Geometry Rules! Chapter 4 Notes. Notes #20: Section 4.1 (Congruent Triangles) and Section 4.4 (Isosceles Triangles)

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1 Geometry Rules! hapter 4 Notes Notes #20: Section 4.1 (ongruent Triangles) and Section 4.4 (Isosceles Triangles) ongruent Figures orresponding Sides orresponding ngles *** parts of triangles are *** Practice: If T = DOG, then complete: (draw a picture first) m = T GD T = O ODG K JOE a) Name three pairs of corresponding angles: b) Name three pairs of corresponding sides:

2 The two triangles shown are congruent; complete. (It will help to rotate the triangles first, to get them in corresponding positions) a) RV b) R c) EV = d) m = e) NV = f) VR E 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 Equilateral 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, m = a+ b and m = 2a b. Find a and b. 9.) In equiangular, = 2x + y, = 6x 2y, and = 10. Solve for x and y. 10.) hat can you conclude from the picture? F 10 cm 10 cm 10 cm 10 cm E G

4 1 Given: is the midpoint of 1 2 D 1 2 Prove: D D Definition of Midpoint 1 Given: 1 4 Prove: Substitution

5 Notes #21: Sections 4.2 and 4.5 (Methods of Proving Triangles ongruent) Q: How can we prove that two triangles are congruent to each other? : Five ways: SSS, SS, S, S, HL SSS: - - Postulate SS: - - Postulate S: - - Postulate S: - - Postulate HL: - -( ) Postulate

6 re the triangles congruent? If so, write the congruence and name the postulate used. Redraw your triangles so they line up ou need three congruent pairs of sides/angles to follow: SSS, SS, S, S, or HL P Q Look for hidden pieces in: - vertical angles - overlapping sides - congruent angles formed by parallel lines - bisected angles - ITT/onverse of ITT - midpoints R V O T S U by by S 80 5 in 7 in R T 7 in 5 in 80 by by 5.) E G 6.) E G F F D H by D H F is the midpoint of DG and EH by

7 7.) V 8.) M T U by H MT bisects MH and TH by 9.) 10.) V D by U by 1 Given:, Prove:

8 1 Given:, Prove: Notes #22: More Proofs and Section 4.3 (Using ongruent Triangles) re the triangles congruent? If so, write the congruence and name the postulate used.,,,,

9 5.) omplete: a) because b) = because c) = E because. Then is the midpoint of by. d) because. Then = ED because. omplete the proofs: follow these key steps 1. Re-draw and label your picture; mark congruencies 2. Find and list 3 congruencies: shared sides (reflexive) vertical angles alternate interior/corresponding angles (when lines are ) angle bisectors midpoints ITT 3. State by SSS, SS, S, S, or HL 4. State part part by PT D E 6.) Given:, Prove:

10 7.) Given:, Prove: If two parallel lines are cut by a transversal, then angles are congruent. 5.) 5.) 8.) Given: is the midpoint of D and E D Prove: D E Definition of Midpoint 5.) 5.)

11 9.) Given: T bisects S and TS Prove: S T S Definition of 5.) 10.) Given: 1 2, is the midpoint of 5.) Prove: 1 2 Definition of

12 Notes #23: Quiz and lgebra Review: (Proof Review orksheet, too) Graph the points and name the quadrant in which each point is found: (-3, 2) (0, -7) (4, -1) D(6, 0) Evaluate for a = -2 and b = 3 -a 2 b(ab 3) y x Simplify: In equilateral, m = 2x+ 4y and m = x+ 5y. Solve for x and y (2 2 1) (-3) 5.) Solve for x and y 6.) Solve for x and y x 80 8 y 3x y x ) hat does PT stand for? 8.) KIM EN omplete: a) IK = b) I c) EN d) IK =

13 Notes #25: Section 4.7 (Special Segments in Triangles) (Proof Review orksheet, too) Median: connects a to the of the opposite side and are medians of is the of is the of = is equidistant from and is equidistant from and ltitude: a segment from a vertex to an opposite side and are of m = 90 m = 90

14 Perpendicular isector: a segment to the of the opposite side and are of is the of is the of = If is on the perpendicular bisector to then is equidistant from and. ngle isector: cuts a into two equal ND is from the sides of the angles. are of m = m is equidistant from and. is equidistant from and.

15 Practice: is a median to. = 5x 3 and = 22. Solve for x. D is a perpendicular bisector to. m D= 3x 15, D = 2y + 6 and D = 4y 14. Solve for x and y. D Name an angle bisector, a median, and an altitude of ngle bisector: Median: ltitude:

16 Notes #26: Test and lgebra Review Solve: Get y alone: 3 ( 1) 3 5 x = x + 2x+ 5y = 8 dd: ( 6x 2 + 4x 3) + ( 2x 2 3x+ 1) Subtract: ( 6x 2 + 4x 3) ( 2x 2 3x+ 1) 5.) Simplify: 6.) Distribute: x(4 x + x) 5 x ( x 7 x) -2xy(4x 2 + 6y - 3) 7.) Evaluate: if a = -2 and b = 4-2a 2 3ab(a + 2b) 8.) Evaluate: if x = -1 and y = 3 -y 2 xy(y x) 9.) FOIL: ( x 3)( x+ 4) 10.) FOIL: (3x 1)(2 x+ 4) 1 FOIL: (5x 2)(2x+ 1) 1 FOIL: 3( xx 4)( x+ 1)

Geometry Rules! Chapter 4 Notes. Notes #22: Section 4.1 (Congruent Triangles) and Section 4.5 (Isosceles Triangles)

Geometry Rules! Chapter 4 Notes. Notes #22: Section 4.1 (Congruent Triangles) and Section 4.5 (Isosceles Triangles) Name: Geometry Rules! hapter 4 Notes - 1 - Period: Notes #: Section 4.1 (ongruent Triangles) and Section 4.5 (Isosceles Triangles) ongruent Figures orresponding Sides orresponding ngles Triangle ngle-sum

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