12.1. Angle Relationships. Identifying Complementary, Supplementary Angles. Goal: Classify special pairs of angles. Vocabulary. Complementary. angles.

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1 . Angle Relationships Goal: Classif special pairs of angles. Vocabular Complementar angles: Supplementar angles: Vertical angles: Eample Identifing Complementar, Supplementar Angles In quadrilateral PQRS, identif all pairs of complementar angles and supplementar angles. Q 5 55 R 5 P 5 S a. Because maq mar, aq and ar are angles. b. Because map maq, ap and aq are angles. c. Because mar mas, ar and as are angles. Copright Holt McDougal. All rights reserved. Chapter Pre-Algebra Notetaking Guide 9

2 Checkpoint Tell whether the angles are complementar, supplementar, or neither.. ma 7. ma. ma5 7 ma 7 ma 8 ma6 5 Adjacent angles that form a right angle are complementar. Adjacent angles that form a straight angle are supplementar. Eample For the diagram shown, a and a are complementar angles, and ma 6. Find ma. Finding an Angle Measure ma ma ma Definition of complementar angles Substitute for ma. ma Subtract from each side. Checkpoint a and a are complementar angles. Given ma, find ma.. ma 6 5. ma 6. ma 8 7. ma 7 9 Chapter Pre-Algebra Notetaking Guide Copright Holt McDougal. All rights reserved.

3 Eample Using Supplementar and Vertical Angles For the diagram shown, ma 65. Find ma, ma, and ma. a. ma ma a and a are supplementar. ma Substitute for ma. ma Subtract from each side. b. ma Vertical angles have same measure. ma Substitute for ma. c. ma Vertical angles have same measure. ma Substitute for ma. Checkpoint 8. a and a are supplementar 9. a and a are supplementar angles, and ma. angles, and ma. Find ma. Find ma. 0. In Eample, suppose that ma 5. Find ma, ma, and ma. Copright Holt McDougal. All rights reserved. Chapter Pre-Algebra Notetaking Guide 95

4 . Angles and Parallel Lines Goal: Identif angles when a transversal intersects lines. Vocabular Transversal: Corresponding angles: Alternate interior angles: Alternate eterior angles: Eample Identifing Angles In the diagram, line t is a transversal. Tell whether the angles are corresponding, alternate interior, or alternate eterior angles. a. a and a5 m n t b. a and a7 c. a and a6 a. a and a5 are angles. b. a and a7 are angles. c. a and a6 are angles. 96 Chapter Pre-Algebra Notetaking Guide Copright Holt McDougal. All rights reserved.

5 Checkpoint In Eample, tell whether the angles are corresponding, alternate interior, or alternate eterior angles.. a and a5. a and a8. a and a8 Angles and Parallel Lines In the diagram, transversal t intersects parallel lines m and n. Corresponding angles ma ma ma ma t m n Alternate interior angles ma ma Alternate eterior angles ma ma Copright Holt McDougal. All rights reserved. Chapter Pre-Algebra Notetaking Guide 97

6 Eample In the diagram, transversal t intersects parallel lines m and n. If ma 00, find the measures of the other numbered angles. Finding Angle Measures t m n ma5,because a and a5 are angles. ma,because a and a5 are angles. ma8,because a and a8 are angles. ma,because a and a are angles. ma6,because a and a6 are angles. ma,because a and a6 are angles. ma7,because a and a7 are angles. Checkpoint. In Eample, if ma 85, find the measures of the other angles. If a transversal intersects two lines so that the corresponding angles have the same measure, then the lines are parallel. Eample Find the value of that makes lines m and n parallel. Finding the Value of a Variable The labeled angles in the diagram are corresponding angles. Lines m and n are when the measures are. ( 5) t m n Set measures equal. Subtract from each side. Divide each side b. 98 Chapter Pre-Algebra Notetaking Guide Copright Holt McDougal. All rights reserved.

7 . Angles and Polgons Goal: Find measures of interior and eterior angles. Vocabular Interior angle: Eterior angle: Measures of Interior Angles of a Conve Polgon The sum of the measures of the interior angles of a conve n-gon is given b the formula (n ) p 80. The measure of an interior angle of a regular n-gon is given b the formula (n ) p 80. n Eample Finding the Sum of a Polgon s Interior Angles Find the sum of the measures of the interior angles of the polgon. For a conve heagon, n. (n ) p 80 ( ) p 80 p 80 Copright Holt McDougal. All rights reserved. Chapter Pre-Algebra Notetaking Guide 99

8 Eample Finding the Measure of an Interior Angle Find the measure of an interior angle of a regular octagon. For a regular octagon, n 8. Measure of an interior angle Write formula. Substitute for n. Simplif. Checkpoint. Find the sum of the measures of the interior angles of a conve 9-gon.. Find the measure of an interior angle of a regular 8-gon. An interior angle and an eterior angle at the same verte form a straight angle. Eample Find ma in the diagram. Finding the Measure of an Eterior Angle The angle that measures forms a straight angle with a, which is the eterior angle at the same verte. ma Angles are supplementar. ma Subtract from each side. 00 Chapter Pre-Algebra Notetaking Guide Copright Holt McDougal. All rights reserved.

9 Checkpoint. In Eample, find ma, ma, ma, and ma5. Each verte of a conve polgon has two eterior angles. If ou draw one eterior angle at each verte, then the sum of the measures of these angles is 60. Eample Using the Sum of Measures of Eterior Angles Find the unknown angle measure in the diagram Add. Answer: The angle measure is. Sum of measures of eterior angles of a conve polgon is 60. Subtract from each side. Checkpoint. Five eterior angles of a conve heagon have measures, 78, 60, 55, and 6. Find the measure of the sith eterior angle. Copright Holt McDougal. All rights reserved. Chapter Pre-Algebra Notetaking Guide 0

10 . Translations Goal: Translate figures in a coordinate plane. Vocabular Transformation: Image: Translation: Tessellation: In a translation, a figure and its image are congruent. Eample Describing a Translation For the diagram shown, describe the translation in words. Think of moving horizontall and verticall from a point on the original figure to the corresponding point on the new figure. For instance, A ou move units to the and units from A(, ) to reach A. A 0 Chapter Pre-Algebra Notetaking Guide Copright Holt McDougal. All rights reserved.

11 Coordinate Notation You can describe a translation of each point (, ) of a figure using the coordinate notation (, ) ( a, b) where a indicates how man units a point moves horizontall, and b indicates how man units a point moves. Move the point (, ) to the right if a is positive and to the if a is. Move the point up if b is positive and if b is. B (, ) A(, ) (, ) B (, ) A (, ) Eample Translating a Figure Draw TABC with vertices A(, ), B(, ), and C(0, ). Then find the coordinates of the vertices of the image after the translation (, ) (, 5), and draw the image. First draw TABC. Then, to translate TABC, -coordinate and each verte. riginal Image (, ) (, 5) A(, ) A from the -coordinate of to the B(, ) B C(0, ) C Finall, draw TA B C. Notice that each point on TABC moves units to the and units. Copright Holt McDougal. All rights reserved. Chapter Pre-Algebra Notetaking Guide 0

12 Checkpoint. Draw quadrilateral PQRS with vertices P(, ), Q(, 0), R(, ), and S(, ). Then find the coordinates of the image after the translation (, ) ( 6, 5), and draw the image. Eample Creating Tessellations Tell whether ou can create a tessellation using onl translations of the given polgon. If ou can, create a tessellation. If not, eplain wh not. a. b. a. You translate a b. You translate the regular octagon to create a rectangle to create a tessellation. Notice in the tessellation. Notice in the design that there design that there gaps and overlaps. gaps or overlaps. 0 Chapter Pre-Algebra Notetaking Guide Copright Holt McDougal. All rights reserved.

13 .5 Reflections and Smmetr Goal: Reflect figures and identif lines of smmetr. Vocabular Reflection: Line of reflection: Line smmetr: Line of smmetr: In a reflection, a figure and its image are congruent. Eample Identifing Reflections Tell whether the transformation is a reflection. If so, identif the line of reflection. a. b. c. a. b. c. Copright Holt McDougal. All rights reserved. Chapter Pre-Algebra Notetaking Guide 05

14 Coordinate Notation You can use coordinate notation to describe the images of figures after reflections in the aes of a coordinate plane. Reflection in the -ais Reflection in the -ais A(, ) B(, ) A(, ) A (, ) A (, ) B (, ) B(, ) B (, ) Multipl the -coordinate b. (, ) Multipl the -coordinate b. (, ) Eample Draw TABC with vertices A(, ), B(, 5), and C(, ). Then find the coordinates of the vertices of the image after a reflection in the -ais, and draw the image. Reflecting a Triangle First draw TABC. Then, to reflect TABC in the -ais, multipl the -coordinate of each verte b. riginal Image (, ) A(, ) A B(, 5) B C(, ) C Finall, draw TA B C. 06 Chapter Pre-Algebra Notetaking Guide Copright Holt McDougal. All rights reserved.

15 Checkpoint. Draw TABC with vertices A(, ), B(, ), and C(, ). Then find the coordinates of the vertices of the image of TABC after a reflection in the -ais, and draw the image. Eample Identifing Lines of Smmetr Draw the lines of smmetr on the figure. Tell how man lines of smmetr the figure has. a. b. c. Checkpoint Draw the lines of smmetr on the figure. Tell how man lines of smmetr the figure has.... Copright Holt McDougal. All rights reserved. Chapter Pre-Algebra Notetaking Guide 07

16 .6 Rotations and Smmetr Goal: Rotate figures and identif rotational smmetr. Vocabular Rotation: Center of rotation: Angle of rotation: Rotational smmetr: In a rotation, a figure and its image are congruent. Eample Identifing Rotations Tell whether the transformation is a rotation about the origin. If so, give the angle and direction of the rotation. a. b. c. a. b. c. 08 Chapter Pre-Algebra Notetaking Guide Copright Holt McDougal. All rights reserved.

17 90 Rotations In this lesson, all rotations in the coordinate plane are centered at the origin. You can use coordinate notation to describe a 90 rotation of a figure about the origin. 90 clockwise rotation 90 counterclockwise rotation A(, ) B(, ) A (, ) B (, ) 5 B (, ) A(, ) A (, ) B(, ) Switch the coordinates, then multipl the new -coordinate b. (, ) Switch the coordinates, then multipl the new -coordinate b. (, ) Eample Draw TABC with vertices A(, ), B(, ), and C(, 0). Then find the coordinates of the vertices of the image after a 90 clockwise rotation, and draw the image. Rotating a Triangle First draw TABC. Then, to rotate TABC 90 clockwise, switch the coordinates and multipl the new -coordinate b. riginal Image (, ) A(, ) A B(, ) B C(, 0) C Finall, draw TA B C. 5 6 Copright Holt McDougal. All rights reserved. Chapter Pre-Algebra Notetaking Guide 09

18 Checkpoint. Draw TABC with vertices A(, ), B(, ), and C(, ). Then find the coordinates of the vertices of the image after a 90 counterclockwise rotation, and draw the image Rotations To rotate a point 80 about the origin, multipl each coordinate b. The image is the same whether ou rotate the figure or. (, ) B (, ) A(, ) B(, ) A (, ) 0 Chapter Pre-Algebra Notetaking Guide Copright Holt McDougal. All rights reserved.

19 Eample Rotating a Triangle Draw TMNP with vertices M(, ), N(, ), and P(, ). Then find the coordinates of the vertices of the image after a 80 rotation, and draw the image. First draw TMNP. Then, to rotate TMNP 80, multipl the coordinates b. riginal Image (, ) M(, ) M N(, ) N P(, ) P Finall, draw TM N P. Eample Identifing Rotational Smmetr Tell whether the figure has rotational smmetr. If so, give the angle and direction of rotation. a. b. c. a. The figure rotational smmetr. b. The figure rotational smmetr. c. The figure rotational smmetr. Copright Holt McDougal. All rights reserved. Chapter Pre-Algebra Notetaking Guide

20 .7 Dilations Goal: Dilate figures in a coordinate plane. Vocabular Dilation: Center of dilation: Scale factor: In a dilation, a figure and its image are similar. Regardless of the value of k (k 0), a point and its image after a dilation are in the same quadrant(s). Dilation In this lesson, the origin of the coordinate plane is the center of dilation. In the diagram, &*& A B is the image of &* AB after a dilation. Because A B, the scale factor is. You can describe a dilation with AB respect to the origin using the notation (, ) (k, k) where k is the. 6 5 A (, ) A(, ) B(, ) B (8, ) Center of dilation Chapter Pre-Algebra Notetaking Guide Copright Holt McDougal. All rights reserved.

21 Eample Dilating a Quadrilateral Draw quadrilateral with vertices A(, ), B(, ), C(, ), and D(, ). Then find the coordinates of the vertices of the image after a dilation having a scale factor of, and draw the image. First draw quadrilateral ABCD. Then, to dilate ABCD, multipl the - and -coordinates of each verte b. riginal Image (, ) A(, ) A B(, ) B 6 5 Notice in Eample that when k >, the new figure is an enlargement of the original figure. C(, ) C D(, ) D Finall, draw quadrilateral A B C D. Checkpoint Draw TDEF with vertices D(, ), E(, ), and F(, ). Then find the coordinates of the vertices of the image after a dilation having a scale factor of, and draw the image Copright Holt McDougal. All rights reserved. Chapter Pre-Algebra Notetaking Guide

22 Eample Using a Scale Factor Less than Draw TPQR with vertices P( 8, ), Q( 6, 6), and R(, ). Then find the coordinates of the vertices of the image after a dilation having a scale factor of 0.5, and draw the image. Draw TPQR. Then, to dilate TPQR,multipl the - and -coordinates of each verte b. Notice in Eample that when k <, the new figure is a reduction of the original figure. riginal (, ) P( 8, ) P Q( 6, 6) Q R(, ) R Finall, draw TP Q R. Eample Image Computer Graphics An artist uses a computer program to enlarge a design, as shown. What is the scale factor of the dilation? The width of the original design is Finding a Scale Factor units. The width of the image is units. So, the units scale factor is, or. units (, ) (9, ) (, ) (, ) Checkpoint. Given CD &* with endpoints C(6, 9) and D(, ), let C D &** with endpoints C (, ) and D, be the image of CD &* after a dilation. Find the scale factor. Chapter Pre-Algebra Notetaking Guide Copright Holt McDougal. All rights reserved.

23 Summar Transformations in a Coordinate Plane Translations In a translation, each point of a figure is moved the in the. (, ) Reflections In a reflection, a figure is over a line. Reflection in -ais: (, ) Reflection in -ais (shown): (, ) Rotations In a rotation, a figure is turned about the origin through a given and. 90 clockwise rotation (shown): (, ) 90 counterclockwise rotation: (, ) 80 rotation: (, ) Dilations In a dilation, a figure with respect to the origin. or (, ),where k is the. Copright Holt McDougal. All rights reserved. Chapter Pre-Algebra Notetaking Guide 5

24 Words to Review Give an eample of the vocabular word. Complementar angles Supplementar angles Vertical angles Transversal Corresponding angles Alternate interior angles Alternate eterior angles Interior angle Eterior angle Transformation 6 Chapter Pre-Algebra Notetaking Guide Copright Holt McDougal. All rights reserved.

25 Image Translation Tessellation Reflection, line of reflection Line smmetr, line of smmetr Rotational smmetr Rotation, center of rotation, angle of rotation Dilation, center of dilation, scale factor Review our notes and Chapter b using the Chapter Review on pages 78 7 of our tetbook. Copright Holt McDougal. All rights reserved. Chapter Pre-Algebra Notetaking Guide 7