Integrated Math, Part C Chapter 1 SUPPLEMENTARY AND COMPLIMENTARY ANGLES

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1 Integrated Math, Part C Chapter SUPPLEMENTARY AND COMPLIMENTARY ANGLES Key Concepts: By the end of this lesson, you should understand:! Complements! Supplements! Adjacent Angles! Linear Pairs! Vertical Angles! What it means to be perpendicular We only have to master a few more basic concepts. Be sure that you check over the Key Concepts so that you really understand them, and if you ever forget things from earlier lessons, don t hesitate to check out what we already covered. Let s see what fun stuff is in this lesson! Section 3- Section 3- concentrates on pairs of angles. Pairs of angles have lots of properties that will help you to understand things we will learn in the next few sections. If you stay on top of things, this should be easy as pie! Postulate 0 Linear Pair Postulate If two angles form a linear pair, then they are supplementary: Theorem 3- Perpendicular Lines Theorem If two intersecting lines form adjacent angles whose measures are equal, then the lines are perpendicular. Theorem 3- Perpendicular to a Plane Theorem Through a given point on a line in a plane, there is exactly one line in that plane perpendicular to the given line.

2 Theorem 3-3 Singular Perpendicular Bisector Theorem In a plane, a segment has exactly one perpendicular bisector. Find the complement of 3.? 3 Remember that two complementary angles must add up to 90. So if the measure of one angle is 3, its compliment must be 90-3 = 67. So the complement of 3 is 67. That wasn t too bad, was it? Find the supplement of 7.? 7 We just learned that supplementary angles add up to 80. So if the measure of an angle is 7, then its supplement must be 80-7 = 08. So the supplement of 7 is 08.

3 The measure of an angle is twice that of its complement. Find the measure of the angle and of the complement. This is where geometry starts to get really fun! Now let s think about this puzzle by drawing a picture. We know that the sum of these two angles is 90. Let s call the smaller angle X and the larger angle X. So this is what we know: X+X=90 3X = 90 X X X = 30 and X = 60. So, the larger angle is 60 degrees which is twice as big as the smaller angle which is 30 degrees. Wasn t that a fun little problem?? Name 4 adjacent angles and linear pairs in this diagram. Let s start first with the adjacent angles. The 3 4 Best way to find the adjacent angles is to count them in a circle. and both share a common ray, but no interior points. The same is true for angles and 4, 4 and 3, and 3 and. So there are 4 pairs of adjacent angles. 3

4 Now let s find the linear pairs. We know that linear pairs are adjacent angles whose measure is 80 degrees. Because the horizontal line makes up two opposite rays, then and & 4 and 3 are linear pairs. Section 3- So, we ve made it to section kind of fun, wasn t it? 3- is showing us that things we learned in algebra also work for geometry. Do you remember the reflexive, symmetric, and transitive properties? Well, they re back! And just as useful; lets see what neat new things we can do with these old properties! Properties Reflexive Property a = a of Symmetric Property If a = b, then b = a Equality Transitive Property If a = b and b = c, then a = c. Name the property of equality that applies to all the problems in this section. If QR = ST and ST = (UV), then QR = (UV). The property that applies here is the transitive property. The reason that it applies: QR = ST ST = (UV) So you can see that QR = ST = (UV) QR = (UV) 4

5 If 5x + = 3x + 4, then x =. The property that applies here is the additive property. The only changes we made to this equation to get it from 5x + = 3x + 4 to x = were adding and subtracting. 5x + = 3x + 4 5x + = 3x + 4 5x = 3x + 5x 3x = 3x + 3x x = m A = m A The property that applies here is the reflexive property. The reflexive property states that an expression is equal to itself. Well, that is pretty much it for lesson three. In the next lesson we are going to learn about conditional statements. Good luck on your submission! 5

6 GLOSSARY ADJACENT ANGLES Adjacent angles are two coplanar angles with the same vertex and a common side but no interior points in common. and are adjacent angles. COMPLEMENTARY ANGLES Complimentary Angles are two angles whose measures have a sum of 90 degrees. Each angle is a complement of the other LINEAR PAIR A linear pair is a pair of adjacent angles such that two of the rays are opposite rays. and are a linear pair. PERPENDICULAR Two lines (rays, segments) are perpendicular if they intersect to form a right angle. C AB C D (line AB is perpendicular to line CD) A B D 6

7 SUPPLEMENTARY ANGLES Supplementary angles are two angles whose measures have a sum of 80 degrees. Each angle is the supplement of the other VERTICAL ANGLES Vertical angles are the nonadjacent angles formed when two lines intersect. and are vertical angles. 3 and 4 are vertical angles