Geometry-Chapter 2 Notes

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Kimberly Morgan
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1 2.1 Perpendicularity Geometry-Chapter 2 Notes Vocabulary: perpendicular symbol parallel symbol Definition #16 Lines, rays or segments that intersect at right angles are perpendicular. Examples and label drawings and write the notations for each. a) draw line a perpendicular b) draw right angle DEF and give to line b. the notation that the sides of the angle are perpendicular c) draw perpendicular rays JM d) draw the angle in b again and ray GH that intersect at K with a different 90 notation. Do not assume perpendicularity or parallelness from a diagram!!!!!!!!! Draw a coordinate plane below. Label both axes, the quadrants, the origin and graph one ordered pair in each quadrant. Check your neighbors work and let them check yours.

2 Given: AB BC A D DC BC 2 Conclusion: B C B C 1) AB BC 1) 2) 2) If two segments are, they form right angles. 3) DC BC 3) 4) C is 4) 5) 5) * * * * * * * * * * * * * * * * * * Given that EH HG H Name all the angles you can prove to be right angles. E F G Given KJ KM J JKO is four times as large as MKO. Find m JKO 4x O K x M

3 Y 3 Given EC to x axis E ( -4,3) C ( 7,3) RT to the x axis Find the area of rectangle RECT? X R (-4,-2) T ( 7,-2)? 2.2 Complementary and Supplementary Angles Def. #17 Complementary angles are two angles whose sum is 90º. Each of the two angles is called a complement of the other. Draw and label the following complementary angles: a) angle A is complementary b) a triangle with 2 complementary angles c) draw and label a right angle divided into 2 complimentary angles with measures of 63º 40 and 26º 20 Def. #18 supplementary angles are two angles whose sum is 180º. Each of the two angles is called the supplement of the other. Draw and label the following supplementary angles a) angle B is supplementary b) a quadrilateral with atleast one supplementary angle

4 4 c) a straight angle divided into 2 angles, one of which is supplementary. One angles is 131º and the other is 48º Given: TVK is a right angle T Prove : 1 is comp. to 2 X 1 2 V K Statement 1) 1) Given 2) 1 is comp. to 2 2) Given: Diagram as shown 1 2 Conclusion: 1 is supp to 2 A B C 1) 1) Given 2) is a straight angle 2) 3) 1 is supp to 2 3) The measure of one of two complementary angles is three more than twice the measure of the other. Find the measure of each.

5 The measure of the supplement of an angle is 60 less than 3 times the measure of the complement of the angle. Find the measure of the complement Drawing Conclusions: 1) Procedure for Drawing Conclusions 2) 3) 4) 5)

6 6 Given: AB bisects CAD C Conclusion: B A D Statement 1) 1) Given 2) 2) Given: A is a right angle A D B is a right angle Conclusion B C Given: E is the midpoint of SG S E G Conclusion:

7 Given: PRS is a right angle Conclusion: P 7 R S 2.4 Congruent supplements and complements Theorem #4 If angles are supplements to the same angle, then they are congruent. Theorem #5 If angles are supplementary to congruent angles, then they are congruent Theorem #6 If angles are complementary to the same angle, they are congruent. Theorem #7 If angles are complementary to congruent angles, then they are congruent. H Sample #3 G Given: Diagram as shown F Hint: # angles Prove: HFE GFJ E J Statement Reason 1) Diagram as shown 1) 2) 2) 3) 3) 4) 4) 5) 5) 6) 6)

8 Given: KM MO K R P PO MO KMR POR Prove: ROM RMO M O 1) 1) 8 2) 2) 3) 3) 4) 4) 5) 5) 6) 6) 2.5 Addition and Subtraction Properties Theorem #8 If a segment is added to two congruent segments the sums are congruent.(addition Property) Theorem #9 If an angle is added to two congruent angles, the sums are congruent. ( Addition Property) Theorem #10 If congruent segments are added to congruent segments, the sums are congruent. (Addition Property) Theorem #11 If congruent angles are added to congruent angles, the sums are congruent. (Addition Property)

9 SUBTRACTION PROPERTY Theorem 12-If a segment(or angle) is subtracted from congruent segments(or angles), the differences are congruent. (Subtraction Property) Theorem 13- If congruent segments ( or angles) are subtracted from congruent segments ( or angles), the differences are congruent. ( Subtraction Property) 1) Using the Addition and Subtraction Properties in Proofs 9 2) Given NOP NPO ROP RPO Prove: NOR NPR N R O P

10 Given: AB CD E D Conclusion C 10 A B F 1) 1) 2) 2) Given : H G Conclusion: HEF GFE E F 1) 1) 2) 2) 3) 3) 4) 4) 5) 5) 6) 6)

11 Multiplication and Division Properties With these properties the multiples and divisions of angles or segments must be congruent to each other. (* You must state their congruence in your proof.) Theorem # 14 : If segments (or angles) are congruent, their like multiples are congruent. ( Multiplication Property) Example A B C D E F G H * In the segments above B, C, F, and G are trisection points. If segment AB is congruent to segment EF and they are both are 3 units long. What can we now say about segments AD and EH? J O S N K P M R Example: Rays KO and PS are angle bisectors. If angle JKO angle NPS and they are both 25º, what can we say about angle JKM and angle NPR? Given: MK FG M K J KG bisects MJ and FH Prove: MJ FH F G H Statement Reason

12 Using the Multiplication and Division Properties in Proofs 12 Theorem # 15 If segments (or angles) are congruent, their like divisions are congruent ( Division Property) Example A B C D E F G H * In the segments above segments AD and EH are congruent. Points B, C, F, G are trisection points. Prove that segment AB is congruent to segment EF. What can we now conclude that will help us prove it? J N O S K M P R Example: Angle JKM angle NPR. Rays KO and PS are angle bisectors. What can we conclude about angle JKO and angle SPR?

13 13 Given: NOP RPO N T S R PT bisects RPO OS bisects NOP NSO is comp. to RTP is comp to 3 O P Prove: NSO RTP Reason 2.7 Transitive and Substitution Properties Theorem # 16 If angles (or segments) are congruent to the same angle (or segment), they are congruent to each other. (Transitive Property) Ex. If angle A angle B, angle A angle C, is angle B angle C? Write as a chain of reasoning: Theorem #17 If angles ( or segments) are congruent to congruent angles (or segments), they are congruent to each other. Make up your own example for this property.

14 14 Substitution: A) You have used substitution this year to solve systems of equations in which you have both an x and y. You substituted a value for X to eliminate one of the variables. Given: angle A angle B, solve for angle A B (x + 10) A (2x -4) When do you substitute in this problem? B) You can also apply substitution with out using any variables. If 1 is comp 2, and 2 3, then is comp by substitution. c) If P R and Q R, then 1) Use a chain of reasoning to complete. 2) P ( x+y+a) R ( 2y +a) Q Express angle Q in terms of x and a P R = = = y Since m P = x + y + a we can substitute in the value equal to Y m P = x + + a m P = therefore, since P R, Q =

15 K J 15 Given: FG KJ GH KJ Prove: KG bisects FH F G H Given : = Prove : = Vertical Angles Definition # 19-2 collinear rays that have a common endpoint and extend in different directions are called opposite rays. 1) Draw: AB and AC are opposite ray 2)Then draw them on this segment D F

16 3) Now draw them as two rays that are not part of the same line. 16 4) PT and RS are not opposite because Vertical Angles: T P R S Definition # 20: Two angles are vertical angles if the rays forming the sides of one and the rays forming the sides of the other are opposite rays Theorem # 18 - Vertical angles are congruent. 4 Which pairs of angles congruent. If 1 = 2x + 5 and 3 = x + 30, find the measure of each angle. Given: Prove:

17 Given: O is comp to 2 O J is comp to 1 H 2 Conclusion: O J 1 K M 17 J

Lesson 2.1 8/5/2014. Perpendicular Lines, Rays and Segments. Let s Draw some examples of perpendicularity. What is the symbol for perpendicular?

Lesson 2.1 8/5/2014. Perpendicular Lines, Rays and Segments. Let s Draw some examples of perpendicularity. What is the symbol for perpendicular? 8/5/04 Lesson. Perpendicularity From now on, when you write a two-column proof, try to state each reason in a single sentence or less. bjective: Recognize the need for clarity and concision in proofs and