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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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2
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°
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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
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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)
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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 ______________
8. 10, 13, and any number between ____________ and ______________
9. 49, 31, 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?
7. If a triangle has sides of length 20 and 30, what is the possible range of values for the third side?
8. If a triangle has sides of length 5 and 5, 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.)
Hold your triangle up to someone else’s triangle. Are they identical? Is it possible for them not to be identical?
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.
Hold your triangle up to someone else’s triangle. Are they identical? Is it possible for them not to be identical?
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Example 4 SSA:
Green: 50 cmYellow: 40 cmAngle between Green and Red: 60°
Hold your triangle up to someone else’s triangle. Are they identical? Is it possible for them not to be identical?
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._________________________
Green: 50 cmYellow: 37 cmAngle between Green and Red: 45°
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.
Green: 30 cmYellow: 50 cmAngle between Green and Red: 90°
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”.
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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.
Example 2: Given a diagram. Prove that the triangles are congruent.
STATEMENT REASON
1. 1.
2. 2.
3. 3.
4.
Example 3: Given a diagram. Prove that the triangles are congruent.
STATEMENT REASON
1. 1.
2. 2.
3. 3.
4.
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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.
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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.
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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:
Directly or Oppositely?
Transformation:
Directly or Oppositely?
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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