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Mathematics: Applications and Concepts, Course 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240 Lesson 10-1 Angles Lesson 10-2 Making Circle Graphs Lesson 10-3 Angle Relationships Lesson 10-4 Triangles Lesson 10-5 Quadrilaterals Lesson 10-6 Similar Figures Lesson 10-7 Polygons and Tessellations Lesson 10-8 Translations Lesson 10-9 Reflections Example 1 Classify Angles Example 2 Classify Angles Example 3 Draw an Angle Classify the angle as acute, obtuse, right, or straight. Answer: The angle is exactly 180, so it is a straight angle. Classify the angle as acute, obtuse, right, or straight. Answer: right Classify the angle as acute, obtuse, right, or straight. Answer: The angle is less than 90, so it is an acute angle. Classify the angle as acute, obtuse, right, or straight. Answer: obtuse Draw an angle with a measure of 140. Classify the angle as acute, obtuse, right, or straight. Draw a ray. Place the center point of the protractor on the endpoint of the ray. Align the ray with 0°. Use the scale that begins with 0°. Locate the mark labeled 140. Then draw the other side of the angle. Answer: Since the angle is greater than 90°, it is obtuse. Draw an angle with a measure of 60. Classify the angle as acute, obtuse, right, or straight. Answer: acute Example 1 Construct a Circle Graph Example 2 Construct a Circle Graph Example 3 Interpret a Circle Graph SPORTS In a survey, a group of middle school students were asked to name their favorite sport. The results are shown in the table. Make a circle graph of the data. Sport football basketball baseball tennis other Percent 30% 25% 22% 8% 15% Find the degrees for each part. Round to the nearest whole degree. football: basketball: baseball: tennis: other: Use a compass to draw a circle with a radius marked as shown. Then use a protractor to draw the first angle, in this case 108. Repeat this step for each section. Check To draw an accurate circle graph, make sure the sum of the angle measures equals 360°. Label each section of the graph with the category and percent. Give the graph a title. Answer: Students’ Favorite Sports ICE CREAM In a survey, a group of students were asked to name their favorite flavor of ice cream. The results are shown in the table. Make a circle graph of the data. Flavor chocolate cookie dough Percent 30% 25% peanut butter strawberry 15% 10% other 20% Answer: MOVIES Gina has the following types of movies in her DVD collection. Make a circle graph of the data. Type of Movie action comedy science fiction Find the total number of DVDs: Number 24 15 7 Find the ratio that compares each number with the total. Write the ratio as a decimal number rounded to the nearest hundredth. Find the number of degrees for each section of the graph. : = : = = Draw the circle graph. Check After drawing the first two sections, you can measure the last section of the circle graph to verify that the angles have the correct measures. Answer: MARBLES Michael has the following colors of marbles in his marble collection. Make a circle graph of the data. Color Black Green Red Gold Number 12 9 5 3 Answer: VOTING The circle graph below shows the percent of voters in a town who are registered with a political party. Use the graph to describe the voters in this town. Answer: The majority of voters are Democrats. Only a very small portion are Independents. SPORTS The circle graph below shows the responses of middle school students to the question, “Should teens be allowed to play professional sports?”. Use the graph to describe the opinions of the middle school students. Answer: The majority of the middle school students felt that teens should be allowed to play professional sports. A small portion of the middle school students had no opinion on this issue. Example 1 Classify Angles Example 2 Classify Angles Example 3 Find a Missing Angle Measure Example 4 Use Angles to Solve A Problem Classify the pair of angles as complementary, supplementary, or neither Answer: The angles are supplementary. Classify the pair of angles as complementary, supplementary, or neither Answer: complementary Classify the pair of angles as complementary, supplementary, or neither x and y form a right angle. Answer: So, the angles are complementary. Classify the pair of angles as complementary, supplementary, or neither Answer: supplementary Angles PQS and RQS are supplementary. If mPQS 56, find mRQS. Since PQS and RQS are supplementary, Write the equation. Replace mPQS with 56. Subtract 56 from each side. Answer: The measure of RQS is 124. Angles MNP and KNP are complementary. If mMNP 23, find mKNP. Answer: 67 GEOMETRY The rectangle shown is divided by a diagonal. Find the value of x. The sum of the two angles created by the diagonal is 90 because the corner of a rectangle is a right angle. Write the equation. Subtract 70 from each side. Answer: The value of x is 20. GRAPHING In the circle graph shown below, find the value of x. Answer: 62 Example 1 Find Angle Measures of Triangles Example 2 Find a Missing Measure Example 3 Classify Triangles Example 4 Classify Triangles PLANES An airplane has wings that are shaped like triangles. Find the missing measure. The sum of the measures is 180. Simplify. Subtract 159 from each side. Answer: The missing measure is 21. SEWING A piece of fabric is shaped like a triangle. Find the missing measure. Answer: 49 ALGEBRA Find mA in ABC if mA mC 80. mB and Draw the triangle. Then write and solve an equation to find mA. The sum of the measures is 180. Replace B with A because the angles are equal. Simplify and replace mC with 80. Subtract 80 from each side. Simplify. Divide each side by 2. Answer: The measure of A is 50. ALGEBRA Find mJ in JKL if mJ mL 100. Answer: 40 mK and Classify the triangle by its angles and its sides. Answer: The triangle has one obtuse angle and two congruent sides. So, it is an obtuse, isosceles triangle. Classify the triangle by its angles and its sides. Answer: right, scalene Classify the triangle by its angles and its sides. Answer: The triangle has all acute angles and no congruent sides. So, it is an acute, scalene triangle. Classify the triangle by its angles and its sides. Answer: acute, equilateral Example 1 Classify Quadrilaterals Example 2 Classify Quadrilaterals Example 3 Find a Missing Measure Classify the quadrilateral using the name that best describes it. Answer: The quadrilateral has 4 right angles and opposite sides are congruent. It is a rectangle. Classify the quadrilateral using the name that best describes it. Answer: parallelogram Classify the quadrilateral using the name that best describes it. Answer: The quadrilateral has one pair of parallel sides. It is a trapezoid. Classify the quadrilateral using the name that best describes it. Answer: square Find the missing angle measure in the quadrilateral. The sum of the measures is 360. Simplify. Subtract 240 from both sides. Answer: The missing angle measure is 120. Find the missing angle measure in the quadrilateral. Answer: 134 Example 1 Find Side Measures of Similar Triangles Example 2 Use Indirect Measurement If ABC DEF, find the length of Since the two triangles are similar, the ratios of their corresponding sides are equal. So, you can write and solve a proportion to find Write a proportion. Let n represent the length of Then substitute. Find the cross products. Simplify. Divide each side by 3. Answer: The length of is 16.5 centimeters. . If JKL MNO, find the length of Answer: 13.5 in. GRID-IN TEST ITEM A rectangular picture window 12-feet long and 6-feet wide needs to be shortened to 9 feet in length to fit a redesigned wall. If the architect wants the new window to be similar to the old window, how wide will the new window be? Read the Test Item To find the width of the new window, draw a picture and write a proportion. Solve the Test Item Write a proportion. Find the cross products. Divide each side by 12. So, the width of the new window will be 4.5 feet. Answer: GRID-IN TEST ITEM Tom has a rectangular garden which has a length of 12 feet and a width of 8 feet. He wishes to start a second garden which is similar to the first and will have a width of 6 feet. Find the length of the new garden. Answer: Example 1 Classify Polygons Example 2 Classify Polygons Example 3 Angle Measures of a Polygon Example 4 Tessellations Determine whether the figure is a polygon. If it is, classify the polygon and state whether it is regular. If it is not a polygon, explain why. Answer: The figure is not a polygon since it has a curved side. Determine whether the figure is a polygon. If it is, classify the polygon and state whether it is regular. If it is not a polygon, explain why. Answer: pentagon, not regular Determine whether the figure is a polygon. If it is, classify the polygon and state whether it is regular. If it is not a polygon, explain why. Answer: This figure has 6 sides which are not all of equal length. It is a hexagon that is not regular. Determine whether the figure is a polygon. If it is, classify the polygon and state whether it is regular. If it is not a polygon, explain why. Answer: not a polygon, sides overlap ALGEBRA Find the measure of each angle of a regular heptagon. Round to the nearest tenth of a degree. Draw all of the diagonals from one vertex and count the number of triangles formed. Find the sum of the measures of the polygon. number of triangles formed 180° sum of angle measures in polygon Find the measure of each angle of the polygon. Let n represent the measure of one angle in the heptagon. There are seven congruent angles. Divide each side by 7. Answer: The measure of each angle in a regular heptagon is about 128.6. Find the measure of each angle in a regular hexagon. Answer: 120 PATTERNS Ms. Pena is creating a pattern on her wall. She wants to use triangles with angles 120, 30, and 30. Can Ms. Pena tessellate with these triangles? The sum of the measures of the angles where the vertices meet must be 360 Both 30 and 120 divide evenly into 360. Therefore, Ms. Pena can arrange the triangles in a way that the angles where the vertices meet make 360. She can tessellate with these triangles. Check You can check if your answer is correct by drawing a tessellation of triangles with angles measuring 120°, 30°, and 30°. Answer: Yes, they can be arranged in a way that the angles where the vertices meet make 360. QUILTING Emily is making a quilt using fabric pieces shaped as equilateral triangles. Can Emily tessellate the quilt with these fabric pieces? Answer: Yes, they can be arranged in a way that the angles where the vertices meet make 360. Example 1 Graph a Translation Example 2 Find Coordinates of a Translation Example 3 Naming Translations with Ordered Pairs Translate ABC 5 units left and 1 unit up. Move each vertex of the figure 5 units left and 1 unit up. Label the new vertices A, B, and C. Connect the vertices to draw the triangle. Answer: The coordinates of the vertices of the new figure are A(–4, –1), B(–1, 2), and C(0, –1). Translate DEF 3 units left and 2 units down. Answer: Trapezoid GHIJ has vertices G(–4, 1), H(–4, 3), I(–2, 3), and J(–1, 1). Find the vertices of trapezoid GHIJ after a translation of 5 units right and 3 units down. Then graph the figure and its translated image. (x + 5, y – 3) Vertices of trapezoid GHIJ G(–4, 1) H(–4, 3) (–4 + 5, 1 – 3) (–4 + 5, 3 – 3) I(–2, 3) J(–1, 1) (–2 + 5, 3 – 3) (–1 + 5, 1 – 3) Vertices of trapezoid GHIJ G(1, –2) H(1, 0) I(3, 0) J(4, –2) Answer: The coordinates of trapezoid GHIJ are G(1, –2), H(1, 0), I(3, 0), and J(4, –2). Triangle MNO has vertices M(–5, –3), N(–7, 0), and O(–2, 3). Find the vertices of triangle MNO after a translation of 6 units right and 3 units up. Then graph the figure and its translated image. Answer: M(1, 0) N(–1, 3) O(4, 6) CLASSROOMS The squares below represent desks in a classroom. Ana is seated at the square marked X. She is moved to the seat marked Y. Describe this translation as an ordered pair. Answer: Ana moved 2 places right and 2 places up. So, the translation can be written (2, 2). CONVENTION The squares below represent booths at a convention center. The Sail Maker Company was located at the booth marked X last year. This year they have been moved to the booth marked Y. Describe this translation as an ordered pair. Answer: (–3, –3) Example 1 Identify Lines of Symmetry Example 2 Identify Lines of Symmetry Example 3 Identify Lines of Symmetry Example 4 Reflect a Figure Over the x-axis Example 5 Reflect a Figure Over the y-axis Determine whether the figure has line symmetry. If so, copy the figure and draw all lines of symmetry. Answer: This figure has line symmetry. There are two lines of symmetry. Determine whether the figure has line symmetry. If so, copy the figure and draw all lines of symmetry. Answer: no symmetry Determine whether the figure has line symmetry. If so, copy the figure and draw all lines of symmetry. Answer: This figure has line symmetry. There is one line of symmetry. Determine whether the figure has line symmetry. If so, copy the figure and draw all lines of symmetry. Answer: There are two lines of symmetry. Determine whether the figure has line symmetry. If so, copy the figure and draw all lines of symmetry. Answer: This figure does not have line symmetry. Determine whether the figure has line symmetry. If so, copy the figure and draw all lines of symmetry. Answer: There is one line of symmetry. Quadrilateral QRST has vertices Q(–1, 1), R(0, 3), S(3, 2), and T(4, 0). Find the coordinates of QRST after a reflection over the x-axis. Then graph the figure and its reflected image. Vertices of Quadrilateral QRST Distance from x-axis Q(–1, 1) R(0, 3) 1 S(3, 2) T(4, 0) 3 2 0 Vertices of Quadrilateral QRST Q(–1, –1) R(0, –3) S(3, –2) T(4, 0) Plot the vertices and connect to form the quadrilateral QRST. The x-axis is the line of symmetry. So, the distance from each point on quadrilateral QRST to the line of symmetry is the same as the distance from the line of symmetry to quadrilateral QRST. Answer: Q(–1, –1) R(0, –3) S(3, –2) T(4, 0) Quadrilateral ABCD has vertices A(–3, 2), B(–1, 5), C(3, 3), and D(2, 1). Find the coordinates of ABCD after a reflection over the x-axis. Then graph the figure and its reflected image. Answer: A(–3, –2) B(–1, –5) C(3, –3) D(2, –1) Triangle XYZ has vertices X(1, 2), Y(2, 1), and Z(1, –2). Find the coordinates of XYZ after a reflection over the y-axis. Then graph the figure and its reflected image. Vertices of XYZ X(1, 2) Y(2, 1) Z(1, –2) Distance from y-axis 1 2 1 Vertices of XYZ X(–1, 2) Y(–2, 1) Z(–1, –2) Plot the vertices and connect to form the triangle XYZ. The y-axis is the line of symmetry. So, the distance from each point on triangle XYZ to the line of symmetry is the same as the distance from the line of symmetry to triangle XYZ. Answer: X(–1, 2), Y(–2, 1), Z(–1, –2) Triangle QRS has vertices Q(3, 4), R(1, 0), and S(6, 2). Find the coordinates of QRS after a reflection over the y-axis. Then graph the figure and its reflected image. Answer: Q(–3, 4) R(–1, 0) S(–6, 2) Explore online information about the information introduced in this chapter. Click on the Connect button to launch your browser and go to the Mathematics: Applications and Concepts, Course 2 Web site. At this site, you will find extra examples for each lesson in the Student Edition of your textbook. When you finish exploring, exit the browser program to return to this presentation. If you experience difficulty connecting to the Web site, manually launch your Web browser and go to www.msmath2.net/extra_examples. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. Click the mouse button or press the Space Bar to display the answers. To navigate within this Interactive Chalkboard product: Click the Forward button to go to the next slide. Click the Previous button to return to the previous slide. Click the Section Back button to return to the beginning of the lesson you are working on. If you accessed a feature, this button will return you to the slide from where you accessed the feature. Click the Main Menu button to return to the presentation main menu. Click the Help button to access this screen. Click the Exit button or press the Escape key [Esc] to end the current slide show. Click the Extra Examples button to access additional examples on the Internet. Click the 5-Minute Check button to access the specific 5-Minute Check transparency that corresponds to each lesson. End of Custom Shows WARNING! Do Not Remove This slide is intentionally blank and is set to auto-advance to end custom shows and return to the main presentation.