Academic CCM2 – Unit 3 – Similarity and Triangles Name: ______________________Notes and Activities Date: _____________ Pd: _____

Date Topic Quiz/TestThursday, February 18 Ways to prove triangles are similar

Friday, February 19 Solving problems with similar trianglesMonday, February 22 Introduction to trianglesTriangle Angle Sum Theorem and Exterior Angle TheoremTuesday, February 23 Quiz on Similar TrianglesIsosceles Triangle TheoremWednesday, February 24 Midsegment Theorem QUIZ

Thursday, February 25 Congruent TrianglesFriday, February 26 Congruent Triangles

Monday, February 29 ReviewTuesday, March 1 Unit 3 Test TEST

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Common Core Standards:Understand congruence in terms of rigid motions.G-CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.G-CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.G-CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.Prove geometric theorems.G-CO.10 Prove theorems about triangles. Make geometric constructionsG-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

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Proportions and Similar FiguresRatios:Example 1: The total number of students who participate in sports programs at central high school is 520. The total number of students in the school is 1850. Find the athlete to student ration.Proportion:

Solve each proportion: Example 2: 618.2

= 9y

Example 3: 4 x−53

=−266

Example 4: A boxcar on a train has a length of 40 feet and a width of 9 feet. A scale model is made with a length of 16 inches. Find the width of the model. Similar Figures:

Notation:

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Example 5: Determine whether the given figures are similarScale Factor:In example 5, the scale factor of 𝜟ABC to 𝜟RST is __________.the scale factor of 𝜟RST to 𝜟ABC is __________.Similarity Statement:You Try!

Practice: A. Solve each proportion. Check your answers with your group.

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B.

C. For each of the following figures determine if they are similar. If yes, find the scale factor.

Solving with Similar Figures

5

6

7

8

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Introduction to TrianglesDefinition of a triangle: a polygon with ____ sidesSides:Vertices:Angles:Exterior Angles: Remote Interior Angles:

1) Classify Triangles by Angles Acute Triangle: all angles are _________ Obtuse Triangle: one angle is _________ Right Triangle: one angle is ___________ Equiangular Triangle: all angles are _____________

*also an _________ triangle!2) Classify Triangles by Sides

Scalene Triangle: ____ two sides congruent Isosceles Triangle: at least ______ two sides congruent Equilateral Triangle: _____ three sides congruent

*note: an ______________ triangle is a special kind of _______________ triangle!Ex 1: Find x, QR, RS, and QS if DQRS is an equilateral triangle.

Ex 2: Find x, JM, MN, and JN if DJMN is an isosceles triangle.

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Q

R

S

4x

6x – 1

2x + 1

J

M

N

2x – 5

x – 2

3x – 9

Triangle Angle-Sum The sum of the angles in a triangle add up to ________________.

The sum of the acute angles in a right triangle add up to _______________.

Each angle in an equilateral triangle is the _________________ and measures _______________.

The measure of an exterior angle is equal to the sum of its remote interior angles.

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Angles in Triangles Practice #1

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Angle Sum Practice #2Solve for the missing angles.1. 2. 3.

4. 5. 6.

7. 8. 9.13

10. 11. 12.

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Isosceles Triangle TheoremWe know that an Isosceles Triangle has two congruent sides (legs). The third side is called the base. The two angles that touch the base are called base angles. The third angle is the vertex angle.

Isosceles Triangle Theorem - If two sides of a triangle are congruent, then the angles opposite those sides are congruent. (If the two legs are congruent, then the base angles are congruent.)If , then

Converse of the Isosceles Triangle Theorem - If two angles of a triangle are congruent, then the sides opposite those angles are congruent. (If the base angles are congruent, then the legs are congruent.)If , then

Examples

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Isosceles Triangles Practice

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50

x

x 54

63

11

10x

2x + 75x - 8

4040

F G

4x - 6

18

16

F

G Hx

98

x

74

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10

10

(10x+6)

(5x -10)(4x + 20)

Isosceles Triangles PracticeFind the value of the variable or question mark.1. 2. 3. 4.

x = ________ x = ________ x = ________ x = ________

5. 6. 7. 8.

x = ________ x = ________ x = ________ x = ________

9. 10. 11. 12.

x = ________ y = ________ ? = ________ x = ________

Draw a picture to help find the missing sides or angles!!13. The vertex angle of an isosceles triangle is 40o. What is the measure of one base angle?14. One base angle of an isosceles triangle is 73o. What is the measure of the vertex angle?15. The degree measure of the vertex angle is (7x - 27)o. The degree measure for each base angle is (9x - 34)o. What is the value of one base angle?

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16. The degree measure of the vertex angle is (3x - 8)o. The degree measure for each base angle is (6x - 41)o. What is the value of vertex angle?17. The degree measure of the vertex angle is (x + 21)o. The degree measure for each base angle is (2x + 17)o. What is the value of x?

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Midsegment of a TriangleMidsegment - A segment that connects the midpoints of two sides of a triangle.Midsegment Theorem - The segment connecting the midpoints of two sides of a triangle is parallel to the third side and HALF AS LONG as that side.

Examples

Mid-segments of D’s Practice

Directions: Find the values of the variables. You must show all work to receive full credit. Figures are not drawn to scale.

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x6

6

4

4

13

x y

z10

20

8

5

x + 1

10

x

5

60°

x2x

31

6x + 11

x + 23 3y - 9

2y + 6

xy

4060

3x - 6

x 2x + 1

y

2521

x

1. x = _____ y = _____ z = _____ 2. x = _____

3. x = _____ 4. x = _____

5. x = _____ 6. x = _____

7. x = _____ y = _____ 8. x = _____ y = _____

9. x = _____ 10. x = _____ y = _____

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Isosceles Triangles and Midsegment PracticeI. Isosceles Triangles: Find the indicated value for each of the following. 1. 2. 3.

4. 5. 6.

II. Midsegment: Find the indicated value.7. x=____________ 8. x=_____________ 9. x=_______________

10. x=________________ 11. x=___________ 12. x=____________

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Triangle CongruenceTriangle ABC is congruent to Triangle FED. Name 6 congruent parts1. 2. 3. 4. 5. 6. IN ORDER FOR TWO TRIANGLES TO BE CONGRUENT ALL CORRESPONDING _________ AND ________ MUST BE __________!Congruency StatementΔABC≅ΔDEFComplete the congruency statement for the following triangles…

ΔPQR≅Δ_______________ ΔPQR≅Δ_______________Corresponding PartsName the corresponding congruent parts for these triangles.1. AB 2. BC 3. AC 4. A 5. B 6. C Do you need all six ? _____________Side-Side-Side (SSS)

1. 2. 3.

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Included angle: The angle _______________ two sides

Side-Angle-Side (SAS)

1. 2. 3.

Included Side: The side between two angles

Practice:

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Angle-Side-Angle (ASA)

1. 2. 3.

Angle-Angle-Side (AAS)

1. 2. 3.

Side Names of TrianglesRight Triangles: side across from right angle is the ________________, the remaining two are _____________.

Examples: Tell whether the segment is a leg or a hypotenuse.

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There are 4 ways to prove triangles are congruent…1. 2. 3. 4. 5.

Warning: No ___________ or ____________ Postulate NO CURSING IN MATH CLASS!

Warning: No AAA Postulate

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B

O

X

FO

X

Congruent Triangles PracticeI. Name the congruent triangles.

1.ΔOGD≅Δ___________ 2. ΔRAC≅Δ___________

3. ΔLIN≅Δ___________ 4. ΔFOX≅Δ___________

II. Name the congruent triangle and the congruent parts..7. ΔFGH≅Δ___________

∠EFI≅∠___________

∠G≅∠_______

∠H≅∠________

Use the congruency statement to fill in the corresponding congruent parts.

8. ΔEFI≅ΔGHK ∠E≅∠_______ ∠I ≅∠_______

∠I≅∠_______

For each pair of triangles, tell which postulates, if any, make the triangles congruent.

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C

A R

G

E

O

L

I

N E

A

R

C

AD

BC

A B D

F

E

A BD

C

A

C

B D

F

E

A B

E

CDD

A

C

BM

A

C

D

B

9. DABC DEFD______________ 10. DABC DCDA ______________

11. DABC DEFD ______________ 12. DADC DBDC ______________

13. DMAD DMBC ______________ 14. DABE DCDE ______________

15. DACB DADB ______________ 16. ______________

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A

C S

H

J

E

State which postulate could be used to prove each set of triangles congruent, then write a mathematical statement for each set of congruent triangles.1. 2. 3.

4. 5. 6.

7. 8. 9.

Given triangles ASC and HJE, if then complete each statement.

10. A _____ 11. SCA ________ 12. _______

13. ______ 14. JEH ________ 15. CAS ______

Answer the following questions. Show ALL Work16. Given , AB = 15, BC = 20, AC = 25, and TS = 3x – 7, determine x.

17. Given , RS = 10, ST = 13, RT = 16, and AC = 4x – 8, determine x.

18. Given , AB = 30, BC = 34, AC = 28, and SR = 5x – 10, determine x.

19. Given , ST = 6, SR = 7, TR = 10, and CB = 6x – 12, determine x.

20. Given , 10 < RS < 14, and BA = 2x + 1, determine x.32

P S T

Q R U

V Z

W X Y L

F G

C D E

S E

D J H R

A B

V W Y L

P

K M

A

E W

Q T

E

T D F K

S

F S

X

H J

Triangle Congruence Picture Questions

I. If the triangles can be proven congruent, give the reason (SSS, SAS, ASA, or AAS). If there is not enough information to prove the triangles congruent, write “none.”

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

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