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Geometry A
2.1A Inductive Reasoning and Conjecture ASSIGNMENT
Name_______________________
Hour ____ Date ______________
1.
Suppose 1 and 2 form a linear pair. What conjecture(s) can you make from this
information?
(Choose all correct answers.)
A. 1 and 2 are supplementary.
B. 1 and 2 are complementary.
C. 1 and 2 are adjacent.
D. 1 and 2 are vertical angles.
2.
Suppose M is the midpoint of AB. What conjecture(s) can you make from this information?
(Choose all correct answers).
A. AM + AB = MB
B. AB = 2(AM)
C. AM = MB
D. AB = MB
3.
Given: A and B are supplementary.
Conjecture: mA = 90 and mB = 90.
Which one of the following is a counterexample to the conjecture?
A. mA = 30 and mB = 60
B. mA = 45 and mB = 45
C. mA = 80 and mB = 100
D. None of the above statements is a counterexample because the conjecture is true.
For #4-7, show that each conjecture is false by finding a counterexample.
The counterexample can be displayed as a drawing or a statement.
4. Given:  1 and  2 form a linear pair.
5. Given: AB , BC , and AC are congruent.
Conjecture:  1   2
Conjecture: A, B, and C are collinear.
Counterexample:
Counterexample:
6. Given: 3 lines a, b, and c lie in the same plane.
7. Given: 2 acute angles
Conjecture: The lines intersect at one point.
Counterexample:
Conjecture: The sum of their measures equals
the measure of an obtuse angle.
Counterexample:
Geometry A
2.1B Inductive Reasoning and Conjecture
Name _______________________
ASSIGNMENT Hour ____ Date ______________
Complete each proof.
1.
Given: -6x – 7 = 41
Prove: x = -8
Statements
Reasons
-6x – 7 = 41
1.
2. -6x – 7 = 41
+7 +7
2.
3. -6x = 48
3.
1.
4.
 6 x 48

6
6
4.
5. x = -8
2.
5.
Given: 4x + 8 = x + 2
Prove: x = -2
Statements
Reasons
1.
1.
2. 4x + 8 = x + 2
–x
–x
2.
3. 3x + 8 = 2
3. Substitution Property
4.
4. Subtraction Property
5.
5. Substitution Property
6.
3x  6

3
3
6.
7. Substitution Property
x5
7
2
Prove: x = 9
3.
1.
Given:
x5
7
2
2. (2)
Statements
Reasons
1.
x5
 7(2)
2
2.
3. x + 5 = 14
3.
4. x + 5 = 14
–5 –5
4.
5. x = 9
5.
For #4-10, select the property that justifies each statement. Write the property on the line
provided.
reflexive property
symmetric property
transitive property
subtraction property
multiplication property
addition property
division property
distributive property
substitution property
4.
If 5x = 15, then 5x + 3 = 15 + 3 _______________________________
5.
2(y – 5) = 2(y) – 2(5) ________________________________
6.
If 6n = 42, then
7.
If 8c = 32, then 32 = 8c ________________________________
8.
17e = 17e ________________________________
9.
If y = 5 and 5 = 2n, then y = 2n ________________________________
10.
If 4m = 15, then 2(4m) = 2(15) _____________________________
6n 42

________________________________
6
6
Geometry A
2.2 Geometric Proof with Congruence
ASSIGNMENT
Name _______________________
Hour ____ Date ______________
For #1-15, select the property, definition, or theorem from the box below that justifies each
statement. Write the property, definition, or theorem on the line provided.
reflexive property
symmetric property
transitive property
addition property
subtraction property
multiplication property
division property
substitution property
distributive property
midpoint theorem
definition of a midpoint
definition of an angle bisector
vertical angles theorem
1.
If m1 = m2, then m2 = m1. ______________________________________
2.
If m1 = 90o and m2 = m1, then m2 = 90o. ____________________________________
3.
If AB = RS and RS = WY, then AB = WY. ____________________________________
4.
If AB = CD, then 3AB = 3CD. ____________________________________
5.
If m1 + m2 = 110o and m2 = m3, then m1 + m3 = 110o. ______________________
6.
RS = RS _________________________________
7.
If AB = RS, then AB + 5 = RS + 5. ____________________________________
8.
If m4  m5 and m5  m6 , then m4  m6 . _______________________________
9.
If 4x = 8, then 4x – 2 = 8 – 2. _______________________________
10.
If 80o = mA , then mA = 80o. _____________________________
11.
If DE  GH and GH  JK , then DE  JK . ________________________________
12.
If E is the midpoint of XY , then XE  EY . ________________________________
13.
If JL bisects AJC , then AJL  CJL . __________________________________
14.
If m3  m4 , then
15.
6(x – 7) = 6(x) – 6(7) ________________________________
17.
Complete the following proof.
Given:
C is the midpoint of BD .
D is the midpoint of CE .
Prove:
m3 m4

. _________________________________
10
10
B
C
D
E
BC  DE
Statements
1. C is the midpoint of BD .
1.
Reasons
2. D is the midpoint of CE .
2.
3. BC  CD
3.
4. CD  DE
4.
5. BC  DE
5.
C
B
A
18.
N
Given: AC  MN
Prove: AB  BC  MN
Statements
AC  MN
AC  AB  BC
MN  AB  BC
AB  BC  MN
M
Reasons
19.
Given: 1 and 3
are vertical angles
m1  3x  5
m3  2x  8
1
2
4
Prove: m1  14
Statements
1 and 3 are vertical angles
m1  m3
m1  3x  5 , m3  2x  8
3x  5  2x  8
3x  5  5  2x  8  5
3x  2x  3
3x  2x  2x  2x  3
x3
m1  3x  5
m1  3(3)  5
m1  14
Reasons
3
Geometry A
2.3 Geometric Proofs with Addition
reflexive property
symmetric property
transitive property
addition property
segment addition postulate
supplement theorem
ASSIGNMENT
subtraction property
multiplication property
division property
substitution property
angle addition postulate
vertical angles theorem
Name _______________________
Hour ____ Date ______________
distributive property
midpoint theorem
definition of a midpoint
definition of an angle bisector
complement theorem
For #1-21, state the property, definition, theorem, or postulate that justifies each statement.
1.
QA = QA. _______________________________
2.
If AB  BC and BC  CE , then AB  CE . _____________________________________
3.
If Q is between P and R, then PQ + QR = PR. ___________________________________
4.
If EF + GH = 14 and GH = 8, then EF + 8 = 14. _________________________________
5.
If MN  PQ , then PQ  MN . _____________________________
6.
If m7  m8  85o and m8  41o , then m7  41o  85o . __________________________
7.
If R is the midpoint of QT , then QR  RT . _______________________________
8.
If m1 = m2, then m1 + 30 = m2 + 30. ______________________________________
9.
If m1 = 23 and m2 = m1, then m2 = 23. ____________________________________
10.
If B is between C and D, then CB + BD = CD. ____________________________________
11.
If AB = CD, then CD  AB . ____________________________________
12.
If m1 + m2 = 110 and m2 = m3, then m1 + m3 = 110.__________________________
13.
If RS = ST, then RS + VW = ST + VW _________________________________
14.
If JL bisects AJC , then AJL  CJL . ____________________________________
15.
If m4 = m5 and m5 = m6, then m4 = m6. _______________________________
16.
If 5x = 30, then
17.
If 100 = mB, then mB = 100. _____________________________
18.
If DE = GH and GH = JK, then DE = JK. ________________________________
19.
If X is the midpoint of BC , then BX  CX . ________________________________
20.
7(x + 3) = 7x + 21 __________________________________
21.
If two angles form a linear pair, then the sum of those two angles will be 180 degrees.
5 x 30

. _______________________________
5
5
_____________________________
22.
If B is in the interior of ACD , then mACB  mBCD  mACD . ______________________
23.
If two angles form a right angle, then the sum of their angles will be 90 degrees.
______________________________
24.
Complete the proof below:
Given: SU = LR, TU = LN
Prove: ST = NR
Statements
1. SU = LR
1.
2. TU = LN
2.
3. ST + TU = SU
3.
4. LN + NR = LR
4.
5. ST + TU = LN + NR
5.
6. ST + LN = LN + NR
6.
7.
ST + LN – LN = LN + NR – LN
8. ST = NR
25.
Reasons
7.
8.
Complete the proof below:
C
Given: ABC  DBE
Prove: ABD  CBE


1.
2. ABC  CBD  ABD
2.

3. DBE  CBD  ABD
3.

4. DBE  CBD  CBE
4.
5. ABD  CBE
5.


E
B
Statements
1. ABC  DBE

D
A
Reasons
2.1A Inductive Reasoning and Conjecture
1.
A and C
2.
B and C
3.
G
4.
Possible answer:  1 = 110o and  2 = 70o
5.
Possible answer: create an equilateral triangle out of the three points and label the corners A, B,
and C. That would make three congruent segments that are not all on the same line.
6.
Possible answer: just make any two of the lines parallel, but still on the same plane.
7.
Possible answer: make both angles 30 degrees. That only adds to 60 degrees, which is acute.
2.1B Inductive Reasoning and Conjecture Complete each proof.
1.
Given: -6x – 7 = 41
Prove: x = -8
Statements
1.
-6x – 7 = 41
Reasons
1. Given
2. -6x – 7 = 41
+7 +7
3. -6x = 48
2. Addition
 6 x 48

6
6
5. x = -8
4. Division
3. Substitution
4.
2.
5. Substitution
Given: 4x + 8 = x + 2
Prove: x = -2
Statements
Reasons
1. 4x + 8 = x + 2
1. Given
2. 4x + 8 = x + 2
–x
–x
2. Subtraction Property
3. 3x + 8 = 2
3. Substitution Property
4. 3x + 8 = 2
- 8 -8
5. 3x = -6
4. Subtraction Property
3x  6

3
3
7. x = -2
6. Division Property
6.
5. Substitution Property
7. Substitution Property
3.
x5
7
2
Prove: x = 9
Given:
x5
7
2
x5
 7(2)
2. (2)
2
3. x + 5 = 14
Statements
1.
4. x + 5 = 14
–5 –5
5. x = 9
4.
5.
6.
7.
8.
9.
10.
Reasons
1.
Given
2.
Multiplication
3.
Substitution
4.
Subtraction
5.
Substitution
Addition Property
Distributive Property
Division Property
Symmetric Property
Reflexive Property
Transitive Property
Multiplication Property
Geometry A 2.2 Geometric Proof with Congruence
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Symmetric Property
Transitive Property
Transitive Property
Multiplication Property
Substitution
Reflexive Property
Addition Property
Transitive Property
Subtraction Property
Symmetric Property
Transitive Property
Midpoint Theorem
Definition of Angle Bisector
Division Property
Distributive Property
16.
Complete the following proof.
Given:
C is the midpoint of BD .
D is the midpoint of CE .
Prove:
B
C
D
E
BC  DE
Statements
1. C is the midpoint of BD .
1. Given
Reasons
2. D is the midpoint of CE .
2. Given
3. BC  CD
3. Midpoint Theorem
4. CD  DE
4. Midpoint Theorem
5. BC  DE
5. Transitive Property
C
B
A
17.
N
Given: AC  MN
M
Prove: AB  BC  MN
Statements
Reasons
AC  MN
Given
AC  AB  BC
Segment Addition Postulate
MN  AB  BC
Substitution
AB  BC  MN
Symmetric Property
18.
Given: 1 and 3
are vertical angles
m1  3x  5
m3  2x  8
1
2
4
Prove: m1  14
Statements
1 and 3 are vertical angles
m1  m3
m1  3x  5 , m3  2x  8
Reasons
Given
Vertical Angle Theorem
Given
3x  5  2x  8
Substitution
3x  5  5  2x  8  5
Subtraction
3x  2x  3
Substitution
3x  2x  2x  2x  3
x3
Subtraction
Substitution
m1  3x  5
Given
m1  3(3)  5
Substitution
m1  14
Substitution
3
Geometry A
2.3 Geometric Proofs with Addition
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
Reflexive Property
Transitive Property
Segment Addition Postulate
Substitution
Symmetric Property
Substitution
Midpoint Theorem
Addition
Transitive Property
Segment Addition Postulate
Symmetric Property
Substitution
Addition Property
Definition of an Angle Bisector
Transitive Property
Division Property
Symmetric Property
Transitive Property
Midpoint Theorem
Distributive Property
Supplement Theorem
Angle Addition Postulate
Complement Theorem
24.
Complete the proof below:
Given: SU = LR, TU = LN
Prove: ST = NR
Statements
Reasons
1. SU = LR
1. Given
2. TU = LN
2. Given
3. ST + TU = SU
3. Segment Addition Postulate
4. LN + NR = LR
4. Segment Addition Postulate
5. ST + TU = LN + NR
5. Substitution
6. ST + LN = LN + NR
6. Substitution
7.
ST + LN – LN = LN + NR – LN
8. ST = NR
7. Subtraction
8. Substitution
25.
Complete the proof below:
C
Given: ABC  DBE
Prove: ABD  CBE
D
A


E
B
Statements
1. ABC  DBE
1. Given
2. ABC  CBD  ABD
2. Angle Addition Postulate

3. DBE  CBD  ABD
3. Substitution

4. DBE  CBD  CBE
4. Angle Addition Postulate
5. ABD  CBE
5. Substitution



Reasons
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