Name: Unit 8 Beaumont Middle School 8th Grade, Advanced Algebra I T Q R U

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1 Unit 8 eaumont Middle School 8th Grade, dvanced lgebra I Name: P T Q R U S I can define ke terms and identif tpes of angles and adjacent angles. I can identif vertical, supplementar and complementar 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 sum of the interior angles of a polgon to determine measurements of angles. I can use the angle relationships involving parallel lines and transversals to determine the measures of corresponding angles, alternate interior angles, and alternate eterior angles. I can identif b name the parts of three dimensional shapes (vertices, lateral faces, bases). I can appl the volume formulas of right prisms, clinders, pramids, cones, and spheres. I can appl the formulas for volume to real-world and mathematical problems.

2 ~~ Unit 8, Page 2 ~~ ngles ngles are made up of two ras with a common endpoint, called the verte. Ras are named starting with the endpoint and then another point on the ra. Ra and ra share a common endpoint (). Notice that both ras are named starting with. Point is called the: The sides of the angle are: and ngles are usuall named b three capital letters. The middle letter names the. If onl 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 evenl 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 ras of the angle is lined up with 0, the other ra of the angle crosses the protractor at the measure of the angle. The angle below has the ra pointing left lined up with 0 (the outside numbers), and the other ra of the angle crossed the protractor at 55. Tpes of ngles Tpe Tpe Tpe Tpe Measure Measure Measure Measure

3 ~~ Unit 8, Page 3 ~~ Using the protractor below, find the measure of the following angles. Then, tell what tpe of angle it is using the information above. # Question Measure Tpe 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 ra as a side. The do not share space inside the angles. Figure D D ) D is adjacent to D. However, D is not adjacent to D because adjacent angles do not share an space inside the angle. Figure ) These two angles are not adjacent. The share a common ra but do not share the same verte. N S O T M Figure ) NOT is not adjacent to SOM. The share space inside the angles. (overlap)

4 ~~ Unit 8, Page 4 ~~ 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.?

5 ~~ Unit 8, Page 5 ~~ Part 2: Fill in the blanks with the correct geometric term. 1) 2) 3) 4) The verte of OD in the drawing above is point 5) 6) 7) 8) 9) O Two acute angles in the figure are O and 10) Part 3: Find the measure of each angle. # Question Measure Tpe 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?

6 ~~ Unit 8, Page 6 ~~ 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, the share a common verte, and the sides of the angles are composed of opposite ras. Use a straight edge. Draw ra O opposite to ra O, and then draw ra O opposite to ra OD. Use what ou 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 alwas have the same measure. Vertical angles are (smbol hint ) ongruent means the 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 G Set Questions F D J H 1) m F = 2) m = 3) m KJ = 4) m G = 6) Figure above is a K 5) m J = 18) m = 7) The proper notation for the figure is 8) The sum of the angles in figure is + + =

7 ~~ Unit 8, Page 7 ~~ omplementar and Supplementar ngles Two angles are complementar if the sum of their angles measure 90. Two angles are supplementar if the sum of their angles measure 180. omplementar and supplementar angle pairs ma be adjacent, but do not need to be. linear pair is a pair of adjacent angles that are supplementar. elow, the angles marked 32 and 148 are a linear pair. Together, these angle pairs form a Sum = Sum = Sum = 148 Sum = Sum = Sum = These angles are: These angles are: PRTIE: alculate the measure of each unknown angle b a 140 c d 119 h e f g G F H O 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 =

8 ~~ Unit 8, Page 8 ~~ 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 = m p z w ) m = 9) m z = 10) m w = 11) m m = 12) m p = 13) + = 14) m + p = 15) w + z = 16) Each of the angle pairs in questions above are angles because their sum is. 17) The sum of the four angles located around ever point in the figure above =

9 ~~ Unit 8, Page 9 ~~ Interior ngles of a Triangle FT: The three interior angles of a triangle alwas 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. 125⁰ 20⁰ + 125⁰ + = Step 2: ombine like terms. + = 180⁰ Step 3: Isolate. 145⁰ + = 180⁰ 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

10 ~~ Unit 8, Page 10 ~~ 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 z 38⁰ (2)⁰ (3 4)⁰ ( + 10)⁰ w 32⁰ 38⁰ z w: : z: : : : : : : z:

11 ~~ Unit 8, Page 11 ~~ Eterior ngles The eterior angle of a triangle is alwas 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 1) Sum s in triangle = 2) = 3) Sum of interior angles opposite of angle = + = 1) = 2) = z = 3) Sum of interior angles opposite of angle = Sum of interior angles opposite of angle z = Eample 2: Find the measure of and. Step 1: Use the rule for eterior angles to write equation. 120⁰ = + 120⁰ = 75⁰ + 45⁰ = 75⁰ 120⁰ 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 + 180⁰ = 75⁰ + 45⁰ + SUPPLEMENTL NGLES 180⁰ = 120⁰ + 60⁰ = 180⁰ = 120⁰ + 60⁰ = Independent Practice

12 ~~ Unit 8, Page 12 ~~ Part 1: Find the measure of the missing angle measures. Show an equation for each angle. 1) 2) 102⁰ 50⁰ 105⁰ 38⁰ 3) 4) 28⁰ 55⁰ 110⁰ 81⁰ 5) 6) 33⁰ 122⁰ 39⁰ 119⁰ 7) w z (2 3) v 133⁰ 8) (2 + 45) 82⁰ 9) 64⁰ 84⁰ ( + 25) w v

13 Interior ngles of Polgons ~~ Unit 8, Page 13 ~~ n Interior ngle is an angle located inside a figure. Triangles: The interior angles of a TRINGLE add up to 180⁰. Interior ngles of POLYGONS 1) State the number of sides for each polgon. 2) From the given point at one verte, draw diagonals to find the number of non-overlapping triangles in each polgon. 3) Use the sum of angles in each triangle to determine the total sum of the interior angles of each figure. Number of sides = Number of s = SUM of int s = Number of sides = Number of s = SUM of int s = Number of sides = Number of s = SUM of int s = D Number of sides = Number of s = SUM of int s = E Number of sides = Number of s = SUM of int s =

14 ~~ Unit 8, Page 14 ~~ Number of sides = Number of s = SUM of int s = F If this polgon had 11 sides (that s an undecagon), then: G How man triangles? What would be the sum of the interior s? The General Rule Make a conjecture about how ou can calculate the sum of the interior angles of an polgon. Test our conjecture: Write an epression that ou could use to calculate the sum of the interior angles of an polgon using n for the number of sides. If the polgon was regular and ou know the sum of angles, how could ou find the measure of each angle?

15 ~~ Unit 8, Page 15 ~~ EXMPLE 1: What is the measure of each angle in a regular DEGON? NSWER: The measure of each angle in a regular decagon = EXMPLE 2; Find the measure of. EXMPLE 3: Find the measures of the missing angles. F Guided Practice: Eample 1: What is the measure of each angle in a regular 20-sided polgon? Eample 2: Find the measure of. Sum of the interior angles: Measure of each angle: Sum of the interior angles: Measure of :

16 ~~ Unit 8, Page 16 ~~ Eample 3: Find the measures of the missing angles. INDEPENDENT PRTIE lassif each polgon b its number of sides and state the sum of the interior angles. 1) 2) 3) lassif: Sum of the interior angles: lassif: Sum of the interior angles: lassif: Sum of the interior angles: 4) a polgon with 8 sides 5) a polgon with 5 sides lassif: Sum of the interior angles: lassif: Sum of the interior angles: What is the measure of each angle in the following regular polgons? 6) 7) Regular Pentagon Sum of the interior angles: Measure of each angle: Sum of the interior angles: Measure of each angle:

17 ~~ Unit 8, Page 17 ~~ Find the measure of the missing angles. 8) 9) Sum of the interior angles: Measure of : Sum of the interior angles: Measure of : 10) 11) Sum of the interior angles: Measure of : Sum of the interior angles: Measure of : 12) 13)

18 ~~ Unit 8, Page 18 ~~ orresponding, lternate Interior, and lternate Eterior ngles If two parallel lines are intersected b another line, how man angles are formed? Number them on the diagram. P T Q R U S PQ RS TU is a transversal The etra arrows on two of the lines mean the 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 b a transversal line have special relationships and are named according to those relationships with one another. ORRESPONDING NGLES 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 ou notice about the angle pairs above? omplete the sentence: If two angles are corresponding angles, then the are:

19 ~~ Unit 8, Page 19 ~~ LTERNTE INTERIOR NGLES 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 man pairs of alternate interior angles are possible? What do ou notice about the angle pairs above? omplete the sentence: If two angles are alternate interior angles, then the are: LTERNTE EXTERIOR NGLES 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 man pairs of alternate eterior angles are possible? What do ou notice about the angle pairs above? omplete the sentence: If two angles are alternate eterior angles, then the are: Look at the diagram below. For each pair of angles, state whether the are corresponding (), alternate interior (I), alternate eterior (E), vertical (V), or supplementar (S). 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, 8) v, w 13) w, s 4) s, t 9) v, z 14) s, v 5) w, 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 = m z =

20 ~~ Unit 8, Page 20 ~~ INDEPENDENT PRTIE Part 1: 1) Parallel lines a and b when cut b transversal l form eight angles, as shown in the diagram below. Use the diagram to find the measures of each of the angles. l a b 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 a and b when cut b transversal l form eight angles, as shown in the diagram below. Use the diagram to find the measures of each of the angles. a b ⁰ 1 2 l 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 =

21 ~~ Unit 8, Page 21 ~~ Part 2: For each pair of angles, state whether the are corresponding (), alternate interior (I), alternate eterior (E), vertical (V), or supplementar (S) angles l m l m 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 a and b when cut b transversal m form eight angles, as shown in the diagram below. Use the diagram below for problems 16 and ) m 2 = 17) m 4 = a b 121⁰ m Parallel lines a and b when cut b transversals m and n. 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 = b 24) m 24 = 25) m 25 = 26) m 26 = 27) m 27 = 28) m 28 = 29) m 29 = n m a

22 Finding Unknown ngle Measures ~~ Unit 8, Page 22 ~~ We will use the angle relationships that are formed when two parallel lines are intersected b a transversal to find the measures of missing angles. ll of the angle relationships will either be supplementar or congruent. Eample : The pair of angles are either vertical angles, alternate interior angles, alternate eterior angles, or corresponding angles; so the are congruent. ll ou have to do is set up and solve an equation where the epressions are congruent. Once ou have solved for, substitute that value back into each epression to find the measure of each angle. #1 Relationship: Equation: = G = D = #2 Relationship: Equation: = D = HF = Eample : Each pair of angles are supplementar to each other, which means the angles add up to 180 o. ll ou have to do is set up and solve an equation where the epressions add up to equal 180 o. Once ou have solved for, substitute that value back into each epression to find the measure of each angle. #3 Relationship: Equation: = FD = HF =

23 ~~ Unit 8, Page 23 ~~ INDEPENDENT PRTIE Part 1: Find the measure of each missing angle in the parallel lines and transversals. Each pair of angles is either supplementar 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 ou've solved for, substitute that value back into each epression to find the measure of each angle. Relationship: 1) Equation: 2) = EF = GFH = Relationship: Equation: = D = FH = 3) Relationship: Equation: = EF = FH = 4) Relationship: Equation: = EF = DF =

24 ~~ Unit 8, Page 24 ~~ 5) Relationship: Equation: = = FH = 6) Relationship: Equation: = = GFH = 7) Relationship: Equation: = D = EF = 8) Relationship: = = GFH = Equation: Part 2: Review lassif each polgon b its number of sides and state the sum of the interior angles. 1) 2) lassif: Sum of the interior angles: lassif: Sum of the interior angles:

25 ~~ Unit 8, Page 25 ~~ What is the measure of each angle in the following regular polgons? 3) 4) Regular Decagon Sum of the interior angles: Measure of each angle: Sum of the interior angles: Measure of each angle: Part 3: The following problems are multiple choice. ircle the letter indicating the best answer for each question. 1) 3) 2) 4)