Geometric Proofs and Angle Relationships
Puzzle 1
1
5
3
Given:
1 and
5 are supplementary.
3 and
5 are supplementary.
Prove: m
1 = m
3
Statements
Reasons
Q
P
R
S
T
U
Given
:
m
SQT =
m
TQU
Prove: m
SQT = m
RQP
Statements
Reasons
Puzzle 2
3
1
2
Given:
1 and
2 are vertical angles.
Prove: m
1 = m
2
(NOTE: This is the proof for the
Theorem – Vertical Angles are Congruent
– Therefore, you CANNOT use this
theorem as a reason within this proof.)
Statements
Reasons
Proof Puzzle 3
Statements
Reasons
1
2
3
4
Given:
1 and
2 are supplementary.
3 and
4 are supplementary.
m
2
=
m
4
Prove: m
1 = m
3
Puzzle 4
Puzzle Proof 5
Given: m
1 = m
3
m
ABC = m
JKL
Prove: m
2 = m
4
2
1
3
4
C
B
K
A
J
L
Puzzle Proof 6
Given: AB = BE
BC = DB
Prove: AC = DE
D
A
B
E
C
Statements
Reasons
Statements
Reasons
D
M
Statements & Reasons for Puzzles 1, 2, and 3
m
5 = m
5
1 and
5 are supplementary.
3 and
5 are supplementary.
m
1 + m
5 = m
3 + m
5
m
1 = m
3
m
1 + m
5 = 180
m
3 + m
5 = 180
m
1 = m
5
1 and
2 are vertical angles.
m
1 = m
2
m
1 + m
3 = 180
m
2 + m
3 = 180
m
1 + m
3 = m
2 + m
3
m
1 + m
2 = m
3
m
3 = m
3
Substitution
Vertical Angles are Congruent
Given
Reflexive
Subtraction
Definition of Supplementary Angles
m
SQT = m
TQU
m
RQP + m
PQU = 180
m
TQU = m
RQP
m
SQT + m
TQU = m
SQU
m
SQT = m
RQP
Puzzle 1
Puzzle 2
Angle Addition Postulate
Given
Vertical Angles are Congruent
Definition of Vertical Angles
Substitution
Definition of Supplementary Angles
Puzzle 3
Angle Addition Postulate
Given
Subtraction
Definition of Vertical Angles
Substitution
Reflexive
Definition of Supplementary Angles
Given
Given
Angle Addition Postulate
Addition Property
Subtraction Property
Definition of Supplementary Angles
Substitution
Substitution
Given
Given
Angle Addition Postulate
Segment Addition Postulate
Addition Property
Subtraction Property
Vertical Angles are Congruent
Substitution
m
1 = m
3
m
1 + m
2 = m
3 + m
4
1 and
2 are supplementary.
3 and
4 are supplementary.
m
1 + m
2 = 180°
m
3 + m
4 = 180°
m
2 = m
4
m
1 = m
4
Statements & Reasons for Puzzles 4, 5, and 6
Puzzle 4
Puzzle 5
Puzzle 6
Given
Given
Definition of Complementary Angles
Definition of Supplementary Angles
Angle Addition Postulate
Substitution
Subtraction
m
1 = m
3
m
2 = m
4
m
1 + m
2 = m
ABC
m
3 + m
4 = m
JKL
m
ABC = m
JKL
m
1 + m
2 = 180
m
3 + m
4 = 180
m
1 + m
2 = m
3 + m
4
AB = BE
BC = DB
AC = DE
AB + BC = AC
BE + DB = DE
AB + BC = BE + DB
m
ABE = m
DBC
various puzzles and proofs involving supplementary angles, vertical angles, congruence, and more in geometry. Understand the statements and reasons used to prove angle relationships convincingly.
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Presentation Transcript
Puzzle 1 Given: 1 and 5 are supplementary. 3 and 5 are supplementary. Prove: m 1 = m 3 1 3 5 Reasons Statements Puzzle 2 S T Given: m SQT =m TQU R Prove: m SQT = m RQP U Q P Statements Reasons
Given:1 and 2 are vertical angles. Prove: m 1 = m 2 Proof Puzzle 3 3 1 2 (NOTE: This is the proof for the Theorem Vertical Angles are Congruent Therefore, you CANNOT use this theorem as a reason within this proof.) Statements Reasons Puzzle 4 Given: 1 and 2 are supplementary. 3 and 4 are supplementary. m 2=m 4 1 2 Prove: m 1 = m 3 3 4 Statements Reasons
Puzzle Proof 5 Given: m 1 = m 3 m ABC = m JKL Prove: m 2 = m 4 D C J M 1 A 2 B 3 4 K L Statements Reasons A Puzzle Proof 6 Given: AB = BE BC = DB Prove: AC = DE E B C D Statements Reasons
Statements & Reasons for Puzzles 1, 2, and 3 m 5 = m 5 Puzzle 1 Substitution 1 and 5 are supplementary. 3 and 5 are supplementary. m 1 + m 5 = m 3 + m 5 Vertical Angles are Congruent Given m 1 = m 3 Reflexive m 1 + m 5 = 180 m 3 + m 5 = 180 Subtraction Definition of Supplementary Angles m 1 = m 5 Angle Addition Postulate Puzzle 2 m SQT = m TQU Given m RQP + m PQU = 180 Vertical Angles are Congruent m TQU = m RQP Definition of Vertical Angles m SQT + m TQU = m SQU Substitution m SQT = m RQP Definition of Supplementary Angles Puzzle 3 Angle Addition Postulate 1 and 2 are vertical angles. Given m 1 = m 2 Subtraction m 1 + m 3 = 180 m 2 + m 3 = 180 m 1 + m 3 = m 2 + m 3 Definition of Vertical Angles Substitution m 1 + m 2 = m 3 Reflexive m 3 = m 3 Definition of Supplementary Angles
Statements & Reasons for Puzzles 4, 5, and 6 m 1 = m 3 Puzzle 4 Given Given m 1 + m 2 = m 3 + m 4 Definition of Complementary Angles 1 and 2 are supplementary. 3 and 4 are supplementary. Definition of Supplementary Angles m 1 + m 2 = 180 m 3 + m 4 = 180 Angle Addition Postulate Substitution m 2 = m 4 Subtraction m 1 = m 4 Puzzle 5 m 1 = m 3 Given Given m 2 = m 4 Angle Addition Postulate m 1 + m 2 = m ABC m 3 + m 4 = m JKL Addition Property Subtraction Property m ABC = m JKL Definition of Supplementary Angles m 1 + m 2 = 180 m 3 + m 4 = 180 Substitution Substitution m 1 + m 2 = m 3 + m 4 Puzzle 6 Given Given AB = BE BC = DB Angle Addition Postulate AC = DE Segment Addition Postulate AB + BC = AC BE + DB = DE Addition Property Subtraction Property AB + BC = BE + DB Vertical Angles are Congruent m ABE = m DBC Substitution
