 + 16x – 4 = 90 24x – 6 = 90 24x = 96 x=4 m m 1+m 2 = 90 Substitute expressions Combine like terms Substitute x into the expressions 1 = 8(4) – 2 = 30 degrees 2 = 16(4) – 4 = 60 degrees 25:00-50:00 Activity: Visual Have students work in pairs and construct a classroom with out any perpendicular lines where the floors, ceiling, doors, windows, and walls meet. Homework: (132) 8-24; 25-28 Nearing Proficient: Proficient: Adv. Proficient: 8-24 (odds) 8-24 (all) 8-24 (all); 25-28 Math 1: Semester 2 4-1 (Parallel Lines & Planes) 0-5:00 5:00-25:00 Lesson- Attendance Objective: Understand how parallel lines form at the intersection of planes. 9 Review: Point, ray, and segment. ActivityHave students place letters to mark the eight corners of the room. List on the board the points. Construct/identify the twelve segments (edges), which make up the outlines of the room. From the points, construct/identify the six planes (faces) that make up the ceiling, floor, and walls. NotesParallel Lines- Lines in the same plane, which do not intersect Skew Lines- Lines that are not in the same plane and they do not intersect B C D A G E F 25:00-50:00 Homework: (145-46) 12-31; 40-45 and worksheet Nearing Proficient: Proficient: Adv. Proficient: 12-21; 40-45 and worksheet 12-21; 24-27; 28-31 (odd); 40-45 and worksheet 12-31; 40-45 and worksheet Math 1: Semester 2 4-2 (Parallel Lines & Transversals) 0-5:00 5:00-25:00 Lesson- Attendance 10 Objective: lines. Understand the relationships of angles formed by transversal and parallel Review: Vertical, supplementary, and complementary angles NotesDay 1: Transversal- a line that intersects two or more lines at two different points Ask: How many angles are formed? 8 Exterior 1 3 2 4 Interior 5 6 7 Interior Exterior : lie between the two lines Alternate Interior : Consecutive Interior Exterior 8 that lie on opposite side of the transversal & in the interior sides of the two lines : that lie on the same side of the transversal & in the interior of the two lines : that lie outside the two lines Alternate Exterior : on the opposite sides of the transversal & in the exterior of the two lines Activity: Ask: Use the diagram to fill in which angles fit which definitions 1 3 2 4 11 5 6 7 Vertical : Interior : Alternate Interior : Consecutive Interior Exterior 8 : : Alternate Exterior : Day 2: Thm. Alternate Interior If two parallel lines are cut by a transversal, then each pair Alt. Int. Ex. ’s are congruent and 12 Thm. Consecutive Interior If two parallel lines are cut by a transversal, then each pair Consecutive Int. supplementary ’s are Ex. Thm. Alternate Exterior If two parallel lines are cut by a transversal, then each pair Alt. Ext. ’s are congruent Ex. 1 3 2 4 5 6 7 8 Corresponding Angles: . All of the pairs are congruent to each other. Thm: If a transversal is perpendicular to on of the two parallel lines, then it is perpendicular to the other line. Examples #1) 1 104° 2 3 1-76° 3-98° 5-76° 6-98° 8-104° 9-82° 11-104° 13-98° 2-82° 4- 104° 7-76° 10-98° 12-76° 14-82° 13 4 5 7 6 8 11 9 12 13 82° 10 14 #2) 26° 124° 56° 56° 124° 25:00-50:00 26° 98° 82° 124° 98° 82° Homework: (152-53) Day 1: 13-26 Day 2- worksheet 4.2 Day 3- worksheet 4.3 Math 1: Semester 2 4-4 (Proving Lines Parallel) 0-5:00 5:00-25:00 Lesson- Attendance Objective: Showing how to lines can be parallel using a transversal. QD. Fill in the numbered angles 14 1 4 104° 2 3 5 6 82° 7 11 8 9 12 13 1-76° 3-98° 5-76° 6-98° 8-104° 9-82° 11-104° 13-98° 2-82° 4- 104° 7-76° 10-98° 12-76° 14-82° 10 14 NotesThm. Corresponding Angles 1 a 2 b Thm. Alternate Interior Angles 1 a 2 b Thm. Alternate Exterior Angles 1 a 2 b Thm. Consecutive Interior Angles 1 a 15 2 b Examples Find x so that #1) a 84° 3x b x = 28 a b #2) 10x° 5x° 10x + 5x = 180 15x = 180 x = 12 25:00-50:00 Homework: (166) Nearing Proficient: Proficient: Adv. Proficient: 9-20(odds) 9-20 (all) 9-20 (all); 21-23 Math 1: Semester 2 5-1 (Classifying Triangles) 0-5:00 5:00-25:00 Lesson- Attendance Objective: triangles. Students will understand all parts of a triangle and how to classify varying 16 NotesTriangle- parts A angle side vertex B ΔABC• • • C The sides are The vertices are A, B, & C The angles are Types of Triangles (by angles)A) Acute B) Obtuse C) Right All acute angles One obtuse angle One Rt. angle Types of Triangles (by sides)A) Scalene B) Isosceles C) Equilateral 17 No sides congruent at least two sides congruent All sides congruent ExamplesClassify each triangle by its angles and sides #1 Right scalene 30° #2 120° Obtuse, isosceles 30° #3 Find all side lengths of the triangle A 2n + 2 B 10 2n – 2 C 25:00-50:00 Activity: Group Have students work in pairs. Student #1 labels an index card for each of the seven possible angle/side classifications for triangles: acute/scalene, acute/isosceles, acute/equilateral, right/scalene, right/isosceles, obtuse/scalene, and obtuse/isosceles. Student #2 prepares seven more cards, each picturing one of the seven triangles possibilities. Shuffle the cards, arrange them face down, and have the students take turns matching one set of cards to the other. Have them keep track of their scores. 18 Homework: (191) 8-16; 18-22; 26,27 Nearing Proficient: Proficient: Adv. Proficient: 8-16; 18-22; 8-16; 18-22; 26, 8-16; 18-22; 26,27 Math 1: Semester 2 5-2 (Angles of a Triangle) 0-5:00 5:00-25:00 Lesson- Attendance Objective: Students will understand the relationship of all angles of a triangle. NotesThm. Angle Sum The sum of the measures of the angles of a triangle is 180° x x° + y° + z° = 180° * This thm. is used to find missing angles measures z y Ex. 1 Find m So, in ΔMNP if m 80 + 45 + m = 180 125 + m = 180 -125 -125 _________________ m = 65° Ex.2 Find the value of each variable in ΔABC 19 75° B x° y° 58° A x is a vertical angle with 75° and so x = 75 last, x° + y° + 58° = 180° 75 + y + 58 = 180 133 + y = 180 -133 -133 y = 47 C Thm. The acute angles of a right triangle are complementary (=90°) x° + y° = 90° x° y° Equiangular Triangle- A triangle, which all three angles are congruent or 60° Ex. Since it is a rt. Triangle both angles J & K equal 180. So, (x + 15)° + (x + 9)° = 90° 2x + 24 = 90 -24 -24 2x = 66 2 2 J (x + 15)° (x + 9)° L x = 33 Now, plug-in 78 into each angle expression K 25:00-50:00 Activity: Homework: (196-97) 8-19, 22, 23; 26-29 Nearing Proficient: Proficient: Adv. Proficient: 8-16, 17, 26-29 8-19, 26-29 8-19, 22, 23; 26-29 20 Math 1: Semester 2 5-3 (Translation, reflection, & rotation) 0-5:00 5:00-25:00 Lesson- Attendance Objective: Students will understand certain transformations move figures in a plane Activity: Auditory Play a simple melody, then play it one octave higher. Ask how they are similar and different. Use this discussion to introduce translations. NotesTranslation- slide a figure; the initial figure is the pre-image & the slided figure is the image Reflection- flipping a figure (the figure has been flipped over the horizontal line) 21 Rotation- turning a figure (the figure has been rotated 90°) Ex. Go over the six examples on pg. 198 (answers: rotation, reflection, translation, reflection, rotation, and translation). Mapping- Each point of a pre-image is paired with exactly one point of an image. When all points have been mapped, this is called a Transformation, this symbol, →, is used to indicate a mapping. Ex. ∆ABC → ∆EFG A B Preimage A B C C → → → Image E F G E F G 25:00-50:00 Homework: (201) Nearing Proficient: Proficient: Adv. Proficient: 9-24 9-24 9-24 Math 1: Semester 2 5-4 (Congruent Triangles) 0-5:00 5:00-25:00 Lesson- Attendance 22 Objective: Students will understand the relationship between segments and angles of congruent triangles. ActivityMaterials: two pieces of graph paper, scissors, & straight-edge Step 1: Have students go to page 203 and copy the two triangles in “hands-on” section at the top of the page. Step 2: After the triangles are constructed, the students will cut them out. Match the same parts up together and compare. Step 3: Have the students answer the following questions. Students should attach the cut-out triangles and questions to turn in. 1) Identify all of the pairs of angles and sides that match or correspond. 2) ∆ABC is congruent to ∆FDE. What is true about their corresponding sides and angles? NotesCorresponding partsparts of congruent triangles that “match” up; slashes (-, =) are used to identify congruent segments and arcs ( )are used for congruent angles. Thm. Congruent Triangles If two triangles are congruent, then all the corresponding parts are congruent. If the corresponding parts of two triangles are congruent, then the two triangles are congruent. Examples #1 If ∆ABC ∆FDE, name the congruent angles and sides. Then draw the triangles, using arcs and slashes marks to show congruent angles and sides. B D 23 A C F E Ex. 2 ∆UVW is congruent to ∆GHI. Since, both triangles are congruent, angle V and H are congruent because they are corresponding parts. So, we can set the measures of both angles equal to each other. 90 = 3x + 15 -15 -15 75 = 3x Subtract 15 from both sides Divide both sides by 3 75 = 3x 3 3 25 = x 25:00-50:00 Homework: Day 1: Homework: (206) 11-25 Day 2: Class activity and worksheet Math 1: Semester 2 5-5 (SSS & SAS) 0-5:00 5:00-25:00 Lesson- Attendance Objective: Students will understand the relationship among sides and angles in showing congruent between triangles ActivityMaterials: one piece of paper, scissors, & ruler Step 1: Have students cut out three segments form the piece of paper: 4” long, 5”, and 6”. 24 Step 2: Ask students to form as many triangles as they can from the three segments and keep track of their constructions on paper. Step 3: Have students compare their triangles with those of other students. Step 4: Have each student make a guess (conjecture) based on their results and observations. NotesPostulate: SSS (side-side-side) If three sides of one triangle are congruent to three corresponding sides of another triangle, then the triangles are congruent. B S A R C T Included Angle- the angle formed by two given sides of a triangle C A B Post: SAS (Side-Angle-Side) If two sides & the included angle of one triangle are congruent to the corresponding sides and included angle of another triangle, then the two triangles are congruent. 25 B S A R C T Examples Write a congruence statement for each pair of triangles #1) Hint: Sketch the two triangles by constructing the segments and angles, which are congruent to each other K N J L M O So, the congruence statement: Tell whether each pair of triangles are congruent; why (which postulate); and write a congruence statement. #2) B A Y C X Z Yes, SAS, #3) D E M F N 26 O Yes, SSS, 25:00-50:00 Homework: (213) Nearing Proficient: Proficient: Adv. Proficient: 8-15; 19; 22 8-16; 19; 22 8-18; 20-22 Math 1: Semester 2 5-6 (ASA & AAS) 0-5:00 5:00-25:00 Lesson- Attendance Objective: Students will understand the relationship among sides and angles in showing congruent between triangles NotesIncluded Side- the side that falls between two angles C A B Postulate: ASA (angle-side-angle) If two angles & the included side of one triangle are congruent to the corresponding angles & the included side of another triangle, then the triangles are congruent. B S 27 A R C T Thm. AAS (Angle-Angle- Side) If two angles & a non-included side of one triangle are congruent to the corresponding angles and non-included side of another triangle, then the two triangles are congruent. (note: the side is not between or included between the angles) B S A R C T Examples Name the needed information to make the triangles congruent #1) Using ASAK J N L M O Needed congruent part: 28 #2) Using AASB A Y C X Z Needed congruent part: #3) Determine whether each pair of triangles are congruent by SSS, SAS, ASA, or AAS D E M F N O Yes, by SSS 25:00-50:00 Activity: Group & visual Have students work in pairs to construct four examples showing two triangles congruent by SSS, SAS, ASA, & AAS. Have them use one sheet of paper for each example. Homework: (218-19) Nearing Proficient: Proficient: Adv. Proficient: 15-22; plus worksheet (evens) 15-22; plus worksheet 15-22; plus worksheet 29