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Geometry - Chapter 1-4
Tehachapi High School Geometry Book
128
Mathematics
10th Grade
10/29/2012
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Term
Space
 Definition
The set of all points
Pg. 6
Term
Collinear
Points
 Definition
Points all in one line
Pg. 6
Term
Coplanar
Points
 Definition
Points all in one plane
Pg. 6
Term
Intersection
 Definition
Where two figures(lines, planes, or the combination
of both) meet or cut.
The set of points that are in both figures. Pg. 6
Term
Segment
 Definition
Shown by two letters(the endpoints) with a
line over it.
Constists of the endpoints and all points between
those endpoints. Pg. 11
Term
Ray
 Definition
Shown by two letters(one an endpoint named first and another) with an arrow going to the right over the top.
Consists of the endpoint and all points to and paste the second letter. Pg. 11
Term
Opposite Ray
 Definition
Rays that start at the same endpoint, but go
in opposite direction. Pg. 11
Term
Length
 Definition
Shown by two letters(the endpoints).
Subtract the coordinates of it endpoints. Pg. 11
Term
Postulate
Axioms
 Definition
Statements that are accepted wihout proof
Pg. 12
Term
Ruler Postulate
 Definition
1. The points on a line can be paired with the real numbers in such a way that any two points can have coordinantes 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. Pg. 12
Term
Segment Addition
Postulate
 Definition
If B is between A and C, then
AB + BC = AC
PG. 12
Term
Congruent and
Congruent Segments
 Definition
Two objects that have the same size and shape.
Two segements that have equal length.
Pg. 13
Term
Midpoint of
a Segment
 Definition
The point that divides the segment into two congruent segements.
Pg. 13
Term
Bisector of a
Segment
 Definition
A line, segment, ray, or plane that intersects the segment at its midpoint.
Pg. 13
Term
Angle
 Definition
The figure formed by two rays that have the same endpoint.
Pg. 17
Term
Sides &Vertex
of an angle
 Definition
Sides - the two rays that form the angle
Vertex - the common endpoint of the rays that form the angle
Pg. 17
Term
Acute Angle
 Definition
Measure between o and 90
Pg. 17
Term
Right Angle
 Definition
Measure of 90
Pg. 17
Term
Obtuse Angle
 Definition
Measure between 90 and 180
Pg. 17
Term
Straight Angle
 Definition
Measure of 180
Pg. 17
Term
Protractor Postulate
 Definition
On line AB in a given plane, choose any point 0 between A and B. Consider line OA and line OB and all the rays that can be drawn from o on one side of line AB. These rays can be paired with the real numbers from 0 to 180 in such a way that:
a. Line OA is paired with 0, and line OB with 180.
b. If line OB is paired with x, and line OQ with y, then m ‹ POQ = ǀx-yǀ Pg. 18
Term
Angle Addition
Postulate
 Definition
If point B lines in the interior of <AOC, then
m< AOB + m< BOC = m< AOC
If < AOC is a straight angle and B is any point not on line AC, then
m< AOC + m< BOC = 180 Pg. 18
Term
Congruent Angles
 Definition
Angles that have equal measures
Pg. 19
Term
Adjacent Angles
 Definition
Two angles in a plane that have a common vertex and a common side but no common interior points
Pg. 19
Term
Bisector of an
Angle
 Definition
The ray that divides that angle inot two congruent adjacent angles.
Pg. 19
Term
Postulate 5
# of points in a line
 Definition
A Line contains at least two points; a plane contains at least three points not all in one line; space contaoins at least four points not all in one plane.
Pg. 23
Term
Postulate 6
Forming a line
 Definition
Through any two points there is exactly one line.
Pg. 23
Term
Postulate 7
# of points in a plane
 Definition
Through any three points there is at least one plane, and through any three noncollinear points there is exactly one plane.
Pg. 23
Term
Postulate 8
Lines in a plane
 Definition
If two points are in a plane, then the line that contains the points is in that plane.
Pg. 23
Term
Postulate 9
Intersecting planes
 Definition
If two planes interest, then their intersection is a line.
Pg. 23
Term
Theorem 1-1
Intersecting lines
 Definition
If two lines intersect, then they intersect in exactly one point.
Pg. 23
Term
Theorem 1-2
Forming a plane
 Definition
Through a line and a point not in a line there is exactly one plane.
Pg. 23
Term
Theorem 1-3 Intersecting lines and planes
 Definition
If two lines intersect, then exactly one plane contains the lines.
Pg. 23
Term
If-Then Statements
Conditional Statements or conditionals
 Definition
Example:
If B is bewtween A and C, then AB + AC =AC
or
If p, then q
Pg. 33
Term
Hypothesis
 Definition
The if part of an if-then statement
Pg. 33
Term
Conclusion
 Definition
The then part of an if-then statement
Pg. 33
Term
Converse
 Definition
When the hypothesis and conclusion are interchanged
Pg. 33
Term
Counterexample
 Definition
When an example can be found where the hypothesis is true and the conclusion is false.
Pg. 33
Term
Biconditional
 Definition
When a conditional and is converse are both true they can be combined into a single statement.
A statement that contains the words "if and only if"
Pg. 34
Term
Addition
Property
 Definition
If a = b and c = d, then a + c = b +d
This can also be used to add a number to both sides.
Example: a + 2 = b + 2
Pg. 37
Term
Subtraction
Property
 Definition
If a = b and c =d, then a - c = b + d
This can also be used to subtract a number from both sides
Example: a - 2 = b - 2
Pg. 37
Term
Multiplication Property
 Definition
If a = b and c = d, then a -c = b -d
This can also be used to multiply both sides of the equation by the same number.
Example: 1/2a = 1/2b
Pg. 37
Term
Division Property
 Definition
If a = b and c ≠ 0, then a/c = b/c
Pg. 37
Term
Subsititution Property
 Definition
If a =b, then either a or b may be substituted for the other in any equation (or inequality)
Pg. 37
Term
Reflexive Property
 Definition
a =a
or
< D ≅ < D
Pg. 37
Term
Symmetric
Property
 Definition
If a = b, then b = a
or
If < D ≅ < E, then < E ≅ < D
Pg. 37
Term
Transitive
Property
 Definition
If a = b and b = c, then a =c
or
If < D ≅ < E and < E ≅ < F, the < D ≅ < F
Pg. 37
Term
Distributive
Property
 Definition
a(d+c) = ad +ac
Pg. 38
Term
Midpoint Theorem
 Definition
If M is the midpoint of seg AB, then AM = 1/2AB and MB = 1/2AB
Pg. 43
Term
Angle Bisector
Theorem
 Definition
If Ray BX is the bisector of < ABC, then
m< ABX = 1/2m< ABC and m< XBC = 1/2m< ABC
Pg. 44
Term
Reasons Used in
Proofs
 Definition
Given Information
Definitions
Postulates
Properties of Equality or Congruency
Theorems (that have already been proved)
Pg. 45
Term
Complentary
Angles
 Definition
Two angles whose measures have the sum of 90
Each angle is called a complement of the other.
Pg. 50
Term
Supplementary
Angles
 Definition
Two angles whose measures have the sum of 180.
Each angle is called asupplement of the other.
Pg. 50
Term
Vertical Angles
 Definition
Two angles that have sides consisting of opposite rays. When two lines interest, they form two pairs of vertical angles.
Pg. 51
Term
Vertical Angle Theorem
 Definition
Vertical angles are Congruent
Pg. 51
Term
Theorem 2-4
If two lines are ┴, then ___________
 Definition
If two lines are perpendicular, then they form congruent adjacent angles.
Pg. 56
Term
Theorem 2-5
If two lines form ≅ adj. <'s, then ________
 Definition
If two lines form congruent adjacent angles, then the lines are perpendicular.
Pg. 56
Term
Theroem 2-6
If the ext. sides of two adj. actue <'s are ┴, then ________
 Definition
If the exterior sides of two adjacent acute angles are perpendicular, then the angls are complementary.
Pg. 56
Term
Perpendicular
Lines
 Definition
Two lines that intersect to form right angles.
Pg. 56
Term
Theorem 2-7
Supplements of ≅ <'s
 Definition
If two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent.
Pg. 61
Term
Theorem 2-8
Complements of ≅ <'s
 Definition
If two angles are complements of congruent angles (or of the same angle), then the two angles are congruent.
Pg. 61
Term
Parrallel Lines
Parrallel Planes
 Definition
Are coplanar lines that do not intersect.

Planes that do not intersect
Pg. 73
Term
Skew Lines
 Definition
Are noncoplanar lines, that are neither parallel or intersecting
Pg. 73
Term
A line and plane
are Parallel
 Definition
If they do not intersect
Pg. 73
Term
Theorem 3-1
If two ǁ planes are cut by a third plane, then _________
 Definition
If two parallel planes are cut by a third plane, then the lines of intersection are parallel.
Pg. 74
Term
Transversal
 Definition
A line that intersects two or more coplanar lines in different points.
Pg. 74
Term
Alternate Interior
Angles
 Definition
Two nonadjacent interior angles on opposite sides of the transversal.
Pg. 74
Term
Same-Side
Interior Angles
 Definition
Two interior angles on the same side of the transversal.
Pg. 74
Term
Corresponding
Angles
 Definition
Two angles in corresponding positions relative to the two lines.
Pg. 74
Term
Postulate 10
Correspoinding <'s
 Definition
If two parallel lines are cut by a transversal, then correspoinding angles are congruent.
Pg. 78
Term
Theorem 3-2
Starting with ǁ lines
 Definition
If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
Pg. 78
Term
Theorem 3-3
Starting with ǁ lines
 Definition
If two parallel lines are cut by a transversal, then same-side interior angles are supplementary.
Pg. 79
Term
Theorem 3-4
If a transversal is ┴ to one of two ǁ lines, then _______
 Definition
If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other one also.
Pg. 79
Term
Postulate 11
Converse to corr. <'s (Post. 10)
 Definition
If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.
Pg. 83
Term
Theorem 3-5
Starting with alt. int. <'s
 Definition
If two lines are cut by a transversal and alt. int. angles are congruent, then the lines are parallel.
Pg. 83
Term
Theorem 3-6
Starting with same-side int. <'s
 Definition
If two lines are cut by a transversal and same-side interior angles are supplementary, then the lines are parallel.
Pg. 84
Term
Theorem 3-7
┴ and ǁ lines
 Definition
In a plane two lines are perpendicular to the same line are parallel.
Pg. 84
Term
Theorem 3-8
Lines and ǁ's
 Definition
Through a point outside a line, there is exactly one line paralllel to the given line.
Pg. 85
Term
Theorem 3-9
Lines and ┴'s
 Definition
Through a point outside a line, there is exactly one line perpendicular to the given line.
Pg. 85
Term
Theorem 3-10
Three ǁ lines
 Definition
Two lines parallell to a third line are parallel to each other.
Pg. 85
Term
Ways to Prove
Two Lines Parallel
 Definition
Show corr. <'s are ≅
Show alt. int. <'s are ≅
Show same-side int. <'s are supp.
Show both lines are ┴ to a third line
Show both lines are ǁto a third line.
Pg. 85
Term
Triangle
 Definition
The figure formed by three segments joining three noncollinear points.
Pg. 93
Term
Vertex & Sides
of a triangle
 Definition
Vertex - each of the three points of the triangle (Pluural: Vertices)
Sides - The segments of a triangle
Pg. 93
Term
Scalene Triangle
 Definition
No sides congruent
Pg. 93
Term
Isosceles Triangle
 Definition
At least two sides congruent
Pg. 93
Term
Equilateral Triangle
 Definition
All sides congruent
Can also be considered isosceles
Equilateral triangle is also equiangular
Pg. 93
Term
Acute Triangle
 Definition
Three actue <'s
Pg. 93
Term
Obtuse Triangle
 Definition
One obtues <
A triangle can not have more then one obtuse <
Pg. 93
Term
Right Triangle
 Definition
One right <
A triangle can not have more than one right <
Pg. 93
Term
Equiangular
Triangle
 Definition
All <'s congruent
Equiangular triangles are also equilaterial
Pg. 93
Term
Theorem 3-11
Sum of the int. <'s of a ∆
 Definition
The sum of the measure of the angles of a triangle is 180
Pg. 94
Term
Corollary 1, 2, 3 & 4
of Theorem 3-11
 Definition
1. If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent
2. Each angle of an equiangular triangle has measure 60
3. In a triangle, there can be at most one right angle or obtuse angle
4. The acute angles of a right triangle are complementary.
Pg. 94
Term
Theorem 3-12
Ext. < of a ∆
 Definition
The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles.
Pg. 95
Term
Auxiliary Line
 Definition
A line, ray, or segement added to a diagram to help in a proof.
Pg. 94
Term
Polygon
 Definition
Means "many angles". Any figure with three or more sides having the two qualitiles below.
1. Each segment intersects exactly two other segements, one at each endpoint
2. No two segments with a common endpoints are collinear
Pg. 101
Term
Convex Polygon
 Definition
A polygon such that no line containing a side of the polygon contains a point in the interior of the polygon.
Pg. 101
Term
Special Polygons
 Definition
3 Triangle 4 Quadrilateral
5 Pentagon 6 Hexagon
8 Octagon 10 Decagon
n n-gon
Pg. 101
Term
Diagonal
 Definition
A segment joining two nonconsecutive vertices of a polygon.
Pg. 102
Term
Theorem 3-13
Sum of int. <'s of a polygon
 Definition
The sum of the measures of the angles of a convex polygon with n sides is (n-2)180
Pg. 102
Term
Theorem 3-14
Sum of ext. <'s of a polygon
 Definition
The sume of the measures of the exterior angles of any convex polygon, one angle at each vertex, is 360.
Pg. 102
Term
Regular Polygon
 Definition
A polygon that is both equiangluar and equlateral.
Pg. 103
Term
Deductive
Reasoning
 Definition
Conclusion based on accepted statements (definitions, postulates, previous theorems, corolaries, and given information)
Conclusion must be true if hypotheses are true.
Pg. 106
Term
Inductive
Reasoning
 Definition
Conclusion based on several past observations
Conclusion is probably true, but not necessarily true.
Pg. 106
Term
Congruent
 Definition
When two figures have the same size and shape.
Pg. 117
Term
Corresponding
Parts
 Definition
When two shapes are ≅, then each < and side of one shape with correspond to an < and side of the other shape.
Pg. 117 - 118
Term
CPCT
 Definition
Corresponding parts of ≅ ∆s are ≅
Used after a polygon is shown ≅ to another polygon.
Pg. 118
Term
SSS Postulate
 Definition
Side Side Side Postulate
If three sides of one ∆ are ≅ to three sides of another ∆ then the ∆s are ≅
Pg. 122
Term
SAS Postulate
 Definition
Side Angle Side Psotulate
If two sides and the included < of one ∆ are ≅ to two sides and the included < of anther ∆, then the ∆s are ≅
Pg. 122
Term
ASA Postulate
 Definition
Angle Side Angle Postulate
If two <s and the included side of one ∆ are ≅ to two <s and the included side of anther ∆, then the ∆s are ≅
Pg. 123
Term
A line and a plane are ┴ IF
 Definition
If and Only If they intersect and the line is perpendicular to all lines in the plane that pass through the point of intersection.
Pg. 128
Term
How to Prove Segments or <s ≅
 Definition
1. Identify two ∆s in which the two segments or <s are corresponding parts.
2. Prove that the triangles are ≅
3. State that the two parts are ≅, using CPCT
Pg. 129
Term
Legs & Base
of an Isosceles ∆
 Definition
*The legs of an isosceles ∆ are the ≅ sides.
*The base is the third side.
*The <s along the base are called base <s
*The < opp. the base is the vertex angle.
Pg. 134
Term
Isosceles ∆ Theorem
 Definition
If two sides of a ∆ are ≅, then the <s opposite those sides are ≅.
Pg. 135
Term
Corollary 1, 2, & 3
of Isosceles ∆ Theorem
 Definition
1. An equilateral ∆ is also equiangluar
2. An equilateral ∆ has three 60⁰ <
3. The bisector of the vertex < of an isosceles ∆ is ┴ to the base at its midpoint.
Pg. 135
Term
Theorem 4-2
Converse of Isosceles ∆ Theorem
 Definition
If two <s of a ∆ are ≅, then the sides opposite those <s are ≅
Pg. 136
Term
Corollary 1
of Theorem 4-2
 Definition
An equiangluar ∆ is also equilateral
Pg. 136
Term
AAS Theorem
 Definition
Angle Angle Side Theorem
If two <s and a non-included side of one ∆ are ≅ to the corresponding parts of anther ∆, then the ∆s are ≅
Pg. 140
Term
Hypotenuse & Legs
of a Right ∆
 Definition
Hypotenuse (hyp.)- the side opposite the right <
Legs - are the other two side or the ┴ sides
Pg. 141
Term
HL Theorem
 Definition
Hypotenuse Leg Theorem
If the hypotenuse and a leg of one right ∆ are ≅ to the corresponding parts of another ∆, then the ∆s are ≅
Pg. 141
Term
Ways to Prove two ∆s ≅
 Definition
All ∆: SSS SAS ASA AAS

Right ∆: HL
Pg. 141
Term
Median
 Definition
A segment from a vertex to the midpoint of the opposite side.
Pg. 152
Term
Altitude
 Definition
A ┴ segment from a vertex to the line that contains the opposite side Pg. 152
*Acute ∆s have all three altitudes inside the ∆
*Right ∆s have two altitudes as legs and the other inside the ∆
*Obtuse ∆s have two altitudes outside and one inside the ∆
Term
Perpendicular
Bisector
 Definition
A segment that is ┴ to another segment at its midpoint
*The segment is both the altitude and median
Pg. 153
Term
Theorem 4-5
Starting with perp. bis
 Definition
If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment.
Pg. 153
Term
Theorem 4-6
Starting with point equidistant from endpts.
 Definition
If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment.
Pg. 153
Term
Distance From
a Point to a Line
 Definition
The length of the ┴ segment from the point to the line or plane.
*Important for theorem 4-7 and 4-8
Pg. 154
Term
Theorem 4-7
Starting with < bis.
 Definition
If a point lies on the bisector of an <, then the point is equidistant from the sides of the <.
Pg. 154
Term
Theorem 4-8
Starting with pt. equidistant from sides of an <
 Definition
If a point is equidistant from the sides of an <, then the point lies on the bisector of the <.
Pg. 154
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