Slide 1 / 185 Slide 2 / 185 Geometry Points, Lines, Planes & Angles Part 2 2014-09-20 www.njctl.org Slide 3 / 185 Slide 4 / 185 Table of Contents for Videos Table of Contents click on the topic to go Part 1 to that section Demonstrating Constructions Introduction to Geometry Points and Lines click on the topic to go to that video Planes Congruent Angles Congruence, Distance and Length Constructions and Loci Angle Bisectors Part 2 Angles Congruent Angles Angles & Angle Addition Postulate Protractors Special Angle Pairs Proofs Special Angles Angle Bisectors Locus & Angle Constructions Angle Bisectors & Constructions Slide 5 / 185 Slide 6 / 185 Angles Definition 8: A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line. Angles A Whenever lines, rays or segments in a plane intersect, they do so at an angle. x Return to Table B C of Contents
Slide 7 / 185 Slide 8 / 185 Angles Angles The measure of angle is the amount that one line, one ray or In this course, angles will be measured with degrees, segment would need to rotate in order to overlap the other. which have the symbol 0 . A For a ray to rotate all the A In this case, Ray BA would way around from BC, as have to rotate through an shown, back to BC would angle of x in order to overlap represent a 360 0 angle. Ray BC. x x B C B C Slide 9 / 185 Slide 10 / 185 Measuring angles in degrees Measuring angles in degrees The use of 360 degrees to represent a full rotation back to the The use of 360 for a full rotation is thought that it come from ancient original position is arbitrary. Babylonia, which used a number system based on 60. Their number system may also be linked to the fact that there are 365 days in a year, which is pretty close to 360. Any number could have been used, but 360 degrees for a full 360 is a much easier number to work with than 365 since it is rotation has become a divided evenly by many numbers. standard. These include 2, 3, 4, 5, 6, 8, 9, 10 and 12. 360 0 Slide 11 / 185 Slide 12 / 185 Right Angles Right Angles Definition 10: When a straight line standing on a straight line Fourth Postulate: That all right angles are equal to one another. makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the Not only are adjacent right angles equal to each other as shown below, all right angles are equal, even if they are not adjacent, for other is called a perpendicular to that on which it stands. instance, all three of the below right angles are equal to one another. The only way that two lines can intersect as A shown and form adjacent A A equal angles, such as shown here where Angle ABC = Angle ABD, is if there are right angles, x x 90 0 x x 90 0 . D B C B C D B C
Slide 13 / 185 Slide 14 / 185 Right Angles Right Angles This definition is unchanged today and should be familiar to you. There is a special indicator of a right angle. Perpendicular lines, segments or rays form right angles. If lines intersect to form adjacent A A It is shown in red in this case equal angles, then they are to make it easy to recognize. perpendicular and the measure of those angles is 90 0 . 90 0 B C B C When perpendicular lines meet, they form equal adjacent angles and their measure is 90 0 . Slide 15 / 185 Slide 16 / 185 Obtuse Angles Acute Angles Definition 11: An obtuse angle is an angle greater than a Definition 12: An acute angle is an angle less than a right angle. right angle. A A 45 0 135 0 B C B C Slide 17 / 185 Slide 18 / 185 Straight Angle Reflex Angle Another modern definition that was not used in The Elements is A definition that we need that was not used in The Elements is that that of a "reflex angle." That is an angle that is greater than 180 0 . of a "straight angle." That is the angle of a straight line. Answer 235 0 B C A B C 2 questions to discuss with a partner: This is also a type A Is this an acute or obtuse angle? of obtuse angle. What is the degree measurement of the angle?
Slide 19 / 185 Slide 20 / 185 Angles 1 This is an example of a (an) ________ angle. Choose all that apply. A acute In the next few slides we'll use our responders to review the B obtuse A 0 0 names of angles by showing angles from 0 0 to 360 0 in 45 0 Answer increments. B C C right Angles can be of any size, not just increments of 45 0 , but this is just to give an idea for what a full rotation looks like. D reflex E straight Slide 21 / 185 Slide 22 / 185 2 This is an example of a (an) ________ angle. 3 This is an example of a (an) ________ angle. Choose all that apply. Choose all that apply. A acute A acute A A Answer Answer B obtuse B obtuse C right C right 45 0 B C 90 0 D reflex D reflex B C E straight E straight Slide 23 / 185 Slide 24 / 185 4 This is an example of a (an) ________ angle. 5 This is an example of a (an) ________ angle. Choose all that apply. Choose all that apply. A acute A acute Answer Answer B obtuse B obtuse A 180 0 C right C right B C A 135 0 D reflex D reflex B C E straight E straight
Slide 25 / 185 Slide 26 / 185 6 This is an example of a (an) ________ angle. 7 This is an example of a (an) ________ angle. Choose all that apply. Choose all that apply. A acute A acute 270 0 235 0 B obtuse B obtuse B Answer C Answer B C C right C right D reflex D reflex A E straight E straight A Slide 27 / 185 Slide 28 / 185 8 This is an example of a (an) ________ angle. 9 This is an example of a (an) ________ angle. Choose all that apply. Choose all that apply. A acute A acute Answer B obtuse C B obtuse 315 0 Answer B A 360 0 C right C right B C D reflex D reflex A E straight E straight Slide 29 / 185 Slide 30 / 185 Naming Angles Interior of Angles An angle has three parts, it has two sides and one vertex, Any angle with a measure of less than 180 0 has an interior where the sides meet. and exterior, as shown below. In this example, the sides are the rays BA and BC side A A and the vertex is B. x vertex Exterior Interior side B C x B C
Slide 31 / 185 Slide 32 / 185 Naming Angles Naming Angles The angle shown can be called ∠ ABC , ∠ CBA, or ∠B . An angle can be named in three different ways: By its vertex (B in the below example) When there is no chance · C of confusion, the angle leg A may also be identified by its vertex B. By a point on one leg, its · x vertex and a point on the vertex other leg (either ABC or B leg C The sides of ∠ABC 32° CBA in the below example) B are rays BC and BA A Or by a letter or number placed inside the angle (x in the below) · The measure of ∠ ABC is 32 degrees, which can be rewritten as m ∠ABC = 32 o . Slide 33 / 185 Slide 34 / 185 Naming Angles Naming Angles Using the vertex to name an angle doesn't work in some What other ways could you name ∠ ABC, ∠ ABD and ∠D BC in the cases. Why would it it would be unclear to use the case below ? (using the side - vertex - side method) vertex to name the angle in the image below? Answer Answer D A D How many angles do A you count in the y x image? y B x C B C How could you name those 3 angles using the letters placed inside the angles? Slide 35 / 185 Slide 36 / 185 Intersecting Lines Form Angles Intersecting Lines Form Angles When an angle is formed by either two rays or segments with a These numbers used have no special significance, but just show shared vertex, one included angle is formed. the 4 angles. When rays or segments intersect but do not have a Shown as x in the below diagram to the left. common vertex, they also create 4 angles. When two lines intersect, 4 angles are formed, they are numbered in the diagram below to the right. A A 2 1 2 1 3 4 x 3 4 B C x B C
Slide 37 / 185 Slide 38 / 185 10 Two lines ________________ meet at more than 11 An angle that measures 90 degrees is __________ one point. a right angle. Always Always A A Answer Sometimes Answer Sometimes B B Never Never C C Slide 39 / 185 Slide 40 / 185 12 An angle that is less than 90 degrees is 13 An angle that is greater than 180 degrees is ___________ obtuse. _______ referred to as a reflex angle. Always Always A A Sometimes Sometimes B B Answer Answer Never Never C C Slide 41 / 185 Slide 42 / 185 Congruence We learned earlier that if two line segments have the same length, they are congruent. a Congruent Also, all line segments Angles with the same length b are congruent. Are these two segments congruent? Return to Table of Contents
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