Focus Standards Draw, construct, and describe geometrical figures and describe the relationships between them. 7.G.A.2 7.G.A.3 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. Describe the two dimensional figures that result from slicing three dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. Solve real life and mathematical problems involving angle measure, area, surface area, and volume. 7.G.B.5 7.G.B.6 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi step problem to write and solve simple equations for an unknown angle in a figure. Solve real world and mathematical problems involving area, volume, and surface area of twoand three dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right pr Foundational Standards Geometric measurement: understand concepts of angle and measure angles. 4.MD.C.7 Recognize angle measure as additive. When an angle is decomposed into non overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure. Solve real world and mathematical problems involving area, surface area, and volume. 6.G.A.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real world and mathematical problems. 2 P age
6.G.A.2 6.G.A.4 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real world and mathematical problems. Represent three dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real world and mathematical problems. Solve real life and mathematical problems involving area, surface area, and volume. 7.G.B.4 Know the formulas for area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. Focus Standards for Mathematical Practice MP.1 MP.3 MP.5 Make sense of problems and persevere in solving them. This mathematical practice is particularly applicable for this module, as students tackle multi step problems that require them to tie together knowledge about their current and former topics of study (i.e., a real life composite area question that also requires proportions and unit conversion). In many cases, students will have to make sense of new and different contexts and engage in significant struggle to solve problems. Construct viable arguments and critique the reasoning of others. In Topic B, students examine the conditions that determine a unique triangle, more than one triangle, or no triangle. They will have the opportunity to defend and critique the reasoning of their own arguments as well as the arguments of others. In Topic C, students will predict what a given slice through a three dimensional figure will yield (or how to slice a threedimensional figure for a given cross section) and must provide a basis for their predictions. Use appropriate tools strategically. In Topic B, students will learn how to strategically use a protractor, ruler, and compass to build triangles according to provided conditions. An example of this is when students are asked to build a triangle provided three side lengths. Proper use of the tools will help them understand the conditions by which three side lengths will determine one triangle or no triangle. Students will have opportunities to reflect on the appropriateness of a tool for a particular task. MP.7 Look for and make use of structure. Students must examine combinations of angle facts within a given diagram in Topic A to create an equation that correctly models the angle relationships. If the unknown angle problem is a verbal problem, such as an example that asks for the measurements of three angles on a line where the values of the measurements are consecutive numbers, students will have to create an equation without a visual aid and rely on the inherent structure of the angle fact. In Topics D and E, students will find area, surface area, and volume of composite figures based on the structure of two and three dimensional figures. 3 P age
Terminology New or Recently Introduced Terms Correspondence (A correspondence between two triangles is a pairing of each vertex of one triangle with one and only one vertex of the other triangle. A triangle correspondence also induces a correspondence between the angles of the triangles and the sides of the triangles.) Identical Triangles (Two triangles are said to be identical if there is a triangle correspondence that pairs angles with angles of equal measure and sides with sides of equal length.) Unique Triangle (A set of conditions for two triangles is said to determine a unique triangle if whenever the conditions are satisfied, the triangles are identical.) Three sides condition (Two triangles satisfy the three sides condition if there is a triangle correspondence that pairs all three sides of one triangle with sides of equal length. The three sides condition determines a unique triangle.) Two angles and the included side condition (Two triangles satisfy the two angles and the included side condition if there is a triangle correspondence that pairs two angles and the included side of one triangle with angles of equal measure and a side of equal length. This condition determines a unique triangle.) Two angles and the side opposite a given angle condition (Two triangles satisfy the two angles and the side opposite a given angle condition if there is a triangle correspondence that pairs two angles and a side opposite one of the angles with angles of equal measure and a side of equal length. The two angles and the side opposite a given angle condition determines a unique triangle.) Two sides and the included angle condition (Two triangles satisfy the two sides and the included angle condition if there is a triangle correspondence that pairs two sides and the included angle with sides of equal length and an angle of equal measure. The two sides and the included angle condition determines a unique triangle.) Two sides and a non included angle condition (Two triangles satisfy the two sides and a nonincluded angle condition if there is a triangle correspondence that pairs two sides and a nonincluded angle with sides of equal length and an angle of equal measure. The two sides and a non included angle condition determines a unique triangle if the non included angle measures 90 or greater. If the non included angle is acute, the triangles are identical with one of two non identical triangles.) Right rectangular pyramid (Given a rectangular region in a plane, and a point not in, the rectangular pyramid with base and vertex is the union of all segments for any point in. It can be shown that the planar region defined by a side of the base and the vertex is a triangular region, called a lateral face. If the vertex lies on the line perpendicular to the base at its center (the intersection of the rectangle s diagonals), the pyramid is called a right rectangular pyramid.) Surface of a pyramid (The surface of a pyramid is the union of its base region and its lateral faces. 4 P age
Familiar Terms and Symbols Vertical angles Adjacent angles Complementary Angles Supplementary Angles Angles on a line Angles at a Point Right rectangular prism 5 P age
Vocabulary Knowledge Rating: Geometry Term Never heard of the term I ve seen or have heard of it I think I know the term I know the term and can explain it Vertical Angels 1 2 3 4 Adjacent Angles 1 2 3 4 Complementary Angles 1 2 3 4 Supplementary Angles 1 2 3 4 Angles on a Line 1 2 3 4 Angles at a Point 1 2 3 4 Right Rectangular Prism 1 2 3 4 Area 1 2 3 4 Surface Area 1 2 3 4 Sphere 1 2 3 4 Cone 1 2 3 4 Pyramid 1 2 3 4 Knowledge Rating on first rating: Knowledge Rating on second rating: Date of first rating: Date of second rating: 6 P age
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Lesson 1: Real World Area Problems Classwork Opening Find the area of each shape based on the provided measurements. Explain how you found each area. Leave your answer in terms of. Lesson 1: Real World Area Problems 14 P age
Example 1 A landscape company wants to plant lawn seed. A 20 lb. bag of lawn seed will cover up to 420 sq. ft. of grass and costs $49.98 plus the 8% sales tax. A scale drawing of a rectangular yard is given. The length of the longest side is 100 ft. The house, driveway, sidewalk, garden areas, and utility pad are shaded. The unshaded area has been prepared for planting grass. How many 20 lb. bags of lawn seed should be ordered, and what is the cost? Exercise 1 A landscape contractor looks at a scale drawing of a yard and estimates that the area of the home and garage is the same as the area of a rectangle that is 100 ft. 35 ft. The contractor comes up with 5,500 ft 2. How close is this estimate? What is the percent error? Lesson 1: Real World Area Problems 15 P age
Example 2 Ten dartboard targets are being painted as shown in the following figure. The radius of the smallest circle is 3 in. and each successive, larger circle is 3 in. more in radius than the circle before it. A tester can of red and of white paint is purchased to paint the target. Each 8 oz. can of paint covers 16 ft 2. Is there enough paint of each color to create all ten targets? Lesson 1: Real World Area Problems 16 P age
Problem Set 1. A farmer has two pieces of unfenced land as shown below in the scale drawing where the dimensions of one side are given. If one acre is equal to 43,560 ft 2, how many acres does he own? 2. The Smith family is renovating a few aspects of their home. The following diagram is of a new kitchen countertop. Approximately how many square feet of counter space is there? Lesson 1: Real World Area Problems 17 P age
3. In addition to the kitchen renovation, the Smiths are laying down new carpet. Everything but closets, bathrooms, and the kitchen will have new carpet. How much carpeting must be purchased for the home? Lesson 1: Real World Area Problems 18 P age
Lesson 2: Mathematical Area Problems Classwork Opening Exercise Patty is interested in expanding her backyard garden. Currently, the garden plot has a length of 4 ft. and a width of 3 ft. a. What is the current area of the garden? Patty plans on extending the length of the plot by 3 ft. and the width by 2 ft. b. What will the new dimensions of the garden be? What will the new area of the garden be? c. Draw a diagram that shows the change in dimension and area of Patty s garden as she expands it. The diagram should show the original garden as well as the expanded garden. d. Based on your diagram, can the area of the garden be found in a way other than by multiplying the length by the width? Lesson 2: Mathematical Area Problems 19 P age
e. Based on your diagram, how would the area of the original garden change if only the length increased by 3 ft.? By how much would the area increase? f. How would the area of the original garden change if only the width increased by 2 ft.? By how much would the area increase? g. Complete the following table with the numeric expression, area, and increase in area for each change in the dimensions of the garden. Dimensions of the garden Numeric expression for the area of the garden Area of the garden Increase in area of the garden Original garden with length of 4 ft. and width of 3 ft. The original garden with length extended by 3 ft. and width extended by 2 ft. The original garden with only the length extended by 3 ft. The original garden with only the width extended by 2 ft. h. Will the increase in both the length and width by 3 ft. and 2 ft., respectively, mean that the original area will increase strictly by the areas found in parts (e) and (f)? If the area is increasing by more than the areas found in parts (e) and (f), explain what accounts for the additional increase. Lesson 2: Mathematical Area Problems 20 P age
Example 1 Examine the change in dimension and area of the following square as it increases by 2 units from a side length of 4 units to a new side length of 6 units. Observe the way the area is calculated for the new square. The lengths are given in units, and the areas of the rectangles and squares are given in units 2. a. Based on the example above, draw a diagram for a square with side length of 3 units that is increasing by 2 units. Show the area calculation for the larger square in the same way as in the example. Lesson 2: Mathematical Area Problems 21 P age
b. Draw a diagram for a square with side length of 5 units that is increased by 3 units. Show the area calculation for the larger square in the same way as in the example. c. Generalize the pattern for the area calculation of a square that has an increase in dimension. Let the side length of the original square be units and the increase in length be by units to the length and width. Use the diagram below to guide your work. Lesson 2: Mathematical Area Problems 22 P age
Example 2 Bobby draws a square that is 10 units by 10 units. He increases the length by units and the width by 2 units. a. Draw a diagram that models this scenario. b. Assume the area of the large rectangle is 156 units 2. Find the value of. Example 3 The dimensions of a square with side length units are increased. In this figure the indicated lengths are given in units, and the indicated areas are given in units 2. a. What are the dimensions of the large rectangle in the figure? Lesson 2: Mathematical Area Problems 23 P age
b. Use the expressions in your response from part (a) to write an equation for the area of the large rectangle, where represents area. c. Use the areas of the sections within the diagram to express the area of the large rectangle. d. What can be concluded from parts (b) and (c)? e. Explain how the expressions 2 3 and 3 2 6 differ within the context of the area of the figure. Lesson 2: Mathematical Area Problems 24 P age
Problem Set 1. Create an area model for each product. Use the area model to write an equivalent expression that represents the area. a. 1 4 b. 5 2 2. Use the distributive property to multiply the following expressions: a. 2 6 2 4 b. 6 4 c. 7 7 d. 3 3 Lesson 2: Mathematical Area Problems 25 P age
Lesson 3: Area Problems with Circular Regions Classwork Example 1 a. The circle to the right has a diameter of 12 cm. Calculate the area of the shaded region. b. Sasha, Barry, and Kyra wrote three different expressions for the area of the shaded region. Describe what each student was thinking about the problem based on their expression. Sasha s expression: 62 Barry s expression: 6 3 4 6 Kyra s expression: 1 2 1 2 6 Lesson 3: Area Problems with Circular Regions 26 P age
Exercise 1 a. Find the area of the shaded region of the circle to the right. 12. b. Explain how the expression you used represents the area of the shaded region. Exercise 2 Calculate the area of the figure below that consists of a rectangle and two quarter circles, each with the same radius. Leave your answer in terms of pi. Lesson 3: Area Problems with Circular Regions 27 P age
Example 2 The square in this figure has a side length of 14 inches. The radius of the quarter circle is 7 inches. a. Estimate the shaded area. b. What is the exact area of the shaded region? c. What is the approximate area using 22 7? Exercise 3 The vertices and of rectangle are centers of circles each with a radius of 5 inches. a. Find the exact area of the shaded region. Lesson 3: Area Problems with Circular Regions 28 P age
b. Find the approximate area using. c. Find the area to the nearest hundredth using your key on your calculator. Exercise 4 The diameter of the circle is 12 in. Write and explain a numerical expression that represents the area. Lesson 3: Area Problems with Circular Regions 29 P age
Lesson 4: Surface Area Classwork Opening Exercise Calculate the surface area of the square pyramid. Example 1 a. Calculate the surface area of the rectangular prism. Lesson 4: Surface Area 30 P age
b. Imagine that a piece of the rectangular prism is removed. Determine the surface area of both pieces. c. How is the surface area in part (a) related to the surface area in part (b)? Lesson 4: Surface Area 31 P age
Exercises 1 5 Determine the surface area of the right prisms. 1. 2. Lesson 4: Surface Area 32 P age
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Problem Set Determine the surface area of the figures. 1. 2. Lesson 4: Surface Area 34 P age
Lesson 5: Surface Area Classwork Example 1 Example 2 a. Determine the surface area of the cube. Lesson 5: Surface Area 35 P age
b. A square hole with a side length of 4 inches is drilled through the cube. Determine the new surface area. Example 3 A right rectangular pyramid has a square base with a side length of 10 inches. The surface area of the pyramid is 260 in 2. Find the height of the four lateral triangular faces. Lesson 5: Surface Area 36 P age
Determine the surface area of each figure. Assume all faces are rectangles unless it is indicated otherwise. In addition to your calculation, explain how the surface area was determined. Lesson 5: Surface Area 37 P age
In addition to your calculation, explain how the surface area was determined. A hexagonal prism has the following base and has a height of 8 units. Determine the surface area of the prism. Lesson 5: Surface Area 38 P age
Determine the surface area of each figure. a. b. A cube with a square hole with 3 m side lengths has been drilled through the cube. c. A second square hole with 3 m side lengths has been drilled through the cube. Lesson 5: Surface Area 39 P age
The figure below shows 28 cubes with an edge length of 1 unit. Determine the surface area. The base rectangle of a right rectangular prism is 4 ft. 6 ft. The surface area is 288 ft 2. Find the height. Let be the height in feet. Lesson 5: Surface Area 40 P age
Problem Set Determine the surface area of each figure. 1. In addition to the calculation of the surface area, describe how you found the surface area. 32 m Lesson 5: Surface Area 41 P age
2. Determine the surface area after two square holes with a side length of 2 m are drilled through the solid figure composed of two rectangular prisms. 3. The base of a right prism is shown below. Determine the surface area if the height of the prism is 10 cm. Explain how you determined the surface area. Lesson 5: Surface Area 42 P age
Lesson 5B: Angles Lesson 5B: Angles 43 P age
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Lesson 5B: Complementary and Supplementary Angles Classwork Opening As we begin our study of unknown angles, let us review key definitions. Term Definition Two angles and, with a common side, are in the interior of. angles if is When two lines intersect, any two non adjacent angles formed by those lines are called angles, or angles. Two lines are if they intersect in one point, and any of the angles formed by the intersection of the lines is 90. Two segments or rays are if the lines containing them are lines. Complete the missing information in the table below. In the Statement column, use the illustration to write an equation that demonstrates the angle relationship; use all forms of angle notation in your equations. Angle Relationship Abbreviation Statement Illustration Adjacent Angles The measurements of adjacent angles add. B A c a b C D Vertical Angles Vertical angles have equal measures. D a C b G F E Lesson 5C: Complementary and Supplementary Angles 50 P age
Angles on a Line If the vertex of a ray lies on a line but the ray is not contained in that line, then the sum of measurements of the two angles formed is 180. a b A B D C Angles at a Point Suppose three or more rays with the same vertex separate the plane into angles with disjointed interiors. Then the sum of all the measurements of the angles is 360. C B b A c a D Angle Relationship Definition Diagram 30 Complementary Angles 60 30 60 Supplementary Angles 120 60 120 60 Lesson 5C: Complementary and Supplementary Angles 51 P age
Exercise 1 In a complete sentence, describe the relevant angle relationships in the diagram. Write an equation for the angle relationship shown in the figure and solve for. Confirm your answers by measuring the angle with a protractor. x 22 Example 1 The measures of two supplementary angles are in the ratio of 2: 3. Find the two angles. Exercises 2 4 2. In a pair of complementary angles, the measurement of the larger angle is three times that of the smaller angle. Find the measurements of the two angles. Lesson 5C: Complementary and Supplementary Angles 52 P age
3. The measure of a supplement of an angle is 6 more than twice the measure of the angle. Find the two angles. 4. The measure of a complement of an angle is 32 more than three times the angle. Find the two angles. Example 2 Two lines meet at the common vertex of two rays. Set up and solve an appropriate equation for and. x y 16 Lesson 5C: Complementary and Supplementary Angles 53 P age
Problem Set 1. Two lines meet at the common vertex of two rays. Set up and solve the appropriate equations to determine and. 55 y x 2. Two lines meet at the common vertex of two rays. Set up and solve the appropriate equations to determine and. x 32 y 3. Two lines meet at the common vertex of two rays. Set up and solve an appropriate equation for and. x y 28 Lesson 5C: Complementary and Supplementary Angles 54 P age
4. Set up and solve the appropriate equations for and. s 79 19 t 5. Two lines meet at the common vertex of two rays. Set up and solve the appropriate equations for and. 38 43 m n 6. The supplement of the measurement of an angle is 16 less than three times the angle. Find the angle and its supplement. 7. The measurement of the complement of an angle exceeds the measure of the angle by 25%. Find the angle and its complement. Lesson 5C: Complementary and Supplementary Angles 55 P age
8. The ratio of the measurement of an angle to its complement is 1: 2. Find the angle and its complement. 9. The ratio of the measurement of an angle to its supplement is 3: 5. Find the angle and its supplement. 10. Let represent the measurement of an acute angle in degrees. The ratio of the complement of to the supplement of is 2: 5. Guess and check to determine the value of. Explain why your answer is correct. Lesson 5C: Complementary and Supplementary Angles 56 P age
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Lesson 5D: Solving for Unknown Angles Using Equations Classwork Opening Exercise Two lines meet at a point. In a complete sentence, describe the relevant angle relationships in the diagram. Find the values of,, and. r 25 s t Example 1 Two lines meet at the vertex of a ray. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of and. r 28 p 16 Exercise 1 Three lines meet at a point. In a complete sentence, describe the relevant angle relationship in the diagram. Set up and solve an equation to find the value of. 144 a a Lesson 5D: Solving for Unknown Angles Using Equations 59 P age
Example 2 Three lines meet at a point. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of. z 19 Exercise 2 Three lines meet at a point; 144. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to determine the value of. E c C A 144 O B F D Example 3 Two lines meet at the vertex of a ray. The ray is perpendicular to one of the lines as shown. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of. 26 t Lesson 5D: Solving for Unknown Angles Using Equations 60 P age
Exercise 3 Two lines meet at the vertex of a ray. The ray is perpendicular to one of the lines as shown. In a complete sentence, describe the relevant angle relationships in the diagram. You may add labels to the diagram to help with your description of the angle relationship. Set up and solve an equation to find the value of. 46 v Example 4 Three lines meet at a point. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of. Is your answer reasonable? x 8 Exercise 4 Two lines meet at the common vertex of two rays. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of. Find the measurements of and. A 2x O 25 B 3x C Lesson 5D: Solving for Unknown Angles Using Equations 61 P age
Exercise 5 a. In a complete sentence, describe the relevant angle relationships in the diagram. Set up and solve an equation to find the value of. Find the measurements of and. A O 5x B (2x+20) C b. Katrina was solving the problem above and wrote the equation 7 20 90. Then she rewrote this as 7 20 70 20. Why did she rewrite the equation in this way? How does this help her to find the value of? Lesson 5D: Solving for Unknown Angles Using Equations 62 P age
Problem Set 1. Two lines meet at the vertex of a ray. Set up and solve an equation to find the value of. c 17 2. Two lines meet at the vertex of a ray. Set up and solve an equation to find the value of. Explain why your answer is reasonable. 49 33 a 3. Two lines meet at the vertex of a ray. Set up and solve an equation to find the value of. 125 w Lesson 5D: Solving for Unknown Angles Using Equations 63 P age
4. Two lines meet at the common vertex of two rays. Set up and solve an equation to find the value of. m 24 68 5. Three lines meet at a point. Set up and solve an equation to find the value of. 34 r 122 6. Three lines meet at the vertex of a ray. Set up and solve an equation to find the value of each variable in the diagram. 51 43 v w x z y Lesson 5D: Solving for Unknown Angles Using Equations 64 P age
7. Set up and solve an equation to find the value of. Find the measurement of and of. 8. Set up and solve an equation to find the value of. Find the measurement of and of. 9. Set up and solve an equation to find the value of. Find the measurement of and of. B (4x+5) (5x+22) A O C 10. Write a verbal problem that models the following diagram. Then solve for the two angles. 10x 2x Lesson 5D: Solving for Unknown Angles Using Equations 65 P age
Adjacent Angles Share a vertex and a common side. lines. Vertical Angles Are formed by intersecting Vertical angles are congruent. 3 4 5 6 1 2 Supplementary Angles Complementary Angles 2 angles that have a sum of 180. 2 angles that have a sum of 90. D A B C m 1 m 2 Solve for x. Find the measure of each angle. m ABD = 6x B m DBC = 3x m 1 = 3x m 2 = 2x 66 P age
Vertical angles are formed when two lines intersect. The angles opposite each other are called vertical angles, and they are congruent. Example: Solve for x. Find the measure of each angle. Solution: (5x 1) (3x + 11) Complementary angles are two angles whose sum is 90. Example: Solve for x. Find the measure of each angle. Solution: (3x) (x) Supplementary angles are two angles whose sum is 180. Example: Solve for x. Find the measure of each angle. Solution: 67 P age
FOOD FOR THOUGHT Directions: Each pair of angles is either vertical, complementary, or supplementary. Find the degree measure of each angle. On the way, you will find the variable in each problem. To find the answer to the question, write the variable on its matching line at the bottom of the back. 1. 2. (4t + 16)º (3e 32)º (e + 16)º (5t 21)º 3. 4. (4o 28)º (8i 14)º (o + 13)º (2i 6)º 68 P age
5. 6. (6f + 3)º (3f + 6)º (2s)º (7s 18)º 7. (4L + 60)º (L + 20)º QUESTION: What meal can you get from standing on a hot sidewalk? 19 11 20 20 49 37 21 19 12 21 20 49 69 P age
Lesson 6: Angle Problems and Solving Equations Classwork Name of Angle Relationship Angle Facts and Definitions Angle Fact Diagram B Adjacent Angles A a b C D Vertical Angles (vert. ) D F a C b G E C Angles on a Line ( on a line) B a A b c D E B Angles at a Point ( at a point) C b A c a D Lesson 6: Angle Problems and Solving Equations 70 P age
Opening Exercise Find all the angles and complete the chart to follow. Name the Angles that are Vertical Adjacent Angles on a Line Angles at a Point Example 1 Estimate the value of. In a complete sentence, describe the angle relationship in the diagram. C B x 132 A D Write an equation for the angle relationship shown in the figure and solve for. Then find the measures of. Lesson 6: Angle Problems and Solving Equations 71 P age
Exercise 1 In a complete sentence, describe the angle relationship in the diagram. C D B 3x 2x A E Find the measurements of and. Example 2 In a complete sentence, describe the angle relationship in the diagram. 144 x L A E B y K Write an equation for the angle relationship shown in the figure and solve for and. Find the measurements of and. Lesson 6: Angle Problems and Solving Equations 72 P age
Exercise 2 In a complete sentence, describe the angle relationships in the diagram. J 3x N 16 M E 85 K L Write an equation for the angle relationship shown in the figure and solve for. Example 3 In a complete sentence, describe the angle relationships in the diagram. E G K 135 x F Write an equation for the angle relationship shown in the figure and solve for. Find the measurement of. Lesson 6: Angle Problems and Solving Equations 73 P age
Exercise 3 In a complete sentence, describe the angle relationships in the diagram. F E 167 103 A 59 (x+1) G Find the measurement of. H Example 4 Two lines intersect in the following figure. In the figure, the ratio of the measurements of the obtuse angle to the acute angle in any adjacent angle pair is 2: 1. In a complete sentence, describe the angle relationships in the diagram. 2x x Label the diagram with expressions that describe this relationship. Write an equation that models the angle relationship and solve for. Find the measurements of the acute and obtuse angles. Lesson 6: Angle Problems and Solving Equations 74 P age
Exercise 4 The ratio of to is 2: 3. In a complete sentence, describe the angle relationships in the diagram. E H G 2x 3x F Find the measures of and. Relevant Vocabulary Adjacent Angles: Two angles and with a common side are adjacent angles if belongs to the interior of. Vertical Angles: Two angles are vertical angles (or vertically opposite angles) if their sides form two pairs of opposite rays. Angles on a Line: The sum of the measures of adjacent angles on a line is 180. Angles at a Point: The sum of the measures of adjacent angles at a point is 360. Lesson 6: Angle Problems and Solving Equations 75 P age
Problem Set For each question, use angle relationships to write an equation in order to solve for each variable. Determine the indicated angles. 1. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measurement of. D E x C A 65 F 2. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measurement of. 3. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measurements of and. D E C B 2x 103 3x 10 12 Q F G 4. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measure of. x 71 x 71 71 x 71 x Lesson 6: Angle Problems and Solving Equations 76 P age
5. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measure of and. B D E 25 x K F y C A 6. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measure of and. E F C K J y y H x A y F x 24 G D 7. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measure of and. Lesson 6: Angle Problems and Solving Equations 77 P age
8. In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measure of. G C 3x 56 Q 155 F D E 9. The ratio of the measures of a pair of adjacent angles on a line is 4: 5. a. Find the measures of the two angles. b. Draw a diagram to scale of these adjacent angles. Indicate the measurements of each angle. 10. The ratio of the measures of three adjacent angles on a line is 3: 4: 5. Find the measures of the three angles. a. Find the measures of the three angles. b. Draw a diagram to scale of these adjacent angles. Indicate the measurements of each angle. Lesson 6: Angle Problems and Solving Equations 78 P age
Lesson 7: Angle Problems and Solving Equations Classwork Opening Exercise a. In a complete sentence, describe the angle relationship in the diagram. Write an equation for the angle relationship shown in the figure and solve for. x 14 b. and are intersecting lines. In a complete sentence, describe the angle relationship in the diagram. Write an equation for the angle relationship shown in the figure and solve for. F C 147 D E y 51 c. In a complete sentence, describe the angle relationship in the diagram. Write an equation for the angle relationship shown in the figure and solve for. 59 41 b 65 Lesson 7: Angle Problems and Solving Equations 79 P age
d. The following figure shows three lines intersecting at a point. In a complete sentence, describe the angle relationship in the diagram. Write an equation for the angle relationship shown in the figure and solve for. z z 158 e. Write an equation for the angle relationship shown in the figure and solve for. In a complete sentence, describe the angle relationship in the diagram. Find the measurements of and. Lesson 7: Angle Problems and Solving Equations 80 P age
Example 2 In a complete sentence, describe the angle relationships in the diagram. You may label the diagram to help describe the angle relationships. Write an equation for the angle relationship shown in the figure and solve for. x 77 Exercise 2 In a complete sentence, describe the angle relationships in the diagram. Write an equation for the angle relationship shown in the figure and solve for and. y x 27 Lesson 7: Angle Problems and Solving Equations 81 P age
Example 3 In a complete sentence, describe the angle relationships in the diagram. Write an equation for the angle relationship shown in the figure and solve for. Find the measures of and. H J 2x G 225 A 3x F Exercise 3 In a complete sentence, describe the angle relationships in the diagram. Write an equation for the angle relationship shown in the figure and solve for. Find the measure of. L J 24 x G 5x K M Lesson 7: Angle Problems and Solving Equations 82 P age
Example 4 In the accompanying diagram, is four times the measure of. a. Label as and as. Write an equation that describes the relationship between and. A C F x B 50 y D G b. Find the value of. E c. Find the measures of,,,,. d. What is the measure of? Identify the angle relationship used to get your answer. Lesson 7: Angle Problems and Solving Equations 83 P age
Problem Set In a complete sentence, describe the angle relationships in each diagram. Write an equation for the angle relationship(s) shown in the figure, and solve for the indicated unknown angle. 1. Find the measure of,, and. D E 6x 4x 2x 30 F C A G 2. Find the measure of. a 126 a 26 3. Find the measure of and. y x 65 25 Lesson 7: Angle Problems and Solving Equations 84 P age
4. Find the measure of. H E A 81 x 15 J F G 5. Find the measure of and. F 20 D E A 3x H 2x C B 6. The measure of. The measure of is five more than two times. The measure of is twelve less than eight times. Find the measures of,, and. b S T Q (8b 12) P (2b+5) R Lesson 7: Angle Problems and Solving Equations 85 P age
7. Find the measure of and. E C H 2y A 21 y Q B G D F 8. The measures of three angles at a point are in the ratio of 2: 3: 5. Find the measures of the angles. 9. The sum of the measures of two adjacent angles is 72. The ratio of the smaller angle to the larger angle is 1: 3. Find the measures of each angle. Lesson 7: Angle Problems and Solving Equations 86 P age
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