CP Math 3 Page 1 of 34 Common Core Math 3 Notes - Unit 2 Day 1 Introduction to Proofs Properties of Congruence Reflexive A A Symmetric If A B, then B A Transitive If A B and B C then A C Properties of Equality Reflexive A = A Symmetric If A = B, then B = A Transitive If A = B and B = C, then A = C Distributive a(b + c ) = ab + ac Substitution If a + b = c and b = f, Then a + f = c. Addition If a = b, then a + c = b +c Subtraction If a = b, then a c = b c Multiplication If a = b, then 3a = 3b. Division If a = b, then a/3 = b/3.
CP Math 3 Page 2 of 34 Proofs YOU MUST SHOW ALL THE STEPS IN AN ALGEBRAIC PROOF!!!! Examples: 1. Given: 3(x 2) = 42 Prove: x = 16 Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 2. Given: 7m + 3 = 6 4 Prove: m = 3 Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7.
CP Math 3 Page 3 of 34 Practice: 3. Given: -3(a + 3) + 5(3-a) = -50 Prove: a =? : Reasons: 4. Given: Prove: p =? : Reasons:
CP Math 3 Page 4 of 34 5. Given: Prove: a =? : Reasons:
CP Math 3 Page 5 of 34 Common Core Math 3 Notes - Unit 2 Day 2 Segments What s the difference???? Geometry: It s all about the SYMBOLS and the PICTURES!!! AB AB AB Example: Line Segment What do you notice about the segments? M 5units A T 5units H Definition of Equality: Definition of Congruence: Definition of Midpoint midpoint of a segment is the point halfway between the endpoints of the segment If X is the midpoint of AB, then AX = XB Midpoint Theorem If X is the midpoint of AB, then AX XB Ex: X is the midpoint of AB AX = 3y + 7 XB = 4y 2 Hints: Draw and label picture Make sure you answer the question. Find AB.
CP Math 3 Page 6 of 34 Definition of Segment Bisector- any segment, line or plane that intersects a segment at its midpoint To bisect a segment means to Sketch each of the following; be sure to include appropriate marks. CD bisects AB at x AB bisects CD at x AB and CD bisect each other at x B A B A B A Definition of BETWEEN - refers to points on a line, ray or segment T is between S and U Draw & Label picture! 1. Find x if T is between S and U and ST = 7x, SU = 45, TU = 5x 45. 2. A is between B and C. BA = x 2, AC = 6x + 10, and BC = 17. Find x and the length of each segment.
CP Math 3 Page 7 of 34 3. L is between K and M. KL = x 2 10, LM = 5x + 4, and KM = 2x 2 42. Find x. Segment Addition Postulate - D O G Ex 1: Given AC AC = 3y + 1 AB = 2y BC = 21 Find AB. Find the value of the variable and LM if L is between points N and M. Ex 2: NL = 5x, LM = 3x, NL = 15 Ex 3: NL = 5x 3, NM = 2x + 6, LM = x 7
CP Math 3 Page 8 of 34 Common Core Math 3 Notes - Unit 2 Day 3 Angles Ways to name angles: 1. 2. 3. 4. Definition of Equality: Definition of Congruence: Definition of Linear Pair a pair of adjacent angles whose non-common sides are opposite rays Definition of Vertical Angles two nonadjacent angles formed by two intersecting lines. Notice that the definition just describes the picture of the angles, not the relationship. Vertical Angles Theorem - If two angles are vertical angles, then they are congruent. Notice that the theorem describes the relationship between the 2 angles. 1 2
CP Math 3 Page 9 of 34 Angle Addition Postulate If A is on the interior of <DOG then, + = So if you see L O You must be able to write. E V Angle Bisector 1. Label <CAT 2. Draw and label AB such that it bisects <CAT 3. Place the appropriate marks to indicate the bisector 4. Write an equality statement about the two angles. Hmmmm same picture..as above! CAREFUL!!! These are not the same! Solve for x in each. C D C D A T A T 1. PN bisects <MPR 2. <CAT = 10x 3. <CAT is a right angle <MPN = 2x + 14 <CAD = 7x <CAD = 7x <NPR = x + 34 <DAT = 15 <DAT = 15
CP Math 3 Page 10 of 34 4. 5. Definition of Complementary Angles 2 angles whose sum is <A is comp to <B Definition of Supplementary Angles 2 angles whose sum is <J is supp to <D Statement: Statement: Ex: The supplement of an angle measures 78 degrees less than the measure of the angle. What are the measures of the angle and the supplement?
non-common side CP Math 3 Page 11 of 34 Supplement Postulate If two angles form a linear pair, then they are supplementary Complement Postulate If the non-common sides of two adjacent angles for a right angle, then the angles are complementary. If I see this picture I can write non-common side If I see this picture I can write Supplementary Angles Theorem Angles supplementary to the same angle or to congruent angles are congruent Complementary Angles Theorem Angles complementary to the same angle or to congruent angles are congruent. A C A D Y B Given: <A is sup to <C and A B Given: <D is comp to <Y and A D Prove: <B is sup to <C Prove: <A is comp to <Y Statement Reason Statement Reason 1 1. 1. 1. 2. 2. 2. 2. 3. 3. 3. 3. 4. 4. 4. 4. 5. 5. 5. 5. 6. 6. 6. 6.
CP Math 3 Page 12 of 34 Definition of Perpendicular Lines: Lines that form right angles Sketch: Symbol: Slopes: Theorem: Perpendicular lines intersect to form 4 right angles Theorem: All right angles are congruent. If you know that <A is a right angle and <B is right angle, prove that <A is congruent to <B. Statement Reason 1. 1. 2. 2. 3. 3. 4. 4.
CP Math 3 Page 13 of 34 Common Core Math 3 Unit 2 Day 4 Segment Proofs!!! Getting Ready for Proofs For each of the following givens state the conjecture and the reason. # Given Conjecture Reason 1. <A is complementary to <B 2. T is the midpoint of BE 3. T is the midpoint of BE 4. <A is supplementary to <B 5. ID= OL 6. ID = OL 7. ID = OL OL = ME 8. m<1 = m<2 m<4 +m <2 = 180 9. <A is a right angle 10. AB is perpendicular to BC 11. AB CD 12. C H S 13. B U G Y
CP Math 3 Page 14 of 34 14. TA is an angle bisector M A 15. T H 1 2 16. 1 2 17. 1 2 18. S is between P an U 19. AB BC 20. <3 and <4 are vertical PROOFS!!! #1. Given: MN PQ ; PQ RS N P Prove: MN RS Reasons M Q 1. MN PQ 1. R S 2. PQ RS 2. 3. MN RS 3.
E CP Math 3 Page 15 of 34 #2. Given: BC = DE Prove: AB + DE = AC D B C A Reason 1. 1. 2. 2. 3. 3.. #3. Given: Q is between P and R R is between Q and S PR = QS Prove: PQ = RS P Q R S Reason 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7..
CP Math 3 Page 16 of 34 #4. Given: PR QS Prove: PQ RS P Q R S Reason 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. #6. Given: M is the midpoint of AB Prove: 2AM = AB A M B Reason 1. 1. 2. 2. 3. 3. 4. 4. 5. 5.
CP Math 3 Page 17 of 34 Common Core Math 3 Unit 2 Day 5 Angle Proofs Steps for parts of some proofs: To prove two angles = 180⁰ 1. Prove LP 2. Prove Supp 3. If Supp then = 180 To prove angles = 90⁰ 1. Prove perpendicular 2. If perpendicular then right < 3. If right < then = 90⁰ 1. Given: AEC BED Prove: AEB CED Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 2. A B E C D Given: AEB CED Prove: AEC BED Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8.
CP Math 3 Page 18 of 34 Proof of the Vertical Angle Theorem: The Vertical Angle Theorem states: #3. Given: 2 intersecting lines Prove: <1 <3 1 2 3 Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8. 9. 9. 10. 10. #4. Given: <1 and <3 are supplementary <3 and <4 are linear pairs Prove: <1 <4 4 3 2 1 Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8 9. 9. 10. 10.
CP Math 3 Page 19 of 34 #5. Given: <1 and <2 form a linear pair <2 <1 Prove: <1 and <2 are right angles Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8. 9. 9. 10. 10. 11. 11. 12. 12.
CP Math 3 Page 20 of 34 Common Core Math 3 Unit 2 Day 6 Parallel Lines Definition: Sketch: Symbol: Slopes: Skew Lines Transversal Angles formed by 2 lines and a transversal: Alternate interior Alternate exterior 1 2 3 4 5 6 7 8 Consecutive or same side interior Corresponding
CP Math 3 Page 21 of 34 INVESTIGATION You will need two different colored highlighters. Investigation 1: 1. Trace with your highlighter. What letter do you see? 2. Name the typs of angle created? 3. What is the relationshp between those two angles? Investigation 2: 4. Trace with your highlighter. What letter do you see? 5. Name the typs of angle created? 6. What is the relationshp between those two angles? Investigation 3: 7. Trace with your highlighter. What letter do you see? 8. Name the typs of angle created? 9. What is the relationshp between those two angles
CP Math 3 Page 22 of 34 Investigation 4: 10. Trace with your highlighter. What letter do you see? 11. Name the typs of angle created? 12. What is the relationshp between those two angles? Investigation 5: 13. Trace with your highlighter. What letter do you see? 14. Name the typs of angle created? 15. What is the relationshp between those two angles? PRACTICE!! 1. Name alt int <s using line x as the transversal: 2. Name s.s. int <s using line y as the transversal: 3. Name corr <s using line z as the transversal: 4. Name alt ext <s using line y as the transversal: 5. Name alt int <s using line z as the transversal: 6. Name s.s. int <s using line z as the transversal:
CP Math 3 Page 23 of 34
CP Math 3 Page 24 of 34 3. Write an equation and solve for the unknown. State the theorem used to make the equation. a) b) c) d) Use the diagrams to find x and y: e. f. : g.
CP Math 3 Page 25 of 34 Common Core Math 3 Unit 2 Day 7 Proofs with Parallel Lines Theorem: If given a line and a point not on the line, then there exists exactly one line through the point that is parallel to the given line. Example: Find the measure of the angle noted by the?? 60 65 Practice: 1. Solve the crook problem to find the missing angle: 2. Solve the crook problem to find the missing angle:
CP Math 3 Page 26 of 34 Using the theorems presented previously, prove that the sum of the degrees of the angles in a triangle measure 180 degrees. Given: Triangle CHS Prove: m<c + m<h + m<s = 180 o Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. Theorem Perpendicular Transversal Theorem In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.
1. Given: 1 5; Prove: 4 is supplementary to 6 CP Math 3 Page 27 of 34 m 1 2 3 4 z p 5 7 8 6 z 2. Given: 4 is supplementary to 6 Prove 3 7 m 1 2 3 4 p 5 7 8 6 z m 1 2 3 4 3. Given: 3 is supplementary to 8 Prove: 4 5 p 5 7 8 6
CP Math 3 Page 28 of 34 m 1 2 z 5 6 9 10 y 13 14 4. Given: 5 13, 10 15 Prove: 2 12 p 4 3 8 7 11 12 15 16 z y m 1 2 p 3 7 4 8 5 6 11 12 9 10 15 16 13 14 5. Given: 2 is supplementary to 3, 1 is supplementary to 13 Prove: 4 10
CP Math 3 Page 29 of 34 Common Core Math 3 Unit 2 Day 8 Proving Triangles Congruent Side Side Side Congruence: If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. Angle Angle Side Congruence: If two angles and a non-included side of one triangle are congruent to the corresponding two angles and a side of a second triangle, then the two triangles are congruent. Side Angle Side Congruence: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Hypotenuse-Leg Congruence: If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and the corresponding leg of another triangle then the triangles are congruent. Abbreviation: Angle Side Angle Congruence: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. Not all triangles are congruent: What about AAA?? What about SSA??
Proving Triangles congruent Reasons CP Unit 2 Notes Page 30 Picture/Diagram required! Include tic marks, arc marks, color, A or S label. 1) 1) Given 2) angle or side stated 2) 3) angle or side stated 3) 4) angle or side stated 4) 5) Δ Δ 5) SSS, SAS, ASA, AAS, or HL Congruent Triangle Proof Examples ACD ACB 1. Given: CD CB A Prove: ACD ACB Reasons B C D ACD ACB 1. CD CB 1. 2. 2. 3. ACD ACB 3. ------------------------------------------------------------------------------------------------------------ R RS US 2. Given: T ST SV Prove: RSV UST S V Reasons U 1. RS US ST SV 1. 2. 2. 3. RSV UST 3.
CP Unit 2 Notes Page 31 3. L Given: Q M O N LP NP NQP LOP Prove: NQP LOP Reasons P 1. LP NP NQP LOP 1. Redraw each triangle and label 2. 2. 3. NQP LOP 3. Proving Triangle PARTS C P C T C Given: Prove: part part Reasons 1) 1) Given 2) angle or side stated 2) 3) angle or side stated 3) 4) angle or side stated 4) 5) Δ Δ 5) SSS, SAS, ASA, AAS, or HL 6) part part 6) CPCTC
CP Unit 2 Notes Page 32 Congruent Triangle PARTS Proof Examples 4. Given: BC AD and AB CD Prove: <B <D B C Reasons 1) 1) 2) 2) A D 3) 3) 4) 4) 5) 5) 6) 6) 5. Given: AB FG AB FG B G B G Prove: AF FE A Reasons 1) 1) F E 2) 2) 3) 3) 4) 4) 5) 5) 6) 6)
CP Unit 2 Notes Page 33 6. Given: FON MNO FO MN F G M Prove: F M 1) FON MNO ; FO MN 1) O Reasons N 2) 2) 3) 3) 4) 4) Isosceles Triangle Theorem - If two sides of a triangle are congruent, then the angles opposite those sides are congruent AC BC Given: Prove: A B (hint: make CD an angle bisector) Statement Reason 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8.
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