Unit 8 eaumont Middle School 8th Grade, 2016-2017 Introduction to lgebra Name: P R U T S Q Lines, angles, triangles, and More I can define key terms and identify types of angles and adjacent angles. I can measure angles. I can identify vertical, supplementary and complementary angles. I can determine the measure of an interior angle of a triangle given two angle measurements. I can determine an interior angle of a triangle given an interior and eterior angle measurement. I can use the angle relationships involving parallel lines and transversals to determine the measures of corresponding angles, alternate interior angles, alternate eterior angles.
~~ Unit 8, Page 2 ~~ Lines Segments, and Rays Draw the diagram that goes with each geometric term below, and then write a definition. ngle Diagonal line segment Horizontal line segment Intersecting line segments Line Line segment Parallel Perpendicular Point Ray Skew Verte (Vertices is plural) Vertical line segment
~~ Unit 8, Page 3 ~~ ngles ngles are made up of two rays with a common endpoint, called the verte. Rays are named starting with the endpoint and then another point on the ray. Ray and ray share a common endpoint (). Notice that both rays are named starting with. Point is called the: The sides of the angle are: and ngles are usually named by three capital letters. The middle letter names the. If only one angle is located at a verte, then the angle can be named using the verte letter alone. nd if there is a lower case letter between the two sides, the angle can also be referred to using the lower case letter. The angle above can be named: NGLE MESURES protractor is used to measure angles. The protractor is divided evenly into a half circle of 180 degrees (180 ). When the middle of the bottom of the protractor is placed on the verte, and one of the rays of the angle is lined up with 0, the other ray of the angle crosses the protractor at the measure of the angle. The angle below has the ray pointing left lined up with 0 (the outside numbers), and the other ray of the angle crossed the protractor at 55. Types of ngles Type Type Type Type Measure Measure Measure Measure
~~ Unit 8, Page 4 ~~ Using the protractor below, find the measure of the following angles. Then, tell what type of angle it is using the information above. # Question Measure Type of ngle 1 What is the measure of RF? 2 What is the measure of RF? 3 What is the measure of DRF? 4 What is the measure of RD? djacent ngles djacent ngles - djacent angles are two angles that have the same verte and share one ray as a side. They do not share space inside the angles. Figure D ) D is adjacent to D. However, D is not adjacent to D because adjacent angles do not share any space inside the angle. Figure ) These two angles are not adjacent. They share a common ray but do not share the same verte. N S O T M Figure ) NOT is not adjacent to SOM. They share space inside the angles. (overlap)
~~ Unit 8, Page 5 ~~ For each diagram below, name the angle that is adjacent to it. D 1) D is adjacent T U V Z 2) TUV is adjacent R S T P 3) SRP is adjacent J R Q P 4) PQR is adjacent to to to to Independent Practice Part 1: ircle the correct choice for each question.?
~~ Unit 8, Page 6 ~~ Part 2: Fill in the blanks with the correct geometric term. 1) 2) 3) 4) 5) 6) 7) The verte of in the drawing above is point 8) 9) 10) Two acute angles in the figure are and Part 3: Find the measure of each angle. # Question Measure Type of ngle 1 What is the measure of RF? 2 What is the measure of ERF? 3 What is the measure of R? 4 What is the measure of KR? 5 What is the measure of R? 6 What is the measure of FR?
~~ Unit 8, Page 7 ~~ Part 4: For each angle, circle the best estimate. Part 5: For each diagram below, name the angle that is adjacent to it. Y O Y Z G D P F V X V 1) YOP is adjacent 2) XVY is adjacent 3) DF is adjacent J K L M 4) JKL is adjacent to to to to
~~ Unit 8, Page 8 ~~ Vertical ngles When two lines intersect, two pairs of VERTIL NGLES are formed. Vertical angles are not adjacent. Vertical angles are located across from each other, they share a common verte, and the sides of the angles are composed of opposite rays. Use a straight edge. Draw ray opposite to ray, and then draw ray opposite to ray. Use what you ve learned about the measure of straight angles to prove that the figure contains two pairs of congruent angles. OD O 45 D O Pairs of vertical angles always have the same measure. Vertical angles are (symbol hint ) ongruent means they have the. Set : In the diagram, name the second angle in each pair of vertical angles. Y Q R 1) YPV 4) VPT 2) QPR 5) RPT 3) SPT 6) VPS V P T S Set : Use the information given in the diagram to find the measure of each unknown vertical angle Set Questions G 50 1) m F = H 130 D 2) m = F 35 145 95 85 J K 6) Figure above is a 7) The proper notation for the figure is 3) m KJ = 4) m G = 5) m J = 18) m = 8) The sum of the angles in figure is + + =
~~ Unit 8, Page 9 ~~ omplementary and Supplementary ngles Two angles are complementary if the sum of their angles measure 90. Two angles are supplementary if the sum of their angles measure 180. omplementary and supplementary angle pairs may be adjacent, but do not need to be. linear pair is a pair of adjacent angles that are supplementary. elow, the angles marked 32 and 148 are a linear pair. Together, these angle pairs form a. 35 55 Sum = 23 67 32 Sum = Sum = 148 Sum = Sum = Sum = 45 45 70 110 These angles are: These angles are: PRTIE: alculate the measure of each unknown angle b a 140 c d 119 h e f g 159 61 24 G F H O 53 74 E D The sum of all the central angles = SET The sum of angles e + d + c = SET 1) m a = 5) m e = 2) m b = 6) m f = 9) m O = 10) m OD = 3)m c = 7) m g = 11)m EOF = 4) m d = 8) m h = 12) m OH =
~~ Unit 8, Page 10 ~~ Independent Practice Part 1: In the diagram below, name the second angle in each pair of vertical angles. Set K M 1) MLN 4) GLM L N 2) KLH 5) KLM J H G 3) GLN 6) HLG Use the information given in the diagram to find the measure of each unknown vertical angle. Set 7) m = y 120 60 m p 52 128 z w 133 47 8) m y = 9) m z = 10) m w = 11) m m = 12) m p = 13) + y = 14) m + p = 15) w + z = 16) Each of the angle pairs in questions 13-15 above are angles because their sum is. 17) The sum of the four angles located around every point in the figure above =
Part 2: ~~ Unit 8, Page 11 ~~
~~ Unit 8, Page 12 ~~ Review: Lines and ngles Notes: Identify each type of triangle by its angles and by its sides y y y sides: sides: sides: y y y angles: angles: angles: Part 1: Find the measure of the angles below. K 1) What is the measure of DR? D 2) What is the measure of RF? 3) What is the measure of R? E 4) What is the measure of R? 5) What is the measure of KR? R F Use the following diagram for questions 6 14. H 55 50 D E F Z Y W X V G 6) Which angle is supplementary angle to EDF? 7) What is the measure of GDF? 13) What is the measure of D? 8) Which two angles are right angles? and 9) What is the measure of EDF? 14) Which angles are adjacent to ED? 10) Which angle is adjacent to D? and 11) Which angle is a complementary angle to HD? 12) What is the measure of H?
~~ Unit 8, Page 13 ~~ Part 2: Use what you know about complementary and supplementary angles to find the measures of the following angles. SET S 1) m RMS = R Q 21 M 62 99 P V 50 T 2) m VMT = 3)m QMN = 4) m WPQ = N W The sum of angles located above = SET D 49 D J 55 K L 45 47 H G 5) m JK = 6) m KD = 7)m FKH = F 8) m L = The sum of angles located below = Part 3: lassify each triangle two ways. 1) 2) 3) 4) 5) 6)
~~ Unit 8, Page 14 ~~ Interior ngles of a Triangle FT: The three interior angles of a triangle always add up to ⁰. Eample 1: 45⁰ 30⁰ 60⁰ 45⁰ 60⁰ 60⁰ 60⁰ 45⁰ + 45⁰ + = 30⁰ + 60⁰ + = 60⁰ + 60⁰ + 60⁰ = 3( ) = Eample 2: Find the missing angle in the triangle. 20⁰ Solution: Step 1: Write equation. 20⁰ + 125⁰ + = Step 2: ombine like terms. + = 180⁰ Step 3: Isolate. 125⁰ 145⁰ + = 180⁰ 145 145 Step 4: State the solution. = Step 5: Use solution to answer the original question. The missing angle is Eample 3: Find the missing angle in the triangle. 42⁰
~~ Unit 8, Page 15 ~~ Independent Practice Find the missing angle in the triangles. For each problem, show an equation and solve. 1) 40⁰ 2) 3) 80⁰ 32⁰ 45⁰ 4) 5) 6) 78⁰ 55⁰ 15⁰ 25⁰ 7) 8) 9) 60⁰ 36⁰ 30⁰ 30⁰ 10) Two equations required 11) 12) Two equations required 22⁰ y z 38⁰ (2)⁰ (3 4)⁰ ( + 10)⁰ w 32⁰ y 38⁰ z w: y: z: : : : : : y: z:
~~ Unit 8, Page 16 ~~ Eterior ngles The eterior angle of a triangle is always equal to the sum of the opposite interior angles. Eample 1: Eamine the figures below. Find the measure of the missing angle. Figure Figure 62⁰ 60⁰ 58⁰ z 40⁰ 50⁰ y 1) Sum s in triangle = 2) = 3) Sum of interior angles opposite of angle = + = 1) = 2) y = z = 3) Sum of interior angles opposite of angle y = Sum of interior angles opposite of angle z = Eample 2: Find the measure of and y. Step 1: Use the rule for eterior angles to write equation. 120⁰ = + 120⁰ = 75⁰ + 75⁰ y 120⁰ 45⁰ = Step 2: The sum of the interior angles of a triangle equals 180⁰, and supplements D, so either equation: SUM of INTERIOR NGLES 180⁰ = 75⁰ + 45⁰ + y 180⁰ = 75⁰ + 45⁰ + y SUPPLEMENTL NGLES 180⁰ = 120⁰ + y 60⁰ = y 180⁰ = 120⁰ + y 60⁰ = y
~~ Unit 8, Page 17 ~~ Independent Practice Part 1: Find the measure of the missing angle measures. Show an equation for each angle. 1) 2) 102⁰ 50⁰ 105⁰ y y 38⁰ 3) 4) 28⁰ y 55⁰ 110⁰ 81⁰ y 5) 6) 33⁰ 122⁰ 39⁰ y y 119⁰ 7) w z (2y 3) y v 133⁰ 8) (2 + 45) y 82⁰ 9) 64⁰ y 84⁰ (y + 25) w v
~~ Unit 8, Page 18 ~~ Follow-up, review assignment for homework after p19-22 1) Find the missing angle. 2) Solve for. Then solve for each of the triangle's interior angles. 42⁰ 120⁰ 2 3 3) Find the measures for and y. 4) Find the measures for and y. y 110⁰ 145⁰ 55⁰ 65⁰ y 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
~~ Unit 8, Page 19 ~~ orresponding, lternate Interior, and lternate Eterior ngles If two parallel lines are intersected by another line, how many angles are formed? Number them on the diagram. P T Q R U S The etra arrows on two of the lines mean they are. The line that intersects the two lines is called a. The number of angles formed is. The angles formed when parallel lines are cut by a transversal line have special relationships and are named according to those relationships with one another. ORRESPONDING NGLES 1 2 4 3 5 6 8 7 Definition: Name the corresponding angles for the following. 1) 1 corresponds with 2) 2 corresponds with 3) 3 corresponds with 4) 4 corresponds with What do you notice about the angle pairs above? omplete the sentence: If two angles are corresponding angles, then they are:
~~ Unit 8, Page 20 ~~ LTERNTE INTERIOR NGLES 8 4 5 1 2 7 3 6 Word attack To alternate means INterior means: Definition: Name the alternate interior angle for the following angles. 1) 3 is an alternate interior angle with 2) 4 is an alternate interior angle with How many pairs of alternate interior angles are possible? What do you notice about the angle pairs above? omplete the sentence: If two angles are alternate interior angles, then they are: LTERNTE EXTERIOR NGLES 8 4 5 1 2 7 3 6 Word attack To alternate means Eterior means: Definition: Name the alternate eterior angle for the following angles. 1) 1 is an alternate eterior angle with 2) 2 is an alternate eterior angle with How many pairs of alternate eterior angles are possible? What do you notice about the angle pairs above? omplete the sentence: If two angles are alternate eterior angles, then they are: Look at the diagram below. For each pair of angles, state whether they are corresponding (), alternate interior (I), alternate eterior (E), vertical (V), or supplementary (S). y s u w z F v E t D 1) u, 6) t, 11) t, u 2) w, s 7) w, z 12) w, 3) t, y 8) v, w 13) w, s 4) s, t 9) v, z 14) s, v 5) w, y 10) s, z 15), z 16) If m s = 110⁰, find the measure of the remaining angles. m v = m t = m u = m w = m = m y = m z =
Parallel Lines ut by a Transversal ~~ Unit 8, Page 21 ~~ s eplained in the previous section, when two parallel lines are intersected, or cut, by a transversal, eight angles are formed. ny two angles are either congruent or supplementary! Given the measure of just one of the eight angles, the other seven can be determined. Eample: Lines a and b are parallel. Line m intersects both line a and b. The eight resulting angles are labeled 1 8, and m 1 is given to be 25⁰. Find all angle measures. m 25⁰ 1 2 3 4 a Step 1: Notice the relationships 5 6 7 8 b 1 and 4 are vertical angles and therefore, so m 4 = 25⁰. Other pairs of vertical angles are 2 and 3, 5 and 8, 6 and 7. 1 is supplementary to 2; so the m 2 = 180⁰ - 1 = 180-25⁰ = 155⁰. 1 is also supplementary to 3; so the m 3 is also 155⁰. Notice that 2 and 3are vertical angles, and would have to be to each other. Step 2: orresponding angles have the same relative position, like 1 and 5 are both in the upper left section of the intersecting lines. orresponding angles are always congruent, so m 1and m 5are both 25⁰. 5 and 8 are vertical angles, so m 8 = 25⁰. 6 and 8 form a linear pair, so m 6 = 180⁰ - 25⁰ = 155⁰. 6 and 7 are vertical angles, so m 7 is also 155⁰. nswer: m 1, m 4, m 5 and m 8 (all) = and are angles m 2, m 3, m 6 and m 7 (all) = and are angles
~~ Unit 8, Page 22 ~~ INDEPENDENT PRTIE Part 1: 1) Parallel lines and when cut by transversal form eight angles, as shown in the diagram below. Use the diagram to find the measures of each of the angles. 1 2 5 6 4 3 7 8 108⁰ 1) m 1 = 3) m 3 = 5) m 5 = 7) m 7 = 2) m 2 = 4) m 4 = 6) m 6 = 8) m 8 = 108⁰ 2) Parallel lines and when cut by transversal form eight angles, as shown in the diagram below. Use the diagram to find the measures of each of the angles. 7 8 5 6 3 4 63⁰ 1 2 9) m 1 = 11) m 2 = 13) m 3 = 15) m 4 = 63⁰ 10) m 5 = 12) m 6 = 14) m 7 = 16) m 8 =
~~ Unit 8, Page 23 ~~ Part 2: For each pair of angles, state whether they are corresponding (), alternate interior (I), alternate eterior (E), vertical (V), or supplementary (S) angles. 3 1 4 2 7 5 6 8 1) 1 and 4 6) 6 and 5 11) 3 and 6 2) 2 and 6 7) 2 and 7 12) 4 and 8 3) 1 and 3 8) 1 and 2 13) 1 and 5 4) 5 and 8 9) 4 and 5 14) 2 and 3 5) 5 and 7 10) 6 and 8 15) 6 and 7 Parallel lines and when cut by transversal form eight angles, as shown in the diagram below. Use the diagram below for problems 16 and 17. 16) m 2 = 17) m 4 = 121⁰ 7 5 6 8 2 3 4 1 Parallel lines and when cut by transversals and. Find all of the unknown angle measures. 18) m 18 = 19) m 19 = 20) m 20 = 21) m 21 = 22) m 22 = 23) m 23 = 24) m 24 = 25) m 25 = 26) m 26 = 27) m 27 = 28) m 28 = 29) m 29 = 19 18 20 21 25 22 26 24 23 29 27 28
Finding Unknown ngle Measures ~~ Unit 8, Page 24 ~~ We will use the angle relationships that are formed when two parallel lines are intersected by a transversal to find the measures of missing angles. ll of the angle relationships will either be supplementary or congruent. Eample : The pair of angles are either vertical angles, alternate interior angles, alternate eterior angles, or corresponding angles; so they are congruent. ll you have to do is set up and solve an equation where the epressions are congruent. Once you have solved for, substitute that value back into each epression to find the measure of each angle. #1 Relationship: Equation: = = = #2 Relationship: Equation: = = = Eample : Each pair of angles are supplementary to each other, which means the angles add up to 180 o. ll you have to do is set up and solve an equation where the epressions add up to equal 180 o. Once you have solved for, substitute that value back into each epression to find the measure of each angle. #3 Relationship: Equation: = = =
~~ Unit 8, Page 25 ~~ INDEPENDENT PRTIE Part 1: Find the measure of each missing angle in the parallel lines and transversals. Each pair of angles is either supplementary or congruent (vertical angles, alternate interior angles, alternate eterior angles, or corresponding angles). State the relationship, set up an appropriate equation and solve for. Once you've solved for, substitute that value back into each epression to find the measure of each angle. Relationship: 1) Equation: 2) = = = Relationship: Equation: = = = 3) Relationship: Equation: = = = 4) Relationship: Equation: = = =
~~ Unit 8, Page 26 ~~ 5) Relationship: Equation: = = = 6) Relationship: Equation: = = = 7) Relationship: Equation: = = = 8) Relationship: Equation: = = =
~~ Unit 8, Page 27 ~~ Part 2: The following problems are multiple choice. ircle the letter indicating the best answer for each question. 1) 2) 3) 4) 5) 6)
~~ Unit 8, Page 28 ~~ Review for Unit Test Part 1: Key Terms, Types of ngles, Measuring ngles and djacent ngles N S T D 1) D is adjacent W 2) NWS is adjacent M 3) The verte is: to to G Use the protractor to measure each angle. Indicate whether it is acute, obtuse, right, or straight. R S 4) = ; 5) = ; 6) = ; 7) = ; E F N 8) = ; The following questions are multiple choice. ircle the letter net to the best answer. 9) 10) Which of the following is a correct name for the angle indicated below with the question mark??... D.
~~ Unit 8, Page 29 ~~ Part 2: Vertical, Supplementary and omplementary ngles Find the measure of angle b and classify the angle relationship. 1) 2) 3) lassification: b: lassification: b: lassification: b: Part 3: Interior and Eterior ngles of a Triangle Find the value of in each of the following diagrams. Equation: 1) 2) Equation: : : 3) Equation: 4) Equation: : : 5) Equation: :
~~ Unit 8, Page 30 ~~ Part 5: Parallel Lines and Transversals [orresponding ngles, lternate Interior ngles, lternate Eterior ngles] Find the missing angle measurement. Identify the relationship of the angles. corresponding (), alternate interior (I), alternate eterior (E), vertical (V), or supplementary (S) angles. 1) 2) 3) : Relationship: 4) : Relationship: 5) 6) : Relationship: : Relationship: : Relationship: : Relationship: Identify the relationship of the angles. corresponding (), alternate interior (I), alternate eterior (E), vertical (V), or supplementary (S) angles. Write an equation to solve for. Solve. 7) Relationship: 8) Equation Relationship: Equation : :
~~ Unit 8, Page 31 ~~ Identify the relationship of the angles. corresponding (), alternate interior (I), alternate eterior (E), vertical (V), or supplementary (S) angles. Write an equation to solve for. Solve. 9) Relationship: Equation 10) (4 + 2) Relationship: Equation : : 11) Identify the measures of the indicated angles. The following questions are multiple choice. ircle the letter net to the best answer. 12) 13) 14)