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Geometry-Chapter 2 Notes 2.1 Perpendicularity 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 and ray GH that intersect at K d) draw the angle in “b” again 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. Given: AB ⊥ BC DC ⊥ BC A 2 D Conclusion: ∠ B ≅ ∠ C Statements 1) AB ⊥ BC 2)___________________ 3) DC ⊥ BC 4) ∠ C is_____________ 5)____________________ B C Reasons 1)________________ 2) If two segments are ⊥ , they form right angles. 3)________________ 4)________________ 5)_______________________ * * * * * * * * * * * Given that EH ⊥ HG Name all the angles you can prove to be right angles. * * * H E * F * * G ___________________________________________________ Given KJ ⊥ KM ∠ JKO is four times as large as ∠ MKO. Find m ∠ JKO J K 4x x O M * Given EC || to x axis RT || to the x axis Find the area of rectangle RECT E ( -4,3) Y 3 C ( 7,3) ? 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 c) a straight angle divided into 2 angles, one of which is supplementary. One angles is 131º 27’ 48” and the other is 48º 32’ 12” Given: Prove : TVK is a right angle 1 is comp. to 2 T X 1 Statement 1) _____________________ ∠ 1 is comp. to ∠ 2 2) Given: Diagram as shown Conclusion: ∠ 1 is supp to ∠ 2 Statements 1)_____________________ 2)______ is a straight angle 3) ∠ 1 is supp to ∠ 2 2 V K Reasons 1) Given 2)_______________________ 1 2 A B C Reasons 1) Given 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. ____________________________________________________ 2.3 Drawing Conclusions: Procedure for Drawing Conclusions 1) 2) 3) 4) 5) Given: AB bisects ∠ CAD Conclusion:___________________ 6 C B D A Statement 1) ____________________ 2)_____________________ Given: A is a right angle B is a right angle Conclusion_________________ Statements Reasons 1) Given 2)_________________________ A D B C Given: E is the midpoint of SG Conclusion:___________________ Statements Reasons S E Reasons G Given: ∠ PRS is a right angle Conclusion:________________ 7 P R Reasons Statements 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) Given: KM ⊥ MO PO ⊥ MO ∠ KMR ≅ ∠ POR Prove: ∠ ROM ≅ ∠ RMO 1) Statements K R M 1) 2) 2) 3) 3) 4) 4) 5) 5) 6) 6) Reasons P 8 O ____________________________________________________ 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) Using the Addition and Subtraction Properties in Proofs 1) 2) Given ∠ NOP ≅ ∠ NPO ∠ ROP ≅ ∠ RPO Prove: ∠ NOR ≅ ∠ NPR Statements N R O Reasons P Given: AB ≅ CD Conclusion________________ E C A Statements 1) 2) 1) D B 10 F Reasons 2) ____________________________________________________ Given : H E F Conclusion: ∠ HEF ≅ ∠ GFE Statements Reasons 1) 1) 2) 2) 3) 3) 4) 4) 5) 5) 6) 6) G 11 2.6 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 N O S 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 KG bisects MJ and FH Prove: MJ ≅ FH Statement F G Reason H J 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 K M S 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 Given: ∠ NOP ≅ ∠ RPO PT bisects RPO OS bisects NOP ∠ NSO is comp. to 1 ∠ RTP is comp to 3 Prove: ∠ NSO ≅ ∠ RTP Statements N T S 1 O R 3 P 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 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 =________ K Given: FG ≅ KJ GH ≅ KJ Prove: KG bisects FH F J G 15 H Statements Reasons Given : ∠ 1 + ∠ 2 = 90˚ ∠ 1 ≅ ∠ 3 1 2 Prove : ∠ 3 + ∠ 2 = 90˚ Statements 3 Reasons 2.8 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. 4) PT and RS are not opposite because ________________________ __________________________________________________________ T P R S Vertical Angles: 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. 2 1 3 4 Theorem # 18 - Vertical angles are congruent. Which pairs of angles congruent.____________________________ If ∠ 1 = 2x + 5 and ∠ 3 = x + 30 , find the measure of each angle. 1 Given: ∠ 2 ≅ ∠ 3 2 Prove: ∠ 1 ≅ ∠ 3 3 Statements Reasons 17 Given: ∠ O is comp to ∠ 2 ∠ J is comp to ∠ 1 Conclusion: ∠ O ≅ ∠ J O H 1 J Statements 2 Reasons K M