Jigsaw Activity: We now proceed to investigate some important types of angles pairs. Be sure to include in your booklets all the new vocabulary and theorems. Good luck! 1. Adjacent Angles and the Angle Addition Postulate Next door neighbors, seat partners, table partners, lane 1 and lane 2. Adjacent angles (#VOC): angles that share a common and a common, and do not overlap. These are adjacent angles: Fig. 1 Fig. 2 Fig. 3 These are not: Fig. 4 Fig. 6 Fig. 5 The Angle Addition Postulate (#THM): If BCD and DCE are adjacent, then m BCE = m BCD + m DCE. Therefore, Page 1
2. Vertical Angles X marks the spot! Just as fun as looking for hidden treasure, look for vertical angles at the X. Intersecting lines form an X-shape, and vertical angles are on opposite sides of the X from one another. Two angles are called congruent angles (#VOC) if and only if their measures are the. Vertical angles (#VOC) are angles formed by two lines. Two pairs of vertical angles can be found at the intersection of two lines. 1 2 4 3 Pair 1: Pair 2: Given an angle, its vertical angle is determined by its opposite rays. More than two pairs of vertical angles can be found at the same vertex, depending on how many intersecting lines share the same intersection. Examples: 1 and 4 2 and 5 3 and 6 ( 1 and 2) and ( 4 and 5) ( 2 and 3) and ( 5 and 6) ( 1 and 2 and 3) and ( 4 and 5 and 6) How many more vertical angles can you find in the diagram above? Name them. Page 2
3. Complementary and Supplementary Angles Enhancing, improving, and to make better typical words that might replace complement or supplement. But how do you improve an angle? Complementary angles (#VOC): angles whose is, or together they create a angle (#VOC). On diagrams, you can determine right angles by the small box used to denote the angle (e.g. MLK below). Right angles are necessary in many facets in life corners of Grandma s rooms, your body when standing in perfect posture, and so forth. Caution: angles do not need to be adjacent to be complementary, their measures just need to sum to 90. Supplementary angles (#VOC): angles whose is, or together they create a angle. We learned about straight angles in the very first lesson, and you ll see its importance now and throughout our unit! A special type of supplementary angles is called a linear pair (#VOC), which are two angles (as seen below). Caution: angles do not need to be adjacent to be supplementary. m MNP + m PNO = m MNO = 180 Linear Pair Postulate (#THM) or more angles are considered a linear pair, if their measures sum to. Can you find any supplementary angles in your classroom? How about on Grandma s house plans? (Note: Grandma s house plans can be found in L1 Foundations, page 3.) Page 3
4. Parallel and Perpendicular Lines To cross, or not to cross, that is the question Parallel lines (#VOC) are lines that do not. From a single line and a point not on the line, there can be only one parallel line that passes through the given point. We can use matching arrows on each line to signify that the two lines are parallel, or we can use the!###"!##" notation: MN OQ A line that crosses lines at distinct points is called a transversal (#VOC).!##"!##"!##" DE is a transversal of the parallel lines AB and CD. How many examples of pairs of parallel lines can you find in your classroom? How about on Grandma s house plan? Perpendicular lines (#VOC) are lines that intersect to form angles ( angles). Given a single line and a point not on the line, there can be only one line passing through the point that is perpendicular to the given line. The box drawn in ABC denotes a 90, or right angle.!##"!##" We can also use the notation: AB BC. How many examples of perpendicular lines can you find in your classroom? How about on Grandma s house plans? Page 4
Practice: Determine if the pair of angles ( 1 and 2) is adjacent or not adjacent. 1. 2. 1 2 1 2 3. 4. 1 1 2 2 Find the indicated angle measure. 5. m 1=45, m 2=52. 6. Find m JLM. m ABC=87, m ABD=119. Find m CBD. 7. m 1=50, m 2=68. 8. Find m NOQ. m VWY=112, m YWX=26. Find m VWX. Page 5
9. List all possible vertical angle pairs: For each, find the value of x. 10. 11. x 31 12. 13. 119 x Page 6
14. Given the figure below, name two unique pairs of complementary angles. A E G B D F C 15. Given the figure below, name an angle supplementary to EJF. Name a different angle supplementary to EJF. For each, find the value of x. 16. 17. x - 2 134 x 5x+4 3x+7 18. Create a pair of adjacent angles and label them ABC and CBD. 19. Create a pair of non-adjacent angles that share a vertex, and label them LMN and OMP. Page 7
4.1 Angles Per Date Follow the directions below: i. Determine if each of the angles below is acute (<90 #VOC), obtuse (between 90 and 180 #VOC), or right (=90 #VOC) before using the protractor. ii. Use a protractor to measure each angle. iii. Use a straightedge to extend the two rays into two lines that intersect at the vertex shown, and without the use of a protractor determine the measure of each of the three new angles thus created. Check your answers using a protractor. 1 2 3 4 5 6 7 8 9 Page 8
4.2 Parallel Lines with a Transversal Per Date When two parallel lines are intersected by a transversal, 8 angles are formed. It appears that many pairs of angles are (i.e. have equal ), and we can intuitively verify which angle pairs are congruent using rigid motion transformations. 1. Identify which angle pairs you believe must be congruent, and the rigid motion transformations that appear to justify your conjectures. Pairs of Angles that you believe must be Congruent Transformation or Facts Used to Justify your Result 2. List all the angles you believe are congruent to angle 8. Page 9
4.2 Parallel Lines with a Transversal Per Date 3. With a protractor, measure all eight angles formed by this pair of parallel lines and the transversal, and record them in the box to the right. Round off to the nearest degree. m 1 = m 2 = m 3 = m 4 = m 5 = m 6 = m 7 = m 8 = What do you notice? Are your measurement results consistent with your earlier conjectures? Which pairs of angles are congruent? Are there any angle pairs that are supplementary? List as many unique pairs of congruent and supplementary angles as you can find (for example, pair 1 and 2 would be the same as pair 2 and 1). Congruent Supplementary 4. The angle pairs that we just identified have special names that are related to their position formed by the two parallel lines and the transversal. Page 10
4.2 Parallel Lines with a Transversal Per Date Use the Vocabulary Bank at the bottom to fill in the blanks with the vocabulary word that we use to identify these angles, based on their definitions, and draw an example of the angle pair in the space provided. (#VOC) Be sure to include each in your Vocabulary Booklet. (#VOC) are a pair of non-adjacent angles, one interior and one exterior, that both lie on the same side of the transversal. Draw: (#VOC) are a pair of non-adjacent angles between parallel lines and on opposite sides of the transversal. Draw: (#VOC) are a pair of non-adjacent angles outside of the parallel lines and on opposite sides of the transversal. Draw: (#VOC) are a pair of angles between parallel lines and on the same side of the transversal. Draw: (#VOC) are a pair of angles outside of the parallel lines and on the same side of the transversal. Draw: Vocabulary Bank (#VOC): Alternate exterior angles Corresponding angles Same-side interior angles Alternate interior angles Same-side exterior angles Page 11
4.2 Parallel Lines with a Transversal Per Date We just learned that a pair of parallel lines cut by a transversal will give special angle pairs that are either complementary or supplementary. Now we can determine if two lines cut by a transversal are parallel, based on measurement. Determine if the following pairs of lines are parallel, non-parallel, or cannot be determined with the given information. A. B. C. D. Exit Pass: Explain how you can determine whether two seemingly parallel lines are indeed parallel, and which are not. Page 12
4.3 The Parallel Postulate Per Date There are five foundational postulates of Euclidean (the kind of geometry that we are learning about that deals with lines and points in a plane). Each one is presented below. You will need to include each/all five in your Theorem Booklet. Let s investigate each of these postulates. Write about, draw an example of, and/or show what you can discover about each of the five postulates. Your demonstration can be in the form of explanation, examples, and/or nonexamples that illustrate each postulate. Consider if the postulates would still be true if instead of locating the points on a plane they were located on a sphere, such as Earth (assuming it was a perfect sphere.) 1. A straight line segment can be drawn joining any two points. (#THM) 2. Any straight line segment can be extended indefinitely in a straight line. (#THM) Note: this implies that a line segment can be extended to any finite length. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. (#THM) 4. All right angles are congruent. (#THM) 5. If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles. (#THM) This is equivalent to: Given any straight line and a point not on it, there exists one and only one straight line which passes through that point and does not intersect the first line. Page 13