Pre-Algebra Notes Unit 13: Angle Relationships and Transformations

Transcription

1 Pre-Algebra Notes Unit 13: Angle Relationships and Transformations Angle Relationships Sllabus Objectives: (7.1) The student will identif measures of complementar, supplementar, and vertical angles. (7.2) The student will compute measures of complementar, supplementar, and vertical angles. (7.3) The student will identif angles formed when a transversal intersects two parallel lines. Let s review: angles are measured in degrees ( ). Protractors are used to measure angles. Here are two interactive websites ou might use to show students how to use this measuring tool. We need to also review how to classif an angle b its measure. Acute angles are greater than 0, but less than 90º. In other words, not quite a quarter rotation. Right angles are angles whose measure is 90º. Obtuse angles are greater than 90º, but less than 180º. That s more than a quarter rotation, but less than a half turn. And finall, straight angles measure 180º. acute right obtuse straight We call two angles whose sum is 90º complementar angles. For instance, if mp 40and mq 0, then P and Q are complementar angles. If ma30, then the complement of A measures 60. Two angles whose sum is 180º are called supplementar angles. If mm 100and ms 80, then M and S are supplementar angles. Eample: Name an angle that is complementar to Name an angle that is supplementar to If m20, what is the measure of 1? m If m140, what is the measure of 4? m440. What is the sum of the measures of 1, 2, 3, 4 and? 360 McDougal Littell, Chapter 13 Pre-Algebra Unit 13: Angle Relationships and Transformations Page 1 of 10

2 Adjacent angles are two angles that have a common verte, a common side (ra), and no common interior points. P T Q S R PQR and PQT are adjacent angles. PQT and TQS are adjacent angles. TQS and SQR are adjacent angles. SQR and RQP are adjacent angles. Vertical angles are formed when two lines intersect the are opposite each other. These angles alwas have the same measure. We call angles with the same measure congruent( ). P T Q S R PQR and TQS are vertical angles. PQT and RQS are vertical angles. Angles will be shown as congruent b using tick marks. If angles are marked with the same number of tick marks, then the angles are congruent. P R Q TQP RQS TQS PQR T S Another wa to show congruent angles is with the same number of arc marks, shown below: P R Q T S A transversal is a line that intersects two or more lines. In the figure to the right, line t is a transversal. If I asked ou to look at the figure and find two angles that are on the same side of the transversal, one an interior angle (between the lines), the other an eterior angle, that were not adjacent, could ou do it? 2 and 3are on the same side of the transversal, one interior, the other is eterior, but the are adjacent. These angles do not fit the conditions t m n How about 2 and 6? These two angles do fit the conditions. We call these angles corresponding angles. Can ou name an other pairs of corresponding angles? If ou said 3 and 7, 4 and 8, 1 and, ou d be right. McDougal Littell, Chapter 13 Pre-Algebra Unit 13: Angle Relationships and Transformations Page 2 of 10

3 When a pair of parallel lines is cut b a transversal, there is a special relationship between the corresponding angles. Discover what this relationship is! 1) Use the lines on a piece of graph paper or a piece of lined paper as a guide to draw a pair of parallel lines. You can also use both edges of our straightedge to create parallel lines on a blank sheet of paper. 2) Draw a transversal intersecting the parallel lines. Label the angles with numbers as in the diagram. 3) Measure 1. Calculate the measures of the other three angles that share the same verte. 4) Measure 8 and calculate the measures of the other three angles. Record our findings into the table. Angle Measure Angle Measure Look for patterns. Complete the conjecture below: If two parallel lines are cut b a transversal, then the corresponding angles are. Hopefull, ou discovered that the corresponding angles created when a transversal intersects parallel lines are congruent! This will save us time when finding angle measures. Eample: m20. Find the measure of all other angles. m1 130, since 1 and 2 are adjacent and supplementar m3130, since 1 and 3are vertical angles or 2 and 3 are adjacent and supplementar a 3 4 b a b m40, since 2 and 4 are vertical angles or 1 and 4 are adjacent and supplementar m130, since 1 and are corresponding angles m7 130, m60, and m80 (using same reasoning as above) McDougal Littell, Chapter 13 Pre-Algebra Unit 13: Angle Relationships and Transformations Page 3 of 10

4 We can save even more time b defining two more pairs of angles. When a transversal intersects two lines, the two angles that lie between the two lines on opposite sides of the transversal are called alternate interior angles. Two angles that lie on the outside of the line on opposite sides of the transversal are called alternate eterior angles. In the diagram to the right, an eample of alternate interior angles is 4 and 6 ; another pair is 3 and An eample of alternate eterior angles is 1 and 7 ; another pair is 2 and 8. t m n And once again, if we took two parallel lines and cut them with a transversal, we would discover that the alternate interior angles and the alternate eterior angles are congruent. l Eample: In the diagram, transversal l intersects parallel lines m and n. If m1 12, find the measures of the other numbered angles. The solution is shown below; reasons ma var. m12, because 1 and are corresponding angles m312, because 3 and are alternate interior angles m7 12, because 1 and 7 are alternate eterior angles m2, because 1 and 2 are supplementar angles m6, because 2 and 6 are corresponding angles m4, because 4 and 6 are alternate interior angles m8, because 2 and 8 are alternate eterior angles m n Polgons and Angle Measure Sllabus Objective: (4.) The student will find unknown angle measures using the sum of the interior angles of a polgon. The sum of the angle measures of an triangle is 180. This can be shown with a quick demonstration: 1) Draw and label a large triangle as shown. 2) Cut the triangle out. 3) Tear each angle from the triangle and place them so their vertices meet at a point. c a b c a McDougal Littell, Chapter 13 Pre-Algebra Unit 13: Angle Relationships and Transformations Page 4 of 10 b b a c

5 The sum of the angle measures in an quadrilateral can be found b dividing the figure into 2 triangles. Since the sum of the angle measures in each of these triangles is 180, we can reason that the sum of the angle measures in our quadrilateral would be We can continue this reasoning with other polgons. Finish identifing the polgons below b name; then complete drawing all the diagonals from one verte of the polgon. Triangle Quadrilateral Number of sides of polgon n Triangles formed b diagonals from one verte Sum of measures of angles A formula for finding the sum of the measures of the angles of a conve polgon with n sides is n Eample: Find the unknown angle measure in the quadrilateral. We know the sum of the angle measures in a quadrilateral is The missing angle is 81. McDougal Littell, Chapter 13 Pre-Algebra Unit 13: Angle Relationships and Transformations Page of 10

6 Eample: Find the sum of the angle measures in a decagon. A decagon has 10 sides. Using the formula: n The sum of the interior angles measures of a decagon is We can find the measure of one of the interior angles of a regular polgon b dividing the sum of the measures of the interior angles b the number of sides. The measure of an interior angle n 218 of a regular n-gon is given b the formula 0. Eample: Find the measure of an interior angle of a regular 12-gon (also called a dodecagon). Using the formula: n n The measure of an interior angle of a regular 12-gon is 10. n Transformations Sllabus Objectives: (7.4)The student will translate figures in a coordinate plane. (7.)The student will rotate figures in a coordinate plane. (7.6)The student will dilate figures in a coordinate plane. (7.7)The student will reflect figures in a coordinate plane. (7.8)The student will describe the relationship between the original figure and its transformation or dilation. Geometr not onl looks at figures, it also studies the movement of figures. If ou move all the points of one figure to create a new geometric figure, ou call the new figure the image. Each point of the image matches eactl with a corresponding point of the original figure; in other words, the image is congruent to the original figure. This changing of position is called a transformation. McDougal Littell, Chapter 13 Pre-Algebra Unit 13: Angle Relationships and Transformations Page 6 of 10

7 There are 4 tpes of transformations we are going to stud: translation, rotation, reflection, and dilation. You ma be more familiar with these as being called slide, turn, flip, and stretch/shrink. A translation is the simplest. If ou cop a figure onto a piece of paper, than slide the paper along a straight path without turning it, our slide motion represents a translation. A rotation is a turning motion. Points of the original figure rotate an identical number of degrees around a fied center point. A reflection flips the figure across a line of reflection, creating a mirror image. A dilation stretches or shrinks a figure, with respect to a fied point. McDougal Littell, Chapter 13 Pre-Algebra Unit 13: Angle Relationships and Transformations Page 7 of 10

8 Transformations can also be shown on a coordinate grid. A translation will move each point like an ordered pair. For eample, the translation shown moved the triangle right 3 units and up 4 units, which we could also show as 3, 4. A prime mark (') is used with the label of an image point. The image of point A is shown as point A' (read A prime ). A A' C' - B C - B' A reflection can be shown on a coordinate grid b reflecting the figure across an ais. A B G E F Across the -ais, the figure will follow the rule,. C D - C' D' A' B' - G' E' F' Across the -ais, the figure will follow the rule,. - - McDougal Littell, Chapter 13 Pre-Algebra Unit 13: Angle Relationships and Transformations Page 8 of 10

9 A rotation can be shown on a coordinate grid b moving the figure about a point. Unless it is indicated otherwise, rotation occurs around the origin. This eample shows a rotation of 90 counterclockwise about the origin. F' G' A' E' C' D' B' A - B F G C D E - A rotation can also occur around a given point. This eample rotated the triangle 180 about point X. Z Y X Y' - - Z' A dilation can be shown on a coordinate grid b stretching or shrinking the figure about a point. Unless it is indicated otherwise, the center of dilation is the origin. A scale factor is the ratio of a side length of the image to the corresponding side lenth of the original figure. This eample shows a dilation about the origin, with a scale factor of 2. In dilation, the figure will follow the rule k, k, where k is the scale factor. A' B' C A B - M' C' M L' K' L K - D E' E F H J D' G H' J' F' G' McDougal Littell, Chapter 13 Pre-Algebra Unit 13: Angle Relationships and Transformations Page 9 of 10

10 Geometric Construction Sllabus Objective: (4.18) The student will construct geometric figures using a variet of tools. Geometric constructions use onl a compass and a straightedge. The are valuable in reinforcing our geometric definitions. There are a few constructions addressed in the McDougal Littell tetbook on pages This objective will not be tested at the state level. Use the following websites as reference: McDougal Littell, Chapter 13 Pre-Algebra Unit 13: Angle Relationships and Transformations Page 10 of 10