Geometry Rules! Chapter 4 Notes. Notes #22: Section 4.1 (Congruent Triangles) and Section 4.5 (Isosceles Triangles)
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1 Name: Geometry Rules! hapter 4 Notes Period: Notes #: Section 4.1 (ongruent Triangles) and Section 4.5 (Isosceles Triangles) ongruent Figures orresponding Sides orresponding ngles Triangle ngle-sum Theorem If two of one triangle are to two in another triangle then the third angles in both triangles are. Practice: If T OG, then complete: (draw a picture first) m = ΔT G T = O ΔOG.) ΔZK Δ JO a) Name three pairs of corresponding angles: b) Name three pairs of corresponding sides: - 1 -
2 - - The two triangles shown are congruent; complete. (It will help to rotate the triangles first, to get them in corresponding positions) a) ΔRV b) R c) V = d) m = e) NV f) ΔVR V N R Isosceles Triangles Isosceles Triangle Theorem ( ) If two sides of a triangle are congruent, then the angles opposite them are. onverse of the Isosceles Triangle Theorem ( ) If two angles of a triangle are congruent, then the opposite them are. - -
3 ngle bisector in vertex angle of Isosceles Triangle: quilateral Triangles Practice: Solve for x and y 5.) 3y + 7 x 7y - 5 x 40 y 6.) 7.) x 64 9 y 3x y x ) In equilateral YZ, m = a+ b and m Y = a b. Find a and b. 9.) In equiangular, = x + y, = 6x y, and = 10. Solve for x and y
4 10.) hat can you conclude from the picture? F cm 10 cm 10 cm 10 cm G 1 Given: is the midpoint of 1 1 Prove:.).) efinition of Midpoint Given: 1 4 Prove: ).) Substitution - 4 -
5 lgebra Review: ollecting like terms Simplfy: x 3x+ x x.) y 6y+ 3y 4y 9+ y+ 5 3x 8x 4x x 5 x y 5y 4 7y 5y 10 5.) x y+ xy x+ y xy + x y 6.) xy+ 3xy x + 5y 4xy+ xy 7.) x + 5x+ 15 x x 9 8.) 1x + 3x+ 5+ 3x + 6 3x 7x - 5 -
6 - 6 - Notes #3: Sections 4. and 4.3 (Methods of Proving Triangles ongruent) Q: How can we prove that two triangles are congruent to each other? : Four ways: SSS, SS, S, S SSS: - - Postulate SS: - - Postulate S: - - Postulate S: - - Postulate - 6 -
7 re the triangles congruent? If so, write the congruence and name the postulate used. Redraw your triangles so they line up You need three congruent pairs of sides/angles to follow: SSS, SS, S, or S Look for hidden pieces in: - vertical angles - overlapping sides - congruent angles formed by parallel lines - bisected angles - ITT/onverse of ITT - midpoints.) P Q R V O T S U Δ Δ by Z 80 5 in 7 in Y 7 in 5 in 80 Δ Δ by Δ Δ by S R T Δ Δ by 5.) G 6.) G F F H Δ Δ by H F is the midpoint of G and H Δ Δ by - 7 -
8 - 8-7.) 8.) V M T U Δ Δ by H MT bisects MH and TH Δ Δ by 9.) Given: YZ, Y Z Y Prove: ΔY Δ YZ Z.).) 10.) Given: YZ, YZ Prove: ΔY Δ YZ Y Z.) Y ZY.) Reflexive - 8 -
9 - 9 - Factoring Review: 1. ollect like terms. Factor out any common terms. Practice: 5x + 5x+ 10.) x 10x 9 x + 6x 14+ x 3x + 5x 3 x 3x+ 1 5.) 5x + 5x 6.) 3y 4y ) y 10y 8.) 3x 18x
10 Notes #4: More Proofs and Section 4.4 (Using ongruent Triangles), PT *** parts of triangles are *** re the triangles congruent? If so, write the congruence and name the postulate used..) Y Y YZ, Y Z Z Z YZ, YZ Y Y Z YZ, Y Z YZ, Y Z Z 5.) omplete: a) because b) = because c) = because. Then is the midpoint of by. d) because. Then because
11 omplete the proofs: follow these key steps 1. Re-draw and label your picture; mark congruencies. Find and list 3 congruencies: shared sides (reflexive) vertical angles alternate interior/corresponding angles (only when lines are ) angle bisectors midpoints ITT 3. State by SSS, SS, S, or S 4. State part part by PT 6.) Given: YZ, Y Z Prove: Z Y Z.).) Δ Δ
12 7.) Given: YZ, Y Z Y Prove: Y Z Z.).) angles theorem Δ Δ 5.) 5.) 8.) Given: is the midpoint of and Prove:.) Δ Δ.) efinition of Midpoint 5.) 5.) - 1 -
13 9.) Given: T bisects S and TS Prove: S T S.).) efinition of 5.) 10.) Given: 1, is the midpoint of Y 5.) Y Prove: YZ 1 Z.).) efinition of
14 Factor Review: 1. ombine like terms. Factor out the greatest common factor if possible 3. How many terms? terms Factor using difference of two squares 3 terms Factor using and box 4 terms - Factor using grouping Two terms: 5x Factor our the greatest common factor : 5( x 5). If both terms are perfect squares factor into ( a+ b)( a b) : ( x+ 5)( x 5) x 49.) y 50 3x y ) x 5 6.) x x 4 7.) r 16 8.) w
15 Notes #5: Proof Review: In equilateral Δ, m = x+ 4y and m = x+ 5y. Solve for x and y..) Solve for x and y 80 x y How can you prove triangles congruent? Solve for x and y y 3x x ) hat does PT stand for? 8.) ΔKIM Δ N omplete: a) IK = b) I c) ΔN d) IK =
16 omplete each proof by filling in the blanks Given: 3. Given: is the mdpt of TP and MR T M R P Prove: Prove: TM PR Given.. lt Int ngles theorem Given. T P Given: Prove: 4. Given: 1 4; 3 M is the mdpt. of Reflexive Prove: M lt Int. ngles theorem M M
17 17 5. Given: M M M is the mdpt. of Prove: M M Given: RS RT Prove: 3 4 S Given. 3. ITT R T G 6. Given: O ZO O YO Prove: Z O Y Z 8. Given: M is the mdpt of JK 1 Prove: JG MK K 1 M J KM JM. 3. JM JG
18 lgebra Review 18 Factoring Review: Three terms; x 16x divide all terms by common denominator (x 8x +15). Put quadratic in standard form; oefficient of x must be 1. (x 8x +15) 3. find factors of last term that will add up to middle coefficient (x 5)(x 3) Practice: x + 5x+ 6.) x 10x + 9 x + 5x 14 x x 8 5.) 5x 5x 10 6.) 3y 4y ) y 10y ) 3x + 18x+ 4 18
19 19 Notes 7: Section 4.6 (ongruence in Right Triangles) Section 4.7( Using orresponding Parts of ongruent Triangles) HL: - -( )Postulate Hypotenuse: Side opposite the right angle Leg: Side adjacent to right angle hich of these triangles are congruent? Using the HL Postulate: Δ Δ by 19
20 0 V.) Δ Δ by U Given:, is the perpendicular bisector of. Prove: Given. and are rt ef. of bisector; ef. of midpoint ef. of right triangles
21 1 Proving Overlapping Triangles ongruent For #1-5, complete the following: a) Separate the overlapping triangles. Mark the side or angle that is/was overlapping. b) Mark the congruent segments and congruent angles. c) re the triangles congruent? If yes, state the postulate used to state the triangle congruence (SSS, SS, S, S, or HL) Z Y ZY, Y ZY.) 1
22 M N L O L O, LMN ONM
23 3, 5.) (the triangles to examine are and ) For #6-9, complete the following proofs: 6.) Given: ZY, Y ZY Prove: Y ZY Z Y.).) 3
24 7.) Given: 4 Prove:.).) 8.) Given: L O, LMN ONM M N Prove: LM ON L O.).) 4
25 5 9.) Given:, Prove:.).) 5
26 omplete each proof by filling in the blanks Given: N O bisects N Prove: N 1. Given. O bisects N 3. Given ΔO ΔON ΔOS ΔONS O S 5 6 N.Given: NGI NI 1 Prove: GT T 1. NGI NI 1. Given Given 4. ΔGIN ΔIN ΔGTN ΔTN G N 3 4 T 1 I 6
27 7 3. Given: U H U H 1 Prove: U HL U L 1 H 4. Given: U H U H L Prove: 1 U 1 L H 1. U H 1. Given. U H. Given ΔU ΔH U H Given 7. ΔU ΔHL U H 1. Given U H 4. Given ΔU ΔH L 8. Given 9. ΔU ΔHL
28 8 lgebra Review: Factoring quadratics with an x coefficient not equal to Put in standard form. ivide by greatest common factor if possible 3. Use and box to factor x + 13x+ 0 Practice: x + 7x + 3.) 4x 1x + 5 3x 7x 6 15x + 7x 5.) 6x + 7x 3 6.) x 5x + 3 8
29 Notes 8: hapter 4 Review: Proof Review 9 Given: is the mdpt. of Prove: Given Given: LM JK LM JK M L Prove: JM LK J K 1. LM JK; LM JK 1. Given.. lt. int. angle thm Given: ZY, Y Z Prove: ZY Y Z
30 Given: 1 3 O 30 Prove: ON OP N 1 P re the triangles congruent? If so, write the congruence and name the postulate used. 5.) G 6.) G F F H Δ Δ by 7.) V H F is the midpoint of G and H Δ Δ by 8.) M T U Δ Δ by H MT bisects MH and TH Δ Δ by 9.) 10.) V Δ Δ by U Δ Δ by 30
31 31 lgebra Review Factor: + +.) Factor: 3x 6x 1 5x 15 Factor: x 11x+ 18 Factor: x + 1x 1 5.) Factor: 6x 7x ) Factor: 3x + 4x 4 7.) Factor: ) Factor: 3x 5x 3x 15 x 31
32 9.) Factor: y ) Factor: x 4x 3 3 3
33 hapter 4 Study Guide: 1. Given: ZY, Y Z 3 4 Y 33 Prove: Z 1 Z Given: ZY, Y Z 3 4 Y Prove: Y Z 1 Z PT Given:, is the midpoint of Prove: Given: NO PO, MO QO N O P Prove: M Q M Q
34 34 5. is equilateral. If m = x+ y and m = 4x y, solve for x and y. 6. In YZ, Y YZ. If m = 5x 10 and m Z = x+ 44 solve for m 7. re the pairs of triangles congruent? If so, name the congruence and the postulate used. a) b) c) d) 8. a) Solve for x: b) Solve for y: 3y x + 17 y x Factor:(Show work on separate sheet of paper) 4 a) x 81 b) y + 16y+ 64 c) x 5x 36 d) 3x 7 e) 4x 5 f) 1x 4x 40 g) 3x 7x h) 36x + 48x 15 34
Geometry Rules! Chapter 4 Notes. Notes #20: Section 4.1 (Congruent Triangles) and Section 4.4 (Isosceles Triangles)
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