Math 2 Unit 2 Notes: DAY 1 Review Properties & Algebra Proofs Warm-up Addition Property of equality (add prop =) If Then a = b If 5x-7 = 23 Then If AB = CD Then AB+GH = Subtraction Property of equality If a = b Then If 6y + 5 = -25 Then If EF + AB = CD + AB Then Multiplication Property of equality If a = b Then If m = 3 Then 4m = If (½)m ABC = 45 Then Division Property of equality If a = b Then If 15y = 105 Then If (5) EFG = 50 Then Symmetric Property of equality If a = b Then If 5 = 3x 1 Then If -30 < x Then Substitution Property of equality If a = b and a+3=c Then If 5x + 3 = z and x = -2 Then If cat+3=dog & cat=n Then Transitive Property of equality If a=b and b=c Then If 4+1=2+3 and 2+3=5 Then If AB=CD and CD=EF Then Reflexive Property of equality a = a AB = m ABC = Distributive Property of equality a(b+c) = 4(x-5) = Definition of Congruence If AB = CD Then If m ABC = m DEF Then
Notes A proof is a argument in which each is supported by a reason. * This could be a,,, etc. There are 4 essential parts of a good proof: 1. 2. 3. 4. When writing a reason for a step, you must use one of the following:
There are a few allowed assumptions: Vertical Angles : Reflexive Property: Linear Pair: Any proof should start with the following: Steps to write a proof: 1) 2) 3) 4)
EX 1: Solve 5x 18 = 3x + 2 and write a reason for each step. Statement Reason 5x 18 = 3x + 2 given EX 2: Solve 55z 3(9z + 12) = - 64 and write a reason for each step. Statement Reason 55z 3(9z + 12) = - 64 given Ex) Write the proof. Given: x + 4 = 1 Prove: x = 9 3
Warm-up Math 2 Unit 3 Notes: DAY 2 Review of Parallel Line Finding the Slope of a Line m = m = Give an example of an equation with a positive slope: Give an example of an equation with a negative slope: Give an equation for a vertical line: Give an equation for a horizontal line:
Day 2 Notes Transversal- Draw a picture of a Transversal in the box. --Angles on opposite sides of a transversal and inside two other lines. Draw a picture. * - If a transversal intersects two parallel lines, then alternate interior angles are. -Angles in the same position relative to a transversal and two other lines Draw a picture. * If a transversal intersects two parallel lines, then corresponding angles are.
-Angles on the same side of a transversal and inside two other lines Draw a picture. * If a transversal intersects two parallel lines, then same-side interior angles are. -Angles on opposite sides of a transversal and outside two other lines Draw a picture. * If a transversal intersects two parallel lines, then alternate exterior angles are. Ex) Use the diagram above. Identify which angle forms a pair of same-side interior angles with 1. Identify which angle forms a pair of corresponding angles with 1.
-a pair of non-adjacent angles formed when two lines intersect. Draw a picture. * If two lines intersect then opposite angles are. -A pair of adjacent angles that form a line. Draw a picture. * A pair of adjacent angles are.
For examples below, the figures shows p q. Ex) m 1 = x 5 and m 2 = 2x - 40, find x and m 1. t x = 1 p m 1 = 2 q Ex) m 3 = 6x + 12 and m 4 = 10x + 8, find x and m 4. t x = 3 p m 4= 4 q
We can use a transversal to prove lines parallel and relate parallel and perpendicular lines. We do that using the of the parallel lines theorems.
Ex)
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Warm-up Math 2 Unit 3 Notes: DAY 3 Parallel Line Proofs y a x 1 8 5 7 6 b 4 2 3 15 9 12 14 11 13 16 10 17 18 19 Assume a b. Complete the chart. ANGLES TRANSVERSAL TYPE, SUPPLEMENTARY, OR NONE 1. 1 and 14 2. 2 and 15 3. 7 and 9 4. 9 and 16 5. 10 and 17 6. 16 and 14 7. 9 and 14 8. 18 and 19 9. 1 and 16 10. 3 and 8 11. 6 and 9
Notes Writing a parallel line proof is similar to writing an algebraic proof, the only difference is you use the to justify your reasons. Can you remember all the properties? List them below: 1) 2) 3) 4) 5) 6) 7) If you want to prove that lines are parallel from these properties remember to use the. Ex) Prove the following: a) Given: l // m; s // t Prove: 1 5 1. l // m; s // t 1. 2. 1 3 2. 3. 3 5 3. 4. 1 5 4.
b) Given: l // m; s // t Prove: 2 4 1. l // m ; s // t 1. 2. 2 3 2. 3. 3 4 3. 4. 2 4 4. c) Given: l // m; 1 4 Prove: s // t 1. l // m ; 1 4 1. 2. 3 1 2. 3. 3 4 3. 4. s // t 4. d) Given: l // m ; 2 5 Prove: s // t 1. l // m ; 2 5 1. 2. 2 3 2. 3. 3 5 3. 4. s // t 4.
Ex. Given: l // m ; s // t Prove: 2 4 l 2 1 s t m 3 5 4 Ex. Given: l // m ; s // t Prove: 1 5
Warm-up Math 2 Unit 3 Notes: DAY 4 Congruent Triangles Definitions and Postulates Regarding Segments If C is between A and B, Segment Addition Postulate Then If AB CD, Definition of Segment Congruence Then If AB bisects CD, Definition of Segment Bisector Then If B is the midpoint of A and C, Definition of Midpoint Then Research the properties above and finish the statement. Then illustrate the Definition or Postulate below. Segment Addition Postulate Definition of Segment Congruence Definition of Segment Bisector Definition of Midpoint
Notes Congruent Triangles: triangles that are the same and Each triangle has three and three. If all pairs of the corresponding parts of two triangles are, then the triangles are. Congruent Triangles: Corresponding Congruent Angles: Corresponding Congruent Sides:
Example #1: In the following figure, QR = 12, RS = 23, QS = 24, RT = 12, TV = 24, and RV = 23. Name the corresponding congruent angles and sides. Properties of Triangle Congruence: Reflexive Symmetric Transitive
Example #2: If WXZ STJ, name the congruent angles and congruent sides. Angles Sides Naming Congruent Triangles I. Draw and label a diagram. Then solve for the variable and the missing measure or length. 1. If BAT DOG, and m B = 14, m G = 29, and m O = 10x + 7. Find the value of x m O. x = m O=
2. If COW PIG, and CO = 25, CW = 18, IG = 23, and PG = 7x 17. Find the value of x and PG. x = PG= 3. If DEF PQR and DE = 3x 10, QR = 4x 23, and PQ = 2x + 7. Find the value of x and EF. x = EF =
II. Use the given information and triangle congruence statement to complete the following. 1. ABC GEO, AB = 4, BC = 6, and AC = 8. 2. What is the length of GO? How do you know? 3. BAD LUK, m D = 52, m B = 48, and m A = 80. a. What is the largest angle of LUK? b. What is the smallest angle of LUK?
Side Side Side Congruence: If the of one triangle are congruent to the sides of a second triangle, then the triangles are. Abbreviation: Side Angle Side Congruence: If two sides and the included of one triangle are congruent to two and the included angle of another triangle, then the triangles are. Abbreviation: Example #1: Mark the figure & state if the triangle is congruent by SSS or SAS. Given: FE HI, and G is the midpoint of both EI and FH.
Example #2: Mark the figure & state if the triangle is congruent by SSS or SAS. Given: DE and BC bisect each other. Example #3: Mark the figure & state if the triangle is congruent by SSS or SAS. Given: AB AC and BY CY
Warm-up Math 2 Unit 3 Notes: DAY 5 ASA, AAS & HL Write a 2-column proof. t k 2 1 6 5 l 4 8 3 7 1) Given: k// l Prove: 6 is supp. to 7. 2) Given: k// l Prove: 2 7
Notes Angle Side Angle Congruence: If two and the included of one triangle are congruent to two angles and the included side of another triangle, then the triangles are. Abbreviation: Angle Angle Side Congruence: If two angles and a nonincluded side of one triangle are congruent to the corresponding two and a side of a second triangle, then the two triangles are. Abbreviation:
Hypotenuse Leg Coungruence: If one angle measures 90 degrees and both have a congruent leg and hypotenuse then both triangles are congruent. Abbreviation: Example #1: Mark the figure and state if the triangle is congruent by ASA, AAS, or HL. Given: AB bisects CAD and 1 2
Example #2: Mark the figure and state if the triangle is congruent by ASA, AAS, or HL. Given: AD CB and A C Example #3: Mark the figure and state if the triangle is congruent by ASA, AAS, or HL. Given: V S and TV QS
After Quiz Practice: Triangle Congruence (SSS-SAS-ASA-HL-AAS) SSS: (Side-Side-Side) If the sides of one triangle are congruent to the sides of another triangle, then the two triangles are congruent. SAS: (Side-Angle-Side) If two sides and the included angle in one triangle are congruent to two sides and the included angle in another triangle, then the two triangles are congruent. ASA: (Angle-Side-Angle) If two angles and the included side in one triangle are congruent to two angles and the included side in another triangle, then the two triangles are congruent. ABC DEF ABC DEF ABC DEF AAS: (Angle-Angle-Side) If two angles and a nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of another triangle, then the two triangles are congruent. HL: (Hypotenuse-Leg) If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the two triangles are congruent. ABC DEF ABC DEF Determine whether each pair of triangles can be proven congruent by using the SSS, SAS, ASA, HL or AAS Congruence Postulates. If so, write a congruence statement and identify which postulate is used. If not, write cannot be proven congruent. 1) 2) 3)
4) 5) 6) 7) 8) 9) 10) 11) 12)
Math 2 Unit 3 Notes: DAY 6 Triangle Congruency Proofs Warm-up For each pair of triangles, tell which conjectures, if any, make the triangles congruent. 1. ABC EFD 2. ABC CDA C C B F A B D E D A 3. ABC EFD 4. ADC BDC C F C A B D E A D B 5. MAD MBC 6. ABE CDE D C D C E A M B A B 7. ACB ADB 8. C A B D
Notes Recall the steps for writing a proof: 1) 2) 3) 4) The only difference in writing a proof for Congruent Triangles is the final statement. The final statement should include one of the following:,,,, and. Ex 1) Given: AD DC AC BD Prove: ΔABD ΔCBD B A D C
Ex 2) Given: <E <H G is the midpoint of EH Prove: ΔGFE ΔGIH F H G E I B Ex 3) Given: AB BC AC BD Prove: ΔABD ΔCBD A D C
SCRAMBLE PROOFS: Draw the figure and write final proof below.
Math 2 Unit 3 Notes: DAY 7 Triangle Congruency Proofs Warm-up 1. write a congruency statement for the two triangles at right. C G O A R E 2. List ALL of the congruent parts if EFG HGF 3. Name all the ways to prove triangles congruent. For each pair of triangles, tell: (a) Are they congruent (b) Write the triangle congruency statement. (c) Give the conjecture that makes them congruent. B 4. A C D 5. A E W T R
Practice: Fill in the following proofs with the necessary Statements and Reasons to prove the triangles congruent. 1) Statements Reason 2) Statements Reason
3) Given: O is the midpoint of MQ O is the midpoint of NP Prove: Statements Reasons 4) Statements Reason
5) Given: AD EC BD BC Prove: ABD EBC Statements Reasons
Math 2 Unit 3 Notes: DAY 8 CPCTC Warm-up Given: B C D F M is the midpoint of DF. Prove: BDM CFM Statements Reasons
Notes CPCTC- Corresponding Parts of Congruent Triangles are Congruent *Explanation: To prove that parts (sides or angles) of triangles are congruent to parts of other triangles, first prove the triangles are congruent. Then by CPCTC, all other corresponding parts will be congruent. When writing a proof, should be your reason!!!!! Ex) Given: AB DC ; AD BC Prove: A C Statements Reasons
Ex) Given: MA TA, A is the midpoint of SR Prove: MS TR Statements Reasons Ex) Given: 1 2 ; 3 4 Prove: CB CD Statements Reasons
Ex) Given: MS TR; Prove: MA TA. MS TR Statements Reasons