Download File - Ms. Castoro`s Geometry Classes

Name: ________________________
May, 2015
Conditional Statements
Conclusion
Converse statements
Inverse statements
Contrapositive statements
Contrapositive statements
Statement: ̅̅̅̅
𝐶𝐴 ≅ ̅̅̅̅
𝐶𝐴
Reason: Reflexive Property
Statement: ∠𝐷𝑃𝐴 ≅ ∠𝐶𝑃𝐵
Reason: Vertical angles are
congruent
Proofs and Logic IP
Castoro / Cheung
Proofs and Logic Independent Practice
Logic How-To
If a, then b.
If a.
Negation: to deny
Then b.
If b, then a.
If not a, then not b.
Counter-Example: an example to disprove a statement
If not b, then not a.
Most Common Proof Moves
̅̅̅̅̅
Statement: ∠𝐴𝐵𝐷 ≅
Statement: ∠𝐴𝐷𝐶 ≅ ∠𝐶𝐷𝐵
Statement: 𝐴𝑀 ≅ ̅̅̅̅̅
𝑀𝐵
Reason: Right angles are
∠𝐷𝐵𝐶
Reason: Definition of
Reason: Definition of
always congruent
midpoint
angle bisector
Statement: ∠𝑎 ≅ ∠𝑏
Reason: Alternate interior
angles are congruent
Statement: ∠𝑎 ≅ ∠𝛼
Reason: corresponding
angles are congruent
SSS Congruence Postulate
Congruent Triangles
SAS Congruence Postulate
AAS Congruence Postulate
HL Congruence
AA Similarity Postulate
SSS Similarity
Similar Triangles
SAS Similarity
Statement: ∠𝐷𝐶𝐴, ∠𝐷𝐶𝐵 are
right angles
Reason: Definition of
perpendicular lines
ASA Congruence Postulate
CPCTC – corresponding parts of
congruent triangles are congruent
Corresponding sides of similar
triangles are proportional
The product of the means is
equal to the product of the
extremes
Name: ________________________
May, 2015
Proofs and Logic IP
Castoro / Cheung
Logic Practice
1. Which compound statement is true?
2. A student wrote the sentence “4 is an odd integer.”
1) A triangle has three sides and a quadrilateral
What is the negation of this sentence and the truth value
has five sides.
of the negation?
2) A triangle has three sides if and only if a
1) 3 is an odd integer; true
quadrilateral has five sides.
2) 4 is not an odd integer; true
3) If a triangle has three sides, then a quadrilateral
3) 4 is not an even integer; false
has five sides.
4) 4 is an even integer; false
4) A triangle has three sides or a quadrilateral has
five sides.
3. What is the inverse of the statement “If two triangles
4. What is the converse of the statement "If
,
are not similar, their corresponding angles are not
then
is a right triangle"?
congruent”?
1) If
is a right triangle, then
.
1) If two triangles are similar, their corresponding
2)
if, and only if,
is a right
angles are not congruent.
triangle.
2) If corresponding angles of two triangles are not
3) If
is not a right triangle, then
congruent, the triangles are not similar.
.
3) If two triangles are similar, their corresponding
4) If
angles are congruent.
, then
is not a right
4) If corresponding angles of two triangles are
triangle.
congruent, the triangles are similar.
6. What is the contrapositive of the statement, “If I am tall,
5. If
, which statement is false?
1) x is prime and x is odd.
then I will bump my head”?
2) x is odd or x is even.
1) If I bump my head, then I am tall.
3) x is not prime and x is odd.
2) If I do not bump my head, then I am tall.
4) x is odd and 2x is even.
3) If I am tall, then I will not bump my head.
4) If I do not bump my head, then I am not tall.
7. Given the statement: “If two lines are cut by a
8. Given the statement, "If a number has exactly two
transversal so that the corresponding angles are
factors, it is a prime number," what is the contrapositive of
congruent, then the lines are parallel.” What is true about this statement?
the statement and its converse?
1) If a number does not have exactly two factors,
1) The statement and its converse are both true.
then it is not a prime number.
2) The statement and its converse are both false.
2) If a number is not a prime number, then it does
3) The statement is true, but its converse is false.
not have exactly two factors.
4) The statement is false, but its converse is true.
3) If a number is a prime number, then it has
exactly two factors.
4) A number is a prime number if it has exactly
two factors.
Complete the statements below and state whether they are true or false:
If two numbers are even, then their sum is even.
Hypothesis:__________________________________________________________________________
Conclusion:__________________________________________________________________________
Converse:___________________________________________________________________________
Inverse: _____________________________________________________________________________
Contrapositive:_______________________________________________________________________
Name: ________________________
May, 2015
Proofs and Logic IP
Castoro / Cheung
Proof Practice
State the postulate that proves the following triangles congruent or similar from the word bank below:
D. SAS
A. SSS
B. AAS
C. ASA
E. SAS
F. AA Similarity
G. HL
Similarity
1.
2.
3.
4.
5.
6.
7.
Match the proof move to the appropriate diagram:
A. Vertical Angles
1.
H. SSS
Similarity
8.
2.
3.
5.
6.
B. Reflexive Property
C. Corresponding Angles
D. Angle Bisector
E. Alternate Interior Angles
F. Right Angles
1. In the diagram of
, and
4.
and
.
Which method can be used to prove
1) SSS
2) SAS
3) ASA
4) HL
Proof Multiple Choice Practice
2. In the accompanying diagram of triangles BAT and FLU,
below,
,
and
.
?
Which statement is needed to prove
1)
2)
3)
4)
?
Name: ________________________
May, 2015
Proofs and Logic IP
Castoro / Cheung
3. In the diagram below, four pairs of triangles are shown.
Congruent corresponding parts are labeled in each pair.
Using only the information given in the diagrams, which
pair of triangles can not be proven congruent?
1) A
2) B
3) C
4) D
5. Which condition does not prove that two triangles are
congruent?
1)
2)
3)
4)
7. In the diagram below of
and
intersect at E, such that
and
,
and
.
Triangle DAE can be proved congruent to triangle BCE by
1) ASA
2) SAS
3) SSS
4) HL
4. Given that ABCD is a parallelogram, a student wrote the
proof below to show that a pair of its opposite angles are
congruent.
What is the reason justifying that
?
1) Opposite angles in a quadrilateral are
congruent.
2) Parallel lines have congruent corresponding
angles.
3) Corresponding parts of congruent triangles are
congruent.
4) Alternate interior angles in congruent triangles
are congruent.
6. Which statements could be used to prove that
and
are congruent?
1)
2)
3)
4)
8. As shown in the diagram below,
.
Which method could be used to prove
1) SSS
2) AAA
3) SAS
4) AAS
bisects
and
?
Name: ________________________
May, 2015
9. In the diagram below of
point on
,
Proofs and Logic IP
Castoro / Cheung
, Q is a point on
is drawn, and
, S is a
below,
.
Which reasons can be used to prove
?
1) reflexive property and addition postulate
2) reflexive property and subtraction postulate
3) transitive property and addition postulate
4) transitive property and subtraction postulate
and
,
information would prove
1)
2)
3)
4)
Which statement must be true?
1)
2)
3)
4)
12. In the diagram below,
.
Which statement is not always true?
1)
2)
3)
4)
Which statement must be true?
1)
2)
3)
4)
15. In
.
.
Which reason justifies the conclusion that
?
1) AA
2) ASA
3) SAS
4) SSS
11. The diagram below shows a pair of congruent triangles,
with
and
.
13. In the diagram of
10. In the diagram below,
. Which additional
?
14. When writing a geometric proof, which angle
relationship could be used alone to justify that two angles
are congruent?
1) supplementary angles
2) linear pair of angles
3) adjacent angles
4) vertical angles
16. In triangles ABC and DEF,
,
,
,
, and
. Which method could be used to
prove
?
1)
2)
3)
4)
AA
SAS
SSS
ASA
Name: ________________________
May, 2015
Given: ̅̅̅̅
𝐿𝑂 bisects ̅̅̅̅̅
𝑀𝑃 at N
∠𝑀 ≅ ∠𝑃
Prove: 𝛥𝐿𝑁𝑀 ≅ 𝛥𝑂𝑁𝑃
Statement
̅̅̅̅ bisects 𝑀𝑃
̅̅̅̅̅ atCN
𝐿𝑂
E
∠𝑀 ≅ ∠𝑃
̅̅̅̅
𝐿𝑁 ≅ ̅̅̅̅
𝑁𝑂
∠𝐿𝑁𝑀 ≅ ∠𝑃𝑁𝑂
𝛥𝐿𝑁𝑀 ≅ 𝛥𝑂𝑁𝑃
B
Proofs and Logic IP
Castoro / Cheung
Reason
Given
Given
Definition of bisector
Vertical angles are congruent
ASA Theorem
Why is the circled statement wrong? What is the correct statement?
D
Given: ̅̅̅̅
𝐴𝐷 bisects ̅̅̅̅
𝐵𝐶 at E, ̅̅̅̅
𝐴𝐵 ⏊ ̅̅̅̅
𝐵𝐶 , ̅̅̅̅
𝐷𝐶 ⏊ ̅̅̅̅
𝐵𝐶
̅̅̅̅ ≅ 𝐷𝐶
̅̅̅̅
Prove: 𝐴𝐵
Given: ̅̅̅̅
𝐴𝐵 || ̅̅̅̅
𝐷𝐸
̅̅̅̅
𝐷𝐶
Prove: ̅̅̅̅
𝐵𝐶
̅̅̅̅
𝐷𝐸
= ̅̅̅̅
𝐵𝐴
̅̅̅̅ ≅ 𝐵𝐶
̅̅̅̅ , 𝐵𝐷
̅̅̅̅
̅̅̅̅ bisects line 𝐴𝐶
Given: 𝐴𝐵
Prove: ∆𝐴𝐵𝐷 ≅ ∆𝐶𝐵𝐷
Name: ________________________
May, 2015
1. 4
2. 2
1. B
2. F
1. B
3. 3
3. D
2. C
1. 3
9. 1
2. 1
10. 2
Proofs and Logic IP
Castoro / Cheung
Logic Multiple Choice Answer Key:
4. 2
5. 3
6. 4
Matching Postulates
4. H
5. E
3. F
3. 1
11. 4
Matching Proof Moves
4. D
7. 1
6. G
7. A
8. C
5. A
Proof Multiple Choice Answer Key:
4. 3
5. 2
6. 2
12. 4
13. 2
14. 4
8. 2
6. E
7. 1
15. 3
8. 4
16. 2
Proof Answer Key
Why is the circled statement wrong? What is the correct statement?
̅̅̅̅ bisects 𝑀𝑃
̅̅̅̅̅. The correct statement is: ̅̅̅̅̅
̅̅̅̅
The circled statement is incorrect because 𝐿𝑂
𝑀𝑁 ≅ 𝑁𝑃
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅
Given: 𝐴𝐷 bisects 𝐵𝐶 at E, 𝐴𝐵 ⏊ 𝐵𝐶 , 𝐷𝐶 ⏊ 𝐵𝐶
̅̅̅̅ ≅ 𝐷𝐶
̅̅̅̅
Prove: 𝐴𝐵
Statement
Reason
̅̅̅̅ bisects 𝐵𝐶
̅̅̅̅ at E, 𝐴𝐵
̅̅̅̅ ⏊ 𝐵𝐶
̅̅̅̅ , 𝐷𝐶
̅̅̅̅ ⏊ 𝐵𝐶
̅̅̅̅
Given
𝐴𝐷
̅̅̅̅̅̅̅̅̅̅̅̅
̅̅̅̅
Definition of bisector
̅̅̅̅
𝐵𝐸 ≅ 𝐸𝐶
∠ABE and ∠ECD are right angles
Definition of perpendicular lines
∠ABE ≅∠ECD
All right angles are congruent
∠AEB ≅ ∠CED
Vertical angles are congruent
ASA Congruence
∆𝐴𝐸𝐵 ≅ ∆𝐷𝐸𝐶
̅̅̅̅ ≅ 𝐷𝐶
̅̅̅̅
CPCTC
𝐴𝐵
Given: ̅̅̅̅
𝐴𝐵 || ̅̅̅̅
𝐷𝐸
̅̅̅̅
𝑫𝑪
Prove: ̅̅̅̅
𝑩𝑪
̅̅̅̅
𝑫𝑬
= ̅̅̅̅
𝑩𝑨
Statement
̅̅̅̅ || 𝐷𝐸
̅̅̅̅
𝐴𝐵
∠DCE ≅ ∠ACB
∠EDC ≅ ∠CBA
∆𝐷𝐸𝐶~∆𝐴𝐵𝐶
̅̅̅̅
̅̅̅̅
𝐷𝐶
𝐷𝐸
=
̅̅̅̅
̅̅̅̅
𝐵𝐶
𝐵𝐴
Given: ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐵𝐶 , ̅̅̅̅
𝐵𝐷 bisects line ̅̅̅̅
𝐴𝐶
Prove: ∆𝐴𝐵𝐷 ≅ ∆𝐶𝐵𝐷
Reason
Given
Vertical angles are congruent
Alternate interior angles are congruent
AA Similarity
Corresponding parts of similar triangles are proportional
Statement
̅̅̅̅
̅̅̅̅
𝐴𝐵 ≅ 𝐵𝐶 , ̅̅̅̅
𝐵𝐷 bisects line ̅̅̅̅
𝐴𝐶
̅̅̅̅ ≅ 𝐷𝐶
̅̅̅̅
𝐴𝐷
̅̅̅̅
𝐵𝐷 ≅ ̅̅̅̅
𝐵𝐷
∆𝐴𝐵𝐷 ≅ ∆𝐶𝐵𝐷
Reason
Given
Definition of line bisector
Reflexive property
SSS Congruence
Name: ________________________
May, 2015
Proofs and Logic IP
Castoro / Cheung
Exit Ticket
Which statement is logically equivalent to the statement "If Corey worked last summer, he buys a car"?
1) If Corey does not buy a car, he did not work
last summer.
2) If Corey buys a car, he worked last summer.
3) If Corey did not work last summer, he does
not buy a car.
4) If Corey buys a car, he did not work last
summer.
In the accompanying diagram,
bisects and
.
What is the most direct method of proof that could be used to prove
1)
2)
3)
4)
Given:
Prove:
and
intersect at B,
, and
bisects
.
?