math-mschernov.weebly.commath-mschernov.weebly.com/uploads/3/7/1/7/3717458… · Web view10.1 Classifying Angles and Triangles 10.2 Exterior Angle and Triangle inequality Theorem

UNIT 10 – Math 621

GEOMETRY – PROOFS & CONGUENCE

10.1 Classifying Angles and Triangles

10.2 Exterior Angle and Triangle inequality Theorem

10.3 Corresponding parts in congruent triangles

Quiz on Sections 10.1-10.3

10.4 Triangle Congruence Theorems (AII, SSS, SAS, ASA, AAS)

10.5 Triangle Congruence Theorems (HL) - Day 2

Quiz on Sections 10.4-10.5

10.6 SSA and Introduction to proofs

10.7 Proof – Day 2

10.8 Review

Unit Assessment

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10.1 Class Notes

1. Classifying Angles

Individual angles can be identified by their measures.

Write the type of angle that is illustrated in each example below and also name the angle using three letters and correct notation.

Type: ________________________

Name: _______________________

Type: _________________________

Name: _______________________

Type: _________________________

Name: _______________________

Type: _______________________________________

Name: ________________________

Type _______________________________________

Name: ________________________

2. Classifying Pairs of Angles

Pairs of angles can also be classified in relation to each other.

Adjacent angles: Two angles are adjacent if

B

C

A

64°

3

Complementary angles: Two angles are complementary if

Supplementary angles: Two angles are supplementary if

Linear Pair: Two adjacent angles form a linear pair if

Fact: Linear Angles are supplementary

Vertical Angles (also know as Opposite Angles):

Two non-straight angles are vertical angles if

Fact:Vertical angles are equal in measure.

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3. Classifying Triangles

A triangle is named according to the number of sides of equal length it has and the size of its largest angle. So, each triangle has two ways of being identified. You are responsible for all of these terms.

Names for Triangles based upon number of equal sides

Triangles with no equal sides are called ________________________________________________

Triangles with 2 equal sides are called ________________________________________________

Triangles with 3 equal sides are called ________________________________________________

Names for Triangles based upon the degree measure of the largest angle :

If the largest angle measures less than 90°, then the triangle is called _________________________

If the largest angle measures exactly 90°, then the triangle is called __________________________

If the largest angle measures more than 90°, then the triangle is called ________________________

Identify each of these triangles with two names – one based upon the side lengths and one based on the degree measure of the largest angle. (Go by how the figures look.)

a. b. c.

4. Facts about sums of angles

The sum of the three angles of a given triangle is always ___________________.

Angles at a point sum to _______________________

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10.1 Practice

1. Fill in the blanks to make each statement true.

a) ∠HAK and ___________________ are supplementary angles.

b) ∠SAT and ___________________ are supplementary angles.

c) ∠BAS and ___________________ are complementary angles.

d) ___________________ and ∠SAC are vertical angles.

e) ___________________ and ∠SAT are vertical angles.

f) ___________________ and ∠SAH are adjacent angles.

2. Find the angle measurements.

a) Find the value of “x” if ∠BAD is a right angle.

b) m∠DAC = __________________

c) m ∠BAC = ___________________

3a. Solve for x.

(7x – 21)°(2x + 3)°

D

C

B

A

6

(4x +12)°

1

2 3

4

(6x – 14)

b) _________________

c) ________________

4. Solve for x∧ y . Show all work / reasoning.

5. Solve for x. Then find the measures of each angle. (Not necessarily drawn to scale.)

______________

______________

______________6. Solve for x.

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x = __________________

_________________

7. Solve for x.

x = ___________________

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Section 10.2Exterior Angles & Triangle Inequality Notes & Classwork

1) Draw an isosceles right triangle. 2) Classify the triangle below Use proper notation to identify by angle and side. congruent parts.

3) Solve for x and state the measure of each angle.

Equation: x=_______________________________________

m∠ A= _________________ m∠B=_________________ m∠C= _________________

Exterior Angle Theorem

Discovery Activity Directions: Use the diagram to the right.

a) m∠ A+m∠B+m∠ ACB=_______ because of the

_____________________________ theorem.

b) m∠ ACB+m∠ ACD=________ because they are ______________________________ angles.

c)

d)

d) m∠ A+m∠B=________

e) Can you write a general statement relating , and

D

9

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

Example:Equation: __________________________________________________

x=_________ m∠BCD=_______

m∠BCA=_______

Triangle Inequality Theorem

Try to build the following triangles using Cuisenaire rods. Record if it is possible to make a complete triangle (endpoints of sides must touch the endpoints of the other sides).

Side 1 Side 2 Side 3 Possible? Yes or No

Lime (3) Magenta (4) Magenta (4)

Magenta (4) Yellow (5) Green (6)

Yellow (5) Orange (10) Lime (3)

Black (7) Magenta (4) Brown (8)

Lime (3) Yellow (5) Brown (8)

Magenta (4) Lime (3) Red (2)

Red (2) Magenta (4) Yellow (5)

Blue (9) Green (6) Yellow (5)

Orange (10) Magenta (4) Green (6)

Yellow (5) Lime (3) Orange (10)

10

Identify a pattern that explains why you can or cannot create a triangle. Test it. Explain your theory.

Triangle Inequality Theorem:

Tell whether the numbers can be lengths of the three sides of a triangle.

1. 14, 7, 12 2. 3, 10, 12 3. 2, 3, 5

4. 2, 2, 2 5. 1, 2, 3 6. 26, 28, 40

What possible range of numbers could be a possible side length of the triangle?

7. 5, 12, and any number between ____________ and ______________



10. Is it possible to construct a triangle using these three side lengths? Justify.

(a) 5, 9, 3 (b) 11, 4, 8 (c) 15, 8, 7

11. If the lengths of two sides of a triangle are 4 in. and 7 in., what is the smallest and largest possible measure of the third side?

smallest = _________________ largest = ________________

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12. If two sides of a triangle are 10 and 30, write an inequality to represent the possible measure of the third side.

13. Solve for x.

a. x = _____________

b. x = _____________

c. x = _____________

y = _____________

d. x = _____________

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10.3 Notes Introduction to Congruence

Angles and _________________ that are images of each other under a transformation are

called _________________________________________. If the transformation is a rigid motion transformation, then the two images will be congruent, as will all of their parts.

For example, if ∆ABC ¿∆XYZ, then the following six pairs of parts will be congruent:

∠A ¿ ____________ AB ¿ ____________

∠B ¿ ____________ BC ¿ ____________

∠C ¿ ____________ AC ¿ ____________

The theorem that tells us this is true is called the CPCFC Theorem. Here’s what it stands for:

C ______________________________________________

P ______________________________________________

of C ______________________________________________

F ______________________________________________

are C ______________________________________________

You are responsible for memorizing this theorem and what CPCFC means. You will be asked this on the next quiz.

Unless otherwise stated, corresponding parts of a figure refer to _____________ and _______________.

Example: First complete the congruence statement, with the vertices listed in the correct order:

∆BWD ¿ ∆ _____________

Now, list the 6 pairs of congruent parts.

∠B ¿ ____________ WB ¿ ____________

∠BWD ¿ ____________ BD ¿ ____________

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∠BDW ¿ ____________ DW ¿ ____________

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When figures are congruent and have the same orientation,

we say they are ____________________________________________________________________.

When figures are congruent and have opposite orientations,

we say they are ____________________________________________________________________.

Example:

The two turtles are

___________________________________congruent,

while the two lizards are

___________________________________congruent.

State whether these pairs of figures are directly or oppositely congruent. Also state which rigid motion is illustrated by each pair of figures.

1. 2. 3.

4. 5. 6.

For each problem assume the two figures that appear to be congruent are congruent.

(a) Write a congruence statement for each with vertices listed in correct order.(b) Name the transformation which maps one figure onto the other.

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(c) Tell whether the figures are directly or oppositely congruent.

1.

(a) ∆ABC _______________

(b) _______________________

(c) _______________________

_______________________

2.

(a) ∆BWD _______________

(b) _______________________

(c) _______________________

_______________________

3.

(a) ∆CAT ________________

(b) _______________________

(c) _______________________

_______________________

4.

(a) DEFG ______________

(b) _______________________

(c) _______________________

_______________________

5.

(a) KLMIJ ______________

(b) _______________________

(c) _______________________

_______________________

6.

(a) EQPAS ______________

(b) _______________________

(c) _______________________

_______________________

7. Suppose that ∆ART ∆WEB. What part has measure equal to the given figure?

ATR ________ WB ________ BEW ________

Find the missing angle measures and side lengths.

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8. Given:

a. CN = ____________

b. _____________

c. DL = ____________

d. _____________

9. Given:

a. HG = ____________ d. ______________

b. IG = ____________ e. ____________

c. IH = ____________ f. ____________

10. What does CPCFC stand for?

C ________________________________________________

P ________________________________________________

of C ________________________________________________

F ________________________________________________

are C _______________________________________________

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#11-14: (a) Write the congruence statement (b) Identify the type of rigid motion transformation that maps one

figure onto the other (c) Indicate if the figures are directly or oppositely congruent (d) List all corresponding, congruent parts.

11.

(a) _____________________________________________

(b) _____________________________________________

(c) _____________________________________________

(d) ________ _________ ________ __________

________ _________ ________ __________

________ _________ ________ __________

12.

(a) _____________________________________________

(b) _____________________________________________

(c) _____________________________________________

(d) ________ _________ ________ __________

________ _________ ________ __________

________ _________ ________ __________

13.

(a) _____________________________________________

(b) _____________________________________________

(c) _____________________________________________

(d) ________ _________ ________ __________

________ _________ ________ __________

14.

(a) _____________________________________________

(b) _____________________________________________

(c) _____________________________________________

(d) ________ _________ ________ __________

________ _________ ________ __________

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________ _________ ________ __________ ________ _________ ________ __________

Math 621Triangles and AnglesReview 10.1-10.2

1. Classify these triangles by their sides and angles.a. b. c.

2. Find the values for the variables.a. b. c.

3. Identify the angle types indicated.

a. Name a pair of acute vertical angles. ________________________

b Name a pair of obtuse vertical angles. ______________________

c. Name a linear pair. __________________________________________

d. Name a pair of acute adjacent angles. _____________________

e. Name an angle complementary to _______________

f. Name an angle supplementary to _______________

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4. Could these be lengths of sides of a triangle?

a. 2, 2, 2 _________________ b. 2, 4, 6 ________________ c. 1, 3, 5 _________________

d. 3, 5, 9 ________________ e. 5, 6, 7 ________________ f. 2, 7, 1 ________________

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5. Use the diagrams to set up an equation and solve for x.

a.

b.

c.

d.

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6. If a triangle has sides of length 10 and 12, what is the possible range of values for the third side?



ANSWERS

1a. Isosceles, right 1b. Scalene, obtuse 1c. Equilateral, acute

2a. n = 49° 2b. x = 65° 2c. x – 45°

3a. 3b. 3c. 3d. 3e. 3f.

4a. yes 4b. no 4c. no 4d. no 4e. yes 4f. no

5a. 8 + 6x = 4x + 2 + 30; x = 12 5b. 3x + 105 = 180; x = 255c. 20x – 30 = 180; x = 10.5 5d. 6x – 19 = 3x + 32; x = 17

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6. 2 < side < 22 7. 10 < side < 50 8. 0 < side < 10

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10.4 Triangle Congruence Notes & Practice

Today we are going to build some triangles. You need a yellow, a green and a red plastic side to work with. Open the rods as follows and snap them together at their corners. If the measures change while you are snapping them together, readjust them afterwards. Start fresh each time.

Example 1 SSS:

Yellow: 50 cmRed: 30 cmGreen: 45 cm

Hold your triangle up to someone else’s triangle. Are they identical? Is it possible for them not to be identical?

Example 2 SAS:

Red: 35 cmGreen: 50 cmangle between Red and green: 60°(The yellow side can be whatever it turns out to be.)


Example 3 ASA:

Green: 50 cmAngle between Red and Green: 60°. The angle between the Yellow and green: 45°. The red and yellow side lengths can be whatever they need to be.


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Example 4 SSA:

Green: 50 cmYellow: 40 cmAngle between Green and Red: 60°


This leads us to four Triangle Congruence Theorems.

SSS (Side – Side – Side): Two triangles are congruent if

SAS (Side – Angle – Side): Two triangles are congruent if

ASA (Angle – Side – Angle): Two triangles are congruent if

AAS (Angle – Angle – Side): Two triangles are congruent if

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Practice Problems.

Use the triangle congruence theorems to determine if these pairs of triangles must be congruent. Answer YES or NO. If the answer is YES, write the theorem that is used.

Your choices are: SSS, SAS, ASA or AAS.

1.

2.

3.

4.

5.

6.

7.

8.

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For each example, complete the congruence statement (if possible). Then, give the congruence theorem (SAS, ASA, AAS, or SSS) that applies. If unable to verify congruent for the triangles, write “Not possible.”

9. ABC _____________ 10. ABC _____________

by ________________ by________________

11. ABC _____________ 12. ADC ____________

by ________________ by________________

13. MAD ____________ 14. ABE _____________

by ________________ by________________

E

F

DBA

C

B

C

A E

F

D

B

D A

C

M B

C

A

D

A B

E

CD

B

D

A

C

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15. ADC ______________ 16. ∆KLM ∆ ______________

by ________________ by__________________

17. What additional information would you need if you wanted to be able to say that the

two triangles are congruent by SAS?

18. What additional information would you need if you wanted to be able to say that the

two triangles are congruent by ASA?

19. In the diagram to the right, suppose the triangles are congruent. Give a congruency statement for the two triangles.

C

LM

K

L

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10.5 HL Theorem & More Practice with Triangle Congruence

Vocabulary

Given a right triangle ABC, the side opposite to the

right angle is called the _____________________________________

The others two sides, and , are called_____________

Hypotenuse-Leg triangle congruence criteria (HL): Given two right triangles ABC and A ' B' C 'with right angles ∠B and ∠B' . If AB=A ' B' (Leg) and AC=A ' C ' (Hypotenuse), then the triangles are congruent.

Congruent Triangles by HL

If there is enough information to determine that the given triangles are congruent, justify with a triangle congruence theorem. Otherwise, write not enough info.

1. 2.

Answer: __________________________ Answer: _________________________

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3. 4.

Answer: __________________________ Answer: _________________________

5. 6.

Answer: __________________________ Answer: _________________________

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

Note the use of the Reflexive Property!

1. ABC EFD ______________ 2. ABC CDA ______________

3. ABC EFD ______________ 4. ADC BDC ______________

5. MAD MBC ______________ 6. ACB ADB ______________

E

F

DBA

C

E

F

DB

C

A

C

D BA

B

D A

C

B

D

C

A

M B

C

A

D

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10.5 Practice with Triangle Congruence

State whether each pair of triangles is congruent by SSS, SAS, ASA, AAS or HL. If none of these methods works, write NONE. If congruent, make a congruence statement for the triangles.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

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State what additional information is required in order to know that the triangles are congruent for the reason given. Be specific by naming which part(s) need to be known.11. ASA 12. SAS

13. SAS 14. ASA

15. SAS 16. ASA

17. SSS 18. SAS

19. On #12, what additional information would you need to have in order to use HL to prove the triangles are congruent?

(From Kuta Software – SSS, SAS, ASA, and AAS Congruence worksheet)

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10. 6. Triangle Congruence (continued)SSA Condition and HL Congruence

You need a yellow, a green, and a red plastic side to work with.

SSA Condition: Recall the activity we did to show that the SSA Condition does not guarantee triangle congruence. Construct two different triangles that have the following measures and sketch them below._________________________


Side-Side-Angle (SSA): Observe the diagrams below. Each triangle has a set of adjacent sides of measures 11 and 9, as well as the non-included angle of 23˚. Yet, the triangles are not congruent.

What is given: Side S: TB =9 side S: BA =11 and Angle measure abbreviated as SSA.

Examine the composite made of both triangles. The sides of lengths 9 each have been dashed to show their possible locations.

The pattern of SSA cannot guarantee congruence criteria. In other words, two triangles under SSA criteria might be congruent, but they might not be; therefore we cannot categorize SSA as congruence criterion.

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We can show that the SSA Condition does not always guarantee triangle congruence. However, there is one exception to this rule. Construct a triangle with the following measures.


Draw your triangle here. Can you make a different triangle with the same measures?

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10.6 Review of with Triangle Congruence

There are 5 ways to prove that triangles are congruent.

1. SAS _________________ _________________ _________________

2. ASA _________________ _________________ _________________

3. SSS _________________ _________________ _________________

4. AAS _________________ _________________ _________________

5. HL ____________________________ _________________

(Must be a right triangle)

There are some combinations that don’t work. They are ________________ and ________________.

Identify the congruence theorem used. If the triangles are not congruent, write “not congruent”.

39

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10.7 - Geometric ProofUsing Triangle Congruence Theorems

Example 1: Given a diagram. Prove that the triangles are congruent.

STATEMENT REASON

1. 1.

2. 2.

3. 3.

4.


STATEMENT REASON

1. 1.

2. 2.

3. 3.

4.


STATEMENT REASON

1. 1.

2. 2.

3. 3.

4.

41

PRACTICE: Given a diagram. Prove that the triangles are congruent.

1. STATEMENT REASON

1. 1.

2. 2.

3. 3.

4.

2. STATEMENT REASON

1. 1.

2. 2.

3. 3.

4.

3. STATEMENT REASON

1. 1.

2. 2.

3. 3.

4.

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4. Given the figure as marked, prove the two triangles are congruent.

STATEMENT REASON

1. 1.

2. 2.

3. 3.

4.

5. Given the figure as marked, prove the two triangles are congruent.

STATEMENT REASON

1. 1.

2. 2.

3. 3.

4.

43

Once you have proven two triangles are congruent, you also know that all of the pairs of

corresponding parts are congruent. What theorem tells us this? ______________________________

Use what you know about congruent triangles to complete the following proofs.

Example 4: Complete the following proof.

STATEMENT REASON

1. 1.

2. 2.

3. 3.

4.

5. 5.

Example 5: Complete the following proof.

STATEMENT REASON

1. M is the midpoint 1. Given

of and

2. 2.

3. 3.

4. 4.

5.

6. 6.

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PRACTICE: Prove that the triangles are congruent.

6.

Given: Prove:

STATEMENT REASON

1. 1.

2. 2.

3. 3

4.

5. 5.

7. Given the figure as marked.

Prove

STATEMENT REASON

1. 1.

2. 2.

3. 3.

4.

5. 5.

45

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8. Given: Circle with center O.

Prove:

STATEMENT REASON

1. Circle with center O. 1.

2. 2.

3. 3.

4. 4.

5.

6. 6.

9. Given:

Prove:

STATEMENT REASON

1. Circle with center C. 1. Given

2. 2.

3. 3.

4. 4.

5.

6. 6.

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10.8 Introduction to GeometryStudy Guide & Review for Test

I. You should be able to identify an angle by its size

Classify each angle as acute, obtuse, right, straight or zero.a. b. c.

d.

e. f.

II. You should be able to identify pairs of angles by name and know their characteristics.

Identify each angle pair as vertical, supplementary, complementary, adjacent, linear pair. If more than one term applies, list all.a.

a. Name two angles that are complementary.

b. Name the angle supplementary to

c. Name two angles adjacent to

b.

a. = ______________

b. Which angle forms a

linear pair with ? _________________ c. Name the angle complementary to ________________

c. Give the measure of the complement of an angle that measures 20°

d. Give the measure of the supplement of an angle that measures 20°

e. True or False

If two angles form a linear pair, then they are supplementary.

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III. You should be able to identify a triangle based upon the number of equal length sides

Identify each triangle as scalene, isosceles or equilaterala. b. c.

IV. You should be able to identify a triangle based upon the size of its largest angle

Identify each triangle as acute, obtuse, or right based upon how it looksa. b. c.

V. You should be able to tell if a triangle can be made from sides of 3 given lengths. Also, given two lengths you should be able to state the range of possible values for

the third side.

Indicate whether or not triangles can be made from sides of these lengthsa. 3 in, 3 in, 3 in b. 2 cm, 4 cm, 6 cm c. 1 ft, 2 ft, 5 ft d. 12 m, 20 m, 18 m

Give the range for the possible third side of a triangle with the two side lengths givene. 3 yd & 10 yd f. 20 mm & 70 mm g. 10 ft & 10 ft

VI. You should know that the measure of an exterior angle of a triangle is the same as the sum of the measures of the two remote interior angles and be able to use that

fact to find missing angle measures.

Find the missing angle measuresa. b. c.

?

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VII. You should know that a triangle has measures that add to 180° and be able to use that fact to find missing angle measures.

Find the missing angle measures.a. b. c.

VIII. You should be able to set up an equation and solve for a variable or the number of degrees in missing angles.a. Solve for x. b. Solve for x.

c. Solve for x. d. Solve for x.

e. Solve for x. f. Solve for x.

?

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IX. Given two congruent figures, you should be able to write a congruency statement & name all pairs of congruent parts

Write the triangle congruency statement and then identify all pairs of corresponding, congruent parts.

_____________ _____________ _____________ _____________

_____________ _____________ _____________ _____________

_____________ _____________ _____________ _____________

X. You should be able to state what CPCFC means and you should be able to use it to

justify why two corresponding parts of congruent triangles are congruent.

State what the acronym “CPCFC” stands for:

XI. Given two congruent figures, you should be able to name the transformation (or

transformations) that would map one onto the other.

XII. Given two congruent figures, you should be able to state if they are directly or oppositely congruent.

For each of these pairs of congruent figures, state the transformation that would map one onto the other and state if they are directly or oppositely congruent.

a. b. c.

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Transformation:

Directly or Oppositely?

Transformation:


Transformation:


XIII. You should be able to identify if two triangles are congruent by SSS, SAS, ASA, AAS or HL

Identify which congruence theorem guarantees that the two triangles in each pair are congruent. If there is not enough information or none is true, write “none”.a. b. c.

d. e. f.

XIV. You should be able to write a series of steps (a proof) justifying why two triangles are congruent

Given: P and M are right angles.

Prove:

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ANSWERS:

Ia. Acute b. straight c. right d. zero e. acute f. obtuse

IIa. b. c.

IIb. 90° b. c, IIc. 70° d. 160° e. true

IIIa. isosceles b. equilateral c. scalene

IVa. right b. obtuse c. acute

Va. yes b. no c. no d. yes e. 7 yd < side length < 13 ydVf. 50 mm < side length < 90 mm g. 0 ft < side length < 20 ft.

VIa. 95° b. 100° c. 70° VIIa. 45° 7b. 50° 7c. 35°

VIIIa. x = 10 b. x = 60 c. x = 50 d. 15 e. x = 35 f. x = 25

IX. ; , , , , ,

X. Corresponding Parts of Congruent Figures are Congruent

XIIa. Translation – Directly b. Rotation – Directly c. Reflection – Oppositely

XIIIa. ASA b. HL c. SAS d. AAS or SAA e. SSS f. none

XIV. Statements Reasons

1. and are right angles 1. Given

2. 2. Given

3. 3. Given

4. 4. All right angles are congruent

5. 5. H-L Theorem

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Additional Practice - Triangle Proofs.

Example 1:

Given:

Prove:

Example 2:

Given:

Prove:

Example 3:

Given:

Prove:

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Documents

math-mschernov.weebly.commath-mschernov.weebly.com/uploads/3/7/1/7/3717458… · Web view10.1 Classifying Angles and Triangles 10.2 Exterior Angle and Triangle inequality Theorem