Honors Geometry
Name:____________________
BIG TIME REVIEW
Date:____________ Pd:____
Midterm Postulate, Theorem, Corollary, Definition, Property Review
Ruler Postulate:
1. The points on a line can be paired with the real numbers in such a way that any two points can have
coordinates 0 and 1.
2. Once a coordinate system has been chosen in this way, the distance between any two points equals the
absolute value of the difference of their coordinates.
Segment Addition Postulate: If B is between A and C, then AB+BC=AC
A
B
C
Definition of a Midpoint: The point on a segment that divides the segment into two congruent segments.
Definition of Complementary Angles: Two angles whose sum is 90˚
Definition of Supplementary Angles: Two angles whose sum is 180˚
⃡ in a given plane, chose any O between A and B. Consider 𝑂𝐴 and 𝑂𝐵 and all
Protractor Postulate: On 𝐴𝐵
𝑟𝑎𝑦𝑠 that can be drawn from O on one side of ⃡𝐴𝐵 . These rays can be paired with real numbers from 0 to 180 in
such a way that:
a. 𝑂𝐴 is paired with O, and 𝑂𝐵 with 180.
b. If 𝑂𝑃 is paired with x and 𝑂𝑄 with y, then m∠POQ=|x-y|
See page 18 in your text book for a diagram.
Angle Addition Postulate: If point B lies on the interior of ∠AOC,
then m∠AOB + m∠BOC = m∠AOC
C
O
25˚
35˚
B
If ∠AOC is a straight angle and B is any point not on ̅̅̅̅
𝐴𝐶 , then m∠AOB + m∠BOC = 180
A
Definition of an Angle Bisector: Ray that intersects the vertex of an angle and splits it into two congruent
angles.
Postulate #5:



A line contains at least 2 points.
A plane contains at least 3 points not all in one line.
Space contains at least 4 points not all in one plane.
Postulate #6: Through any two (2) points there is exactly one line.
Postulate #7:


Through any three points there is at least one (1) plane.
Through any three (3) non-collinear points there is exactly one (1) plane.
Postulate #8: If two points are in a plane, the line that contains those points is in the plane.
Postulate #9: If two planes intersect, then their intersection is a line.
Theorem #1: If two (2) lines intersect, then they intersect in exactly one point. C
A
Theorem #2: Through a line and a point not in the line there is exactly one plane.
E
B
D
B
A
C
Theorem #3: If two (2) lines intersect, then exactly one plane contains the lines.
B
E
C
D
A
Addition Property of Equality: If a=b and c=d, then a+c=b+d
Subtraction Property of Equality: If a=b and c=d, then a-c=b-d
Multiplication Property of Equality: If a=b, then ca=cb
Division Property of Equality: If a=b and c≠0, then (a/c)=(b/c)
Substitution Property of Equality: If a=b, then either a or b may be substituted for the other in any equation
or inequality.
Distributive Property: a(b+c)=ab+ac
Reflexive:

Property of Equality(=): a=a; m∠ABC=m∠ABC; MD=MD

Property of Congruence(≅): ∠ABC≅∠ABC; MD≅MD
Symmetric:

Property of Equality(=): ab=ba; if m∠HBG=m∠ALC, then m∠ALC=m∠HBG; if HG=AC, then AC=
HG

Property of Congruence(≅): if ∠HBG≅∠ALC, then ∠ALC≅∠HBG; if HG≅AC, then AC≅HG
Transitive:

Property of Equality(=): if m∠PHL=m∠LAX and m∠LAX=m∠EWR, then m∠PHL=m∠EWR

Property of Congruence(≅): if ∠PHL≅∠LAX and ∠LAX≅∠EWR, then ∠PHL≅∠EWR
Definition of Congruence(≅): if m∠1=m∠2, then ∠1≅∠2; if ∠1≅∠2, then m∠1=m∠2
̅̅̅̅, then AM=½AB and MB=½AB
Midpoint Theorem: If M is the midpoint of 𝐴𝐵
Given: M is the midpoint of ̅̅̅̅
𝐴𝐵
A
M
Prove: AM=½AB; MB=½AB
Statements:
̅̅̅̅
1. 𝑀 𝑖𝑠 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐴𝐵
2.
3.
4.
5.
6.
7.
𝐴𝑀 ≅ 𝑀𝐵
𝐴𝑀 = 𝑀𝐵
𝐴𝑀 + 𝑀𝐵 = 𝐴𝐵
𝐴𝑀 + 𝐴𝑀 = 𝐴𝐵; 𝑀𝐵 + 𝑀𝐵 = 𝐴𝐵
2𝐴𝑀 = 𝐴𝐵; 2𝑀𝐵 = 𝐴𝐵
𝐴𝑀 = ½𝐴𝐵; 𝑀𝐵 = ½𝐴𝐵
Reason:
1. Given
2. Definition of a Midpoint
3. Definition of Congruence
4. Segment Addition Postulate
5. Substitution Property of Equality
6. Addition Property of Equality
7. Division Property of Equality
B
Definition of an Angle Bisector: ray that intersects a vertex and splits two angles into congruent angles
A
X
B
C
Angle Bisector Theorem: If 𝐵𝑋 is the bisector of ∠ABC, then m∠ABX=½m∠ABC and
m∠XBC=½m∠ABC
Given: 𝐵𝑋 is the bisector of ∠ABC
Prove: m∠ABX=½m∠ABC; m∠XBC=½m∠ABC
Statements:
1.
2.
3.
4.
5.
6.
7.
𝐵𝑋 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠ 𝐴𝐵𝐶
∠𝐴𝐵𝑋 ≅ ∠𝑋𝐵𝐶
𝑚∠𝐴𝐵𝑋 = 𝑚∠𝑋𝐵𝐶
𝑚∠𝐴𝐵𝑋 + 𝑚∠𝑋𝐵𝐶 = 𝑚∠𝐴𝐵𝐶
𝑚∠𝐴𝐵𝑋 + ∠𝐴𝐵𝑋 = 𝑚∠𝐴𝐵𝐶
2(𝑚∠𝐴𝐵𝑋) = 𝑚∠𝐴𝐵𝐶
𝑚∠𝐴𝐵𝑋 = ½𝑚∠𝐴𝐵𝐶
Reason:
1. Given
2. Definition of an Angle Bisector
3. Definition of Congruence
4. Angle Addition Postulate
5. Substitution Property of Equality
6. Addition Property of Equality
7. Division Property of Equality
Definition of Complementary Angles: two angles whose sum is 90˚
Definition of Supplementary Angles: two angles whose sum is 180˚
Definition of Vertical Angles: Two angles whose sides form opposite rays
Vertical Angle Theorem: Vertical angles are congruent
3
Given: ∠1 and ∠2 are vertical angles
1
2
Prove: ∠1≅∠2
Statements:
Reason:
1. ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑎𝑛𝑔𝑙𝑒𝑠
2. 𝑚∠ 1 + 𝑚∠ 3 = 180;
𝑚∠ 2 + 𝑚∠ 4 = 180
3. 𝑚∠1 + 𝑚∠3 = 𝑚∠2 + 𝑚∠3
4. 𝑚∠3 = 𝑚∠3
5. 𝑚∠1 = 𝑚∠3
6. ∠1 ≅ ∠2
1. Given
2. Def. Straight Angle
3. Substitution Property of Equality
4. Reflexive Property of Equality
5. Subtraction Property of Equality
6. Def. of Congruence
Definition of a Right Angle: An angle whose measure is 90˚

All right angles are congruent
Definition of Perpendicular Lines: Two lines that intersect to form right angles
Theorem 2-4: If two lines are perpendicular, then they form congruent adjacent angles.
Given: l and m are perpendicular
l
Prove: ∠1≅∠2≅∠3≅∠4
Statements:
1. 𝑙 𝑎𝑛𝑑 𝑚 𝑎𝑟𝑒 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟
2.
3.
4.
5.
𝑚∠ 1, 𝑚∠ 2, 𝑚∠ 3, 𝑚∠ 4 𝑎𝑟𝑒 𝑟𝑡 𝑎𝑛𝑔𝑙𝑒𝑠
𝑚∠ 1 = 90; 𝑚∠ 2 = 90; 𝑚∠ = 90; 𝑚∠ 4 = 90
𝑚∠ 1 = 𝑚∠ 2 = 𝑚∠ 3 = 𝑚∠4
∠1 ≅∠2 ≅∠3≅∠4
Reason
:
1. Given
2. Def. ⊥ lines
3. Def. right angle
4. Substitution Propery of Equality
5. Definition of Congruence
2
1
3
4
m
Theorem 2-5: If two lines form congruent adjacent angles, then the lines are perpendicular
l
Given: ∠1≅∠2
2
Prove: l ⊥ m
m
Reason:
Statements:
1.
2.
3.
4.
5.
6.
7.
8.
1
∠1 ≅ ∠2
𝑚∠ 1 = 𝑚∠ 2
𝑚∠1 + 𝑚∠ 2 = 180
𝑚∠ 1 + 𝑚∠ 1 = 180
2𝑚∠ 1 = 180
𝑚∠ 1 = 90
𝑚∠ 1 𝑎𝑛𝑑 𝑚∠ 2 𝑎𝑟𝑒 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒𝑠
𝑙⊥𝑚
1. Given
2. Def. of Congruence
3. Angle Addition Postulate
4. Substitution Prop. of Equality
5. Addition Prop. of Equality
6. Division Prop. of Equality
7. Definition of a Right Angle
8. Def. of Perpendicular Lines
Theorem 2-6: If the exterior sides of two adjacent angles are perpendicular, then the angles are complementary
A
̅̅̅̅
Given: ̅̅̅̅
𝑂𝐴⊥𝑂𝐶
B
Prove: Angles 1 and 2 are complementary
Statements:
1.
2.
3.
4.
5.
6.
̅̅̅̅ ⊥ 𝑂𝐶
̅̅̅̅
𝑂𝐴
∠𝐴𝑂𝐶 𝑖𝑠 𝑟𝑡 ∠
𝑚∠𝐴𝑂𝐶 = 90
𝑚∠ 1 + 𝑚∠ 2 = 𝑚∠𝐴𝑂𝐶
𝑚∠ 1 + 𝑚∠ 2 = 90
∠ 1 + ∠ 2 𝑎𝑟𝑒 𝑐𝑜𝑚𝑝. ∠
1
Reason:
O
1. Given
2. Def. Perpendicular Lines
3. Definition of a Right Angle
4. Angle Addition Postulate
5. Substitution Property of Equality
6. Definition of Complementary Angles
2
C
Theorem 2-7: If two angles are supplements of congruent angles (or the same angle), then the two angles are
congruent.
Given: ∠1 and ∠2 are supp
∠3 and ∠4 are supp
∠2≅∠4
1
2
Prove: ∠1≅∠3
3
Statements:
4
Reason:
1. ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝;
∠3 𝑎𝑛𝑑 ∠4 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝; ∠2 ≅ ∠4
2. 𝑚∠1 + 𝑚∠2 = 180; 𝑚∠3 + 𝑚∠4
3. 𝑚∠1 + 𝑚∠2 = 𝑚∠3 + 𝑚∠4
4. 𝑚∠2 = 𝑚∠4
5. 𝑚∠1 = 𝑚∠3
6. ∠1 ≅ ∠3
1. Given
2. Def. Supplementary Angles
3. Substitution Property of Equality
4. Def. of Congruence
5. Subtraction Property of Equality
6. Def. of Congruence
Theorem 2-8: If two angles are complementary of congruent angles (or of the same angle), then the two angles
are congruent.
Given: ∠1 and ∠2 are comp
∠3 and ∠4 are comp
∠2≅∠4
Prove: ∠1≅∠3
Statements:
:
1. ∠1 𝑎𝑛𝑑 ∠2 𝑎𝑟𝑒 𝑐𝑜𝑚𝑝;
∠3 𝑎𝑛𝑑 ∠4 𝑎𝑟𝑒 𝑐𝑜𝑚𝑝; ∠2 ≅ ∠4
2. 𝑚∠1 + 𝑚∠2 = 90;
m∠3 + m∠4 = 90
3. 𝑚∠1 + 𝑚∠2 = m∠3 + m∠4
4. 𝑚∠2 = 𝑚∠4
5. 𝑚∠1 = m∠3
6. ∠1 ≅ ∠3
1
2
3
4
Reason:
1. Given
2. Def. Comp Angles
3. Substitution Property of Equality
4. Def. of Congruence
5. Substitution Property of Equality
6. Def. of Congruence
Definition of Parallel Lines: Are coplanar lines that do not intersect
Definition of Skew Lines: Are non-coplanar lines. Therefore, they are neither parallel nor intersecting.
Definition of Parallel Planes: Planes that do not intersect
Definition of Parallel Line and Plane: They do not intersect.
Definition of a Transversal: A line that intersects two or more coplanar lines at different points.
Definition of Alternate Interior Angles: Two non-adjacent interior angles on opposite sides of the transversal.
Definition of Same-Side Interior Angles: Two interior angles that are on the same side of the transversal.
Definition of Corresponding Angles: Two angles in corresponding positions relative to two lines
Postulate #10: If two parallel lines are cut by a transversal, then corresponding angles are congruent.
Theorem 3-2: If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
t
Given: k║n; transversal t cuts k and n
k
Prove: ∠1 ≅ ∠2
1
Statements:
1.
2.
3.
4.
k║n
∠1 ≅ ∠3
∠3 ≅ ∠2
∠1 ≅ ∠2
3
Reason:
n
2
1. Given
2.Vertical angles are congruent
3. 2║cbt → corres ∠ ≅
4. Transitive Property(2,3)
Theorem 3-3: If two parallel lines are cut by a transversal, then the same-side interior angles are
supplementary.
t
Given: k║n; transversal t cuts k and n
k
Prove: ∠1 is supplementary to ∠4
Statements:
1.
2.
3.
4.
5.
6.
k║n
∠1 ≅ ∠2
m∠2 + m∠4 = 180
m∠1 = m∠2
m∠1+m∠4 = 180
∠1 is supp. to ∠4
1
Reason:
n
4
1. Given
2. 2║cbt → corres ∠ ≅
3. Angle Addition Postulate
4. Definition of Congruence(2)
5. Substitution Property of Equality(3,4)
6. Definition of Supplementary Angles(5)
2
Theorem 3-4: If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other
one also
t
Given: Transversal t cuts l and n; t⊥l; l║n
Statement:
1.
2.
3.
4.
5.
6.
7.
8.
9.
n
Reason:
t⊥l
∠1 is a right angle
m∠1 = 90
l║ n
∠2 ≅∠1
m∠2 = m∠1
m∠2 = 90
∠2 is a Right Angle
t⊥n
1
l
Prove: t⊥n
2
1. Given
2. Definition of Perpendicular Lines
3. Definition of a Right Angle(2)
4. Given
5. 2║cbt → corres ∠ ≅
6. Definition of Congruence(1)
7. Substitution Property of Equality(3,6)
8. Definition of a Right Angle(7)
9. Definition of Perpendicular Lines(8)
Postulate #11: If two lines are cut by a transversal and corresponding angles are congruent, then the lines are
parallel.
Theorem 3-5: If two are cut by a transversal and alternate interior angles are congruent, then the lines are
parallel.
t
3
Given: ∠ 2 ≅ ∠ 1
Prove: 𝑘 ∥ 𝑛
Statement:
1.
2.
3.
4.
∠2≅∠1
∠3 ≅∠2
∠3 ≅∠1
𝑘∥𝑛
k
Reason:
2
n
1. Given
2. Vertical Angle Theorem
3. Transitive Property of Congruence
4. 2 𝑐𝑏𝑡 𝑎𝑛𝑑 𝑐𝑜𝑟𝑟𝑒𝑠. ∠ 𝑎𝑟𝑒 ≅→ 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒𝑠 𝑎𝑟𝑒 ∥
1
Theorem 3-6: If two lines are cut by a transversal and the same-side interior angles are supplementary, then the
lines are parallel.
t
Given: ∠ 1𝑎𝑛𝑑 ∠ 2 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦
Prove: 𝑘 ∥ 𝑛
k
1
2
n
Statement:
1.
2.
3.
4.
5.
∠ 1𝑎𝑛𝑑 ∠ 2 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦
𝑚∠2 + 𝑚∠ 3 = 180
∠ 2 𝑎𝑛𝑑 ∠ 3 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝
∠1 ≅∠3
𝑘∥𝑛
3
Reason:
1. Given
2. Angle Addition Postulate
3. Def. of Supplementary Angles
4. 2∠ 𝑠𝑢𝑝𝑝 𝑠𝑎𝑚𝑒 → ∠ 𝑎𝑟𝑒 ≅
5. 2 𝑙𝑖𝑛𝑒𝑠 𝑐𝑏𝑡, 𝑎𝑛𝑑 𝑎𝑙𝑡 𝑖𝑛𝑡 ∠ 𝑎𝑟𝑒 ≅→ 𝑙𝑖𝑛𝑒𝑠 𝑎𝑟𝑒 ∥
Theorem 3-7: In a plane two lines perpendicular to the same line are parallel
k
n
t
1
2
m
Theorem 3-8: Through a point outside a line, there is exactly one line parallel to the given line.
D
Theorem 3-7: Through a point outside a line, there is exactly one line perpendicular to the given line
P
l
n
l
k
Theorem 3-10: Two lines parallel to a third line are parallel to each other.
Given: 𝑘 ∥ 𝑙; 𝑙 ∥ 𝑛
Prove: 𝑘 ∥ 𝑛
1
Statement:
1.
2.
3.
4.
𝑘 ∥ 𝑙; 𝑘 ∥ 𝑛
∠ 1 ≅ ∠ 2; ∠ 3 ≅ ∠ 1
∠3 ≅∠2
𝑘∥𝑛
l
2
n
3
Reason:
1. Given
2. 2 ∥ 𝑙𝑖𝑛𝑒𝑠 𝑐𝑏𝑡 → 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑎𝑛𝑔𝑙𝑒𝑠 𝑎𝑟𝑒 𝑐𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡
3. Transitive Property of Congruence
4. 2 ∥ 𝑙𝑖𝑛𝑒𝑠 𝑐𝑏𝑡 𝑎𝑛𝑑 𝑐𝑜𝑟𝑟𝑒𝑠. ∠ 𝑎𝑟𝑒 ≅ → 𝑙𝑖𝑛𝑒𝑠 𝑎𝑟𝑒 ∥
Definition of a Triangle: Figure formed by three segments joining three non-collinear points.
B
Theorem 3-11: The sum of the measures of the angles of a triangle is 180˚ D
4
2
5
Given: ∆𝐴𝐵𝐶
Prove: 𝑚∠ 1 + 𝑚∠ 2 + 𝑚∠ 3 = 180˚
1
Statements:
1. ∆𝐴𝐵𝐶
2. 𝑚∠ 4 + 𝑚∠ 2 + 𝑚∠ 5 = 180
̅̅̅̅
̅̅̅̅ ∥ 𝐴𝐶
3. 𝐷𝐵
4. 𝑚∠ 1 + (𝑚∠ 2 + 𝑚∠ 5) = 180
𝑚∠ 3 + (𝑚∠ 4 + 𝑚∠ 2) = 180
5. ∠ 4 ≅ ∠ 1; ∠ 5 ≅ ∠ 3
6. 𝑚∠ 4 = 𝑚∠ 1; 𝑚∠5 = 𝑚∠ 3
7. 𝑚∠ 1 + 𝑚∠ 2 + 𝑚∠ 3 = 180
3
Reason:
1. Given
2. Angle Addition Postulate
3. Through any point not on a line there is one line ∥ to the given line
4. 2║cbt → ssi ∠ sup.
5. 2║cbt → alt. int. ∠ ≅
6. Definition of Congruence
7. Substitution Property of Equality
Definition of a Corollary: A statement that can be proved easily by applying a theorem.
Corollary 3-1: If two angles of one triangle are congruent to two angles of another triangle then the third angles
are congruent.
Corollary 3-2: Each angle of an equilateral triangle has measure 60˚
Corollary 3-3: In a triangle, there can be at most one right angle or one obtuse angle
Corollary 3-4: The acute angles of right triangles are complementary
Theorem 3-12: The measure of an exterior angle of a triangle equals the sum of the measures of the remote
interior angles
Remote Interior
Angles
Definition of a Polygon: A closed figure formed by coplanar segments
1. The sides that have a common endpoint are non-collinear
2. Each side intersects exactly two other sides, but only at their endpoints
Definition of a Convex Polygon: A polygon such that no line containing a side of the polygon contains a point
in the interior of a polygon.
 A polygon that is not convex is nonconvex or concave.
Definition of a Diagonal: A segment joining two nonconsecutive vertices.
Theorem 3-13: The sum of the measures of the angles of a convex polygon with n sides is (n-2)180.
Theorem 3-14: The sum of the measures of the exterior angles of any convex polygon, one angle at each
vertex, is 360˚
Definition of a Regular Polygon: A polygon that is both equiangular and equilateral.
Definition of Deductive Reasoning: A conclusion based on accepted statements (definitions, corollaries,
theorems) based on facts.
Definition of Inductive Reasoning: Conclusions based on several past observations.
Definition of CPCTC: Corresponding Parts of Congruent Triangles are Congruent
Definition of Congruent Polygons: Two polygons are congruent iff (if and only if) their vertices match up so
that corresponding parts are congruent
Side Side Side Postulate (SSS): If three sides of one triangle are congruent to three sides of another triangle,
then the triangles are congruent.
Side Angle Side Postulate (SAS): If two sides and the included angle of one triangle are congruent to the two
sides and the included angle of another triangle, then the triangles are congruent.
Angle Side Angle Postulate (ASA): If two angles and the included side of one triangle are congruent to two
angles and the included side of another triangle, then the triangles are congruent.
Definition of a Line Perpendicular to a Plane: A line and a plane are perpendicular iff they intersect and the
line is perpendicular to all lines in the plane that pass through the point of intersection.
Definition of an Isosceles Triangle: When two sides of a triangle are congruent to each other.
Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite those sides are
congruent.
Corollary 4-1: An equilateral triangle is also equiangular
Corollary 4-2: An equilateral triangle has three 60 degree angles
Corollary 4-3: The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its
midpoint.
Converse of Isosceles Triangle Theorem: If two angles of a triangle are congruent, then the sides opposite
those angles are congruent.
A
Given: ∠ 𝐵 ≅ ∠ 𝐶
̅̅̅̅ ≅ 𝐴𝐶
̅̅̅̅
Prove: 𝐴𝐵
Statement:
Reason:
B
C
D
1.
2.
3.
4.
5.
6.
∠𝐵 ≅∠𝐶
̅̅̅̅ 𝑎𝑡 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓𝐵𝐶
̅̅̅̅
𝐷𝑟𝑎𝑤 𝐴𝐷
̅̅̅̅
̅̅̅̅ ≅ 𝐷𝐶
𝐵𝐷
∠𝐴𝐷𝐵 ≅ ∠𝐴𝐷𝐶
∆𝐴𝐷𝐵 ≅ ∆𝐴𝐷𝐶
̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐴𝐶
1. Given
2. Bisector of vertex of isosc is perpendicular to m.p.
3. Definition of a Midpoint
4. Definition of Perpendicular Lines
5. ASA Postulate
6. CPCTC
Corollary 4-4: An equiangular triangle is also equilateral
AAS Theorem: If two angles and a non-included side of one triangle are congruent to the corresponding parts
of another triangle, then the triangles are congruent,
Definition of a Hypotenuse: The side opposite the right angle in a right triangle
HL Theorem: If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of
another right triangle, then the triangle are congruent.
Definition of a Median (of a triangle): A segment from a vertex to the midpoint of the opposite side.
 3 medians intersect inside the triangle
Definition of an Altitude (of a triangle): The perpendicular segment from a vertex to the line that contains the
opposite side.
Definition of an Acute Triangle: A triangle with at least one angle less than 90˚

Altitudes intersect inside triangle
Definition of a Right Triangle: A triangle with exactly one 90˚

Altitude intersects on the vertex of the right angle
Definition of an Obtuse Triangle: A triangle with exactly one angle more than 90˚
 Altitude intersects outside of triangle
Definition of a Perpendicular Bisector: A segment is a line (or ray or segment) that is perpendicular to the
segment at its midpoint.
 Doesn’t have to go through vertex
Theorem 4-5: If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the
endpoints of the segment.
B
H
G
A
Theorem 4-6: If a point is equidistant from the endpoint of a segment, then the point lies on the perpendicular
bisector of the segment
B
A
C
X
The Distance from a Point to a Line (or Plane): is defined to be the length of the perpendicular segment from
the point to the line (or plane).
B
l
A
Theorem 4-7: If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle.
̅̅̅̅
Given: 𝐵𝑃 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠ABC; ̅̅̅̅
𝑃𝐴 ⊥ ̅̅̅̅
𝐵𝐴; ̅̅̅̅
𝑃𝐶 ⊥ 𝐵𝐶
Prove: 𝐴𝑃 = 𝑃𝐶
Statement:
A
P
Reason:
B
1.
2.
3.
4.
5.
6.
7.
8.
̅̅̅̅ ⊥ 𝐵𝐴
̅̅̅̅; 𝑃𝐶
̅̅̅̅ ⊥ 𝐵𝐶
̅̅̅̅
𝐵𝑃 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠ABC; 𝑃𝐴
∠ABP ≅ ∠CBP
̅̅̅̅
𝐵𝑃 ≅ ̅̅̅̅
𝐵𝑃
∠BAP and ∠BCP are right angles
∠BAP ≅ ∠BCP
∆𝐴𝐵𝑃 ≅ ∆𝐶𝐵𝑃
̅̅̅̅
𝐴𝑃 ≅ ̅̅̅̅
𝑃𝐶
𝐴𝑃 = 𝑃𝐶
1. Given
2. Def. Angle Bisector
3. Reflexive Property of Congruence
4. Def. of Perpendicular Lines
5. All Right Angles are Congruent
6. AAS Theorem
7. CPCTC
8. Def. of Congruence
C
Theorem 4-8: If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle
A
̅̅̅̅ ⊥ 𝐵𝐴
̅̅̅̅; 𝑃𝐶
̅̅̅̅ ⊥ 𝐵𝐶
̅̅̅̅
Given: 𝐴𝑃 = 𝑃𝐶; 𝑃𝐴
Prove: 𝐵𝑃 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠ABC;
Statement:
P
Reason:
B
̅̅̅̅
𝐴𝑃 = 𝑃𝐶; ̅̅̅̅
𝑃𝐴 ⊥ ̅̅̅̅
𝐵𝐴; ̅̅̅̅
𝑃𝐶 ⊥ 𝐵𝐶
̅̅̅̅
𝐴𝑃 ≅ ̅̅̅̅
𝑃𝐶
𝑚∠BAP = 90; m∠BCP = 90
∆𝐵𝐴𝑃 𝑎𝑛𝑑 ∆𝐵𝐶𝑃 𝑎𝑟𝑒 𝑟𝑖𝑔ℎ𝑡 ∆′𝑠
̅̅̅̅ ≅ 𝐵𝑃
̅̅̅̅
𝐵𝑃
1.
2.
3.
4.
5.
6. ∆𝐴𝐵𝑃 ≅ ∆𝐶𝐵𝑃
7. ∠ABP ≅ ∠CBP
1. Given
2. Def. of Congruence
3. Def. of Perpendicular Lines
4. Def. of a Right Triangle
5. Reflexive Property of Congruence
6. HL Theorem
7. CPCTC
8. 𝐵𝑃 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠ABC
8. Def. of an Angle Bisector
C
Definition of a Parallelogram: A quadrilateral with both pairs of opposite sides parallel
Theorem 5-1: Opposite sides of a parallelogram are congruent
Theorem 5-2: Opposite angles of a parallelogram are congruent
Theorem 5-3: Diagonals of a parallelogram bisect each other
Theorem 5-4: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a
parallelogram
Theorem 5-5: If one pair of opposite sides of a quadrilateral are both parallel and congruent then the
quadrilateral is a parallelogram
Theorem 5-6: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a
parallelogram
Theorem 5-7: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram
**By definition, if both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a
parallelogram
Theorem 5-8: If two lines are parallel, then all the points on one line are equidistant from the other line
l
m
A
B
C
D
Theorem 5-9: If three parallel lines cut off congruent segments on one transversal, then they cut off congruent
segments of every transversal
X
A
Y
B
Z
C
Theorem 5-10: A line that contains the midpoint of one side of a triangle and is parallel to another side passes
through the midpoint of third side
C
N
M
A
B
Theorem 5-11: The segment that joins the midpoints of two sides of a triangle
(1) Is the parallel to the third side;
(2) Is half as long as the third side.
A
M
B
N
C
Definition of a Rectangle: A quadrilateral with 2 pairs of opposite sides congruent and parallel
 Theorem 5-12: The diagonals of a rectangle are congruent.
 Theorem 5-16: If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle.
Definition of a Rhombus: A quadrilateral with four congruent sides.
 Theorem 5-13: The diagonals of a rhombus are perpendicular.
 Theorem 5-14: Each diagonal of a rhombus bisects two angles of the rhombus.
Definition of a Square: A qudrilateral with four right angles and four congruent sides
 Theorem 5-17: If two consecutive sides of a parallelogram are congruent, then the parallelogram is a
rhombus.
 Theorem 5-15: The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices.
Definition of a Trapezoid: A quadrilateral with exactly one pair of parallel sides.
Definition of Isosceles Trapezoid: A trapezoid with congruent legs.
 Theorem 5-18: Base angles of an isosceles trapezoid are congruent.
Definition of a Median (of a Trapezoid): The segment that joins the midpoints of the leg.
 Theorem 5-19: The median of a trapezoid:
(1) Is parallel to the base;
(2) has length equal to the average of the base lengths
Addition Property of Inequality: 𝐼𝑓 𝑎 > 𝑏 𝑎𝑛𝑑 𝑐 ≥ 𝑑, 𝑡ℎ𝑒𝑛 𝑎 + 𝑐 > 𝑏 + 𝑑
Multiplication/Division Property of Inequality: 𝐼𝑓 𝑎 > 𝑏 𝑎𝑛𝑑 𝑐 > 0, 𝑡ℎ𝑒𝑛 𝑎𝑐 > 𝑏𝑐 𝑎𝑛𝑑 𝑎⁄𝑐 > 𝑏⁄𝑐
Multiplication/Division Property of Inequality: 𝐼𝑓 𝑎 > 𝑏 𝑎𝑛𝑑 𝑐 > 0, 𝑡ℎ𝑒𝑛 𝑎𝑐 < 𝑏𝑐 𝑎𝑛𝑑 𝑎⁄𝑐 < 𝑏⁄𝑐
Transitive Property of Inequality: 𝐼𝑓 𝑎 > 𝑏 𝑎𝑛𝑑 𝑏 > 𝑐, 𝑡ℎ𝑒𝑛 𝑎 > 𝑐
Definition of Inequality: 𝐼𝑓 𝑎 = 𝑏 + 𝑐 𝑎𝑛𝑑 𝑐 > 0, 𝑡ℎ𝑒𝑛 𝑎 > 𝑏
Exterior Angle Inequality Theorem: The measure of an exterior angle of a triangle is greater than the measure
of either remote interior angle.
Definition of Logic: Classification of arguments: Study of methods and principles that allow you to classify
arguments
Definition of Statements: Statements that are either true or false
Definition of Compound Statements: Joining of two or more statements
 Conjunction: 𝑝 𝛬(𝑎𝑛𝑑) 𝑞
 Disjunction: 𝑝 v(or) 𝑞
Definition of a Negation: “NOT” take the opposite of the given statement (not false, negative)
Definition of Truth Tables: Tell the conditions under which a conjunction or disjunction is true or false
Definition of Conditionals: This can be read as “if p, then q” or “p implies q” or “p follows from q”
Related If-Then Statements




Given Statement: If p, then q
Contrapositive: If not q, then not p
Converse: If q, then p
Inverse: If not p, then not q
Definition of a Valid Argument: Conditional whose truth table contains a tautology (all true)
Definition of Logically Equivalent: Two statements that are either both true or both false.

The conditional and its contrapositive are logically equivalent
Venn Diagrams: You can use Venn diagrams to represent a conditional
q
p
Steps for Writing an Indirect Proof
1. Assume temporarily that the conclusion is not true
2. Reason logically until you reach a contradiction of a known fact.
3. Point out the temporarily assumption must be false, and that the conclusion must then be true.
Theorem 6-2: If one side of a triangle is longer than a second side, then the angle opposite the first side is
larger than the angle opposite the second side.
Theorem 6-3: If one angle of a triangle is larger than a second angle, then the side opposite the first angle is
longer than the side opposite the second angle.
Corollary 6-1: The perpendicular segment from a point to a line is the shortest segment from the point to the
line.
Corollary 6-2: The perpendicular segment from a point to a plane is the shortest segment from the point to the
plane.
Theorem 6-4 – The Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is
greater than the length of the third side.
Theorem 6-5 – SAS Inequality Theorem: If two sides of one triangle are congruent to two sides of another
triangle, but the included angle of the first triangle is larger than the included angle of the second, then the third
side of the first triangle is longer than the third side of the second triangle.
Theorem 6-6: SSS Inequality Theorem: If two sides of one triangle are congruent to two sides of another
triangl, but the third side of the first triangle is longer than the third side of the second triangle, then the
included angle of the first triangle is larger than the included angle of the second.
Test is on January 29TH , 2013
Created By: The Hilarious Geometrically Magnificient Hillary Gendler, Medical Doctor Matthew Davis, and
The Amazingly Brilliant LeighAnn Burke