Download Geometry Theorems Chapters 1-4

Name:____________________
Geometry Theorems Chapters 1-4
KEEP THIS IN YOUR NOTES FOR EVER!!!!! IT IS SUPER USEFUL
Postulate 1: If there are two points, then there is_____________________________ that contains them.
Postulate 3: If there are three points, then there is_____________________________ that contains
them.
The Ruler Postulate:
 Points on a line can be numbered so that:_____________________________________________
 Distance between points can be calculated by:__________________________________________
Betweeness of Points Theorem: If B is between points A and C then we can write the equation:
𝐴𝐶 =_____+______
Linear Pairs Theorem: If two angles are a linear pair then they are ____________________________
Vertical Angles Theorem: If two angles are vertical angles then they are________________________
Pythagorean Theorem:__________________________________________________________________
Substitution Property of Equality: If two things are equal, then they can be____________________ for
each other.
Transitive Property of Equality:______________________________________________________
Addition Property of Equality:_______________________________________________________
Subtraction Property of Equality:_____________________________________________________
Multiplication Property of Equality:___________________________________________________
Division Property of Equality:________________________________________________________
Square Root Property of Equality:____________________________________________________
Reflexive Property of Equality:_______________________________________________________
Theorem 2.2: If two angles are congruent and supplementary, then each is a _______________________
Theorem 2.3: If two angles are both supplementary to a third angle, then they’re___________________
to each other.
Theorem 2.4: If two angles are both complementary to a third angle, then they’re ___________________
to each other
Alternate Interior Angles Theorem: Two lines are _______________ if and only if their alternate
interior angles are_____________________.
Alternate Exterior Angles Theorem: Two lines are _______________ if and only if their alternate
exterior angles are_____________________.
Corresponding Angles Theorem: Two lines are _______________ if and only if their corresponding
angles are_____________________.
Same Side Interior Angles Theorem: Two lines are _______________ if and only if their same side
interior angles are_____________________.
Same Side Exterior Angles Theorem: Two lines are _______________ if and only if their same side
exterior angles are_____________________.
Transitivity of Parallel Lines Theorem: If 𝑎 ∥ 𝑏 and 𝑏 ∥ 𝑐 then ________________
Theorem 3.8: If 𝑚 ⊥ 𝑡 and 𝑛 ⊥ 𝑡 then _________________________
Triangle Sum Theorem: The sum of the three interior angles of a triangle is always________________
Exterior Angle Theorem: An exterior angle equals the ______________ of the remote interior angles.
Corresponding Parts of Congruent Triangles are Congruent Theorem (CPCTC Thm): If two
triangles are congruent, then all their corresponding parts are___________________________
Third Angle Theorem: If two triangles have two angles that are congruent, then their third angles are also
__________________
Reflexive Property of Congruence: Any figure is always __________________ to itself.
Transitive Property of Congruence: If shape A is congruent to shape B and shape B is congruent to shape
C then________________________________________________________________________
SSS Postulate: If two triangles have all three corresponding sides congruent, then the triangles themselves
are________________________________
ASA Postulate: If two triangles have a corresponding ____________, _____________ and __________
congruent, then the triangles are congruent.
AAS Postulate: If two triangles have a corresponding ____________, _____________ and __________
congruent, then the triangles are congruent.
SAS Postulate: If two triangles have a corresponding ____________, _____________ and __________
congruent, then the triangles are congruent.
Some Useful Vocabulary that is also used in proofs
Bisect: Something is cut into two _________________ pieces
Midpoint: The point that divides a line segment into two ____________________ pieces
Right Angle: An angle that measures_________ degrees. NOTE: ALL RIGHT ANGLES ARE
CONGRUENT TO EACH OTHER!!!
Perpendicular Bisector: A line segment that cuts another line into two congruent pieces and does it at a 90
degree angle.
Name:_____________________ Date:__________________
Classwork
4-3
Recognizing Theorems and using them in Proofs
(1) Look through your list of theorems and decide which theorem justifies each conclusion below.
Given: 𝑥 + 3 = 7
Conclusion 𝑥 = 4
Given: ∠1 = 25°
Conclusion: ∠2 = 25°
Reason:________________
2
Reason:_______________
Given:
Given:
Given Line 𝐴𝐵 ∥ 𝐶𝐷 and line
𝐶𝐷 ∥ 𝐸𝐹
Conclusion: 𝐴𝐵 ∥ 𝐸𝐹
1
Reason:__________________
𝑎
40°
Conclusion: ∠𝑎 = 110°
Given: ∠1 is supplementary to
∠2 and ∠2 is supplementary to
∠3
Conclusion: 𝑚∠1 = 𝑚∠3
Reason:__________________
Reason:__________________
Given: Line 𝐴𝐵 ⊥ 𝐶𝐷 and line
𝐶𝐷 ⊥ 𝐸𝐹
Conclusion: 𝐴𝐵 ∥ 𝐸𝐹
Given: Point 𝐵 is between
points 𝐷 and 𝐽
Conclusion: 𝐷𝐵 + 𝐵𝐽 = 𝐷𝐽
Given: Ray ⃗⃗⃗⃗⃗
𝐴𝐵 bisects ∠𝐶𝐴𝑇
Conclusion: 𝑚∠𝐶𝐴𝐵 = 𝑚∠𝐵𝐴𝑇
Reason:__________________
Reason:__________________
Reason:__________________
Given: ∠𝐴𝐵𝐶 is complementary
to ∠𝐷𝐸𝐹 and ∠𝐷𝐸𝐹 is
complementary to ∠𝐺𝐻𝐼
Conclusion: 𝑚∠𝐴𝐵𝐶 = 𝑚∠𝐺𝐻𝐼
Given: ∠1 = 45°
Conclusion: ∠2 = 135°
Given: the diagram
Conclusion: The
triangles are
congruent
Reason:__________________
Reason:______________
Reason:________
Given:
𝑚∠1 = 𝑚∠2
Conclusion: 𝑎 ∥ 𝑏
Given: The triangles are
congruent
Conclusion:
2
𝑚∠3 = 𝑚∠2
3
Reason:_______
Given: the diagram
Conclusion:
∠𝑎 = 60
Conclusion: The triangles are
congruent.
Reason:_________________
𝑎
1
𝑏
2
Reason:_________________
Given: A is the midpoint of CR
C
A
Conclusion: 𝐶𝐴 = 𝐴𝑅
R
Reason:_________________
Given: 𝑎 ∥ 𝑏
Conclusion:
𝑚∠1 = 𝑚∠2
70°
1
2
𝑎
1
45°
75° 𝑎
Reason:__________________
𝑏
2
Reason:_________________
Given: The diagram
Conclusion:
𝑚∠𝑎 = 𝑚∠𝑏
33°
67° 𝑎
𝑏
67°
33°
Reason:________________
(2) Come up with your own conclusions AND the reasons.
Given Line 𝐸𝐴 ∥ 𝐺𝐷 and line
𝐺𝐷 ∥ 𝐵𝐶
Conclusion:______________
Given: 𝑎 ∥ 𝑏
Conclusion:
__________
__________
1
𝑎
Given:
𝑚∠1 = 𝑚∠2
Conclusion:
__________
𝑏
2
1
𝑎
𝑏
2
Reason:__________________
Reason:_________________
Reason:_________________
Given: diagram
Conclusion
__________
__________________
Given:
Given: Line 𝐺𝑇 ⊥ 𝐻𝐼and line
𝐵𝑌 ⊥ 𝐴𝐶
Conclusion:
Reason:___________
Reason:__________________
Reason:__________________
Given: the diagram
Conclusion:
______________
Given: ∠1 = 32°
Conclusion:_________
Given: E is the midpoint of BT
82°
𝑎
34°
Conclusion:______________
1
52°
68° 𝑎
B
E
T
Conclusion:______________
2
Reason:_______________
Reason:__________________
Reason:_________________
Given: ∠1 = 36°
Given: Ray ⃗⃗⃗⃗
𝐼𝐺 bisects ∠𝑅𝐼𝑇
Conclusion:_______________ Conclusion:__________ 1
________________________
2
Given:
Conclusion: _____________
_______________________
Reason:__________________
Reason:______________
Reason:_________________
Given: The diagram
Given: The triangles are
congruent
Given: 𝑥 2 = 49
Conclusion:
____________
𝑐
60°
80°
60° 80°
𝑑
Reason:________________
Given: ∠1 is supplementary to
∠2 and ∠2 is supplementary to
∠3
C
Conclusion_______________
Conclusion:
______________
A
Reason:_________
D
B
Given: Point 𝐵 is between
points 𝐴 and 𝑇
Conclusion:_______________
Conclusion:_______________
Reason:__________________
Reason:________________
Given: ∠1 is complementary to
∠2 and ∠2 is complementary to
∠3
Conclusion:_______________
Reason:__________________
Reason:__________________