Geometry 1
Terms
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- Addition property of equality
- If a = b, then a + c = b + c
- Line Intersection Theorem
- Two different lines intersect in at most one point. p.43
- Linear Pair Theorem
- If two angles form a linear pair, then they are supplementary. p.140
- Vertical Angles Theorem:
- If two angles are vertical angles, then they have equal measures. p.141
- Parallel Lines and Slopes Theorem
- Two nonvertical lines are parallel if and only if they have the same slope. p.158
- Transitivity of Parallelism Theorem
- In a plane, if l m and m n, then l n. p. 158
- Two Perpendiculars Theorem
- If two coplanar lines l and m are each perpendicular to the same lines, then they are parallel to each other. p. 162
- Perpendicular to Parallels Theorem1
- In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other.
- Perpendicular Lines and Slopes Theorem
- Two nonvertical lines are pependicular if and only if the product of their slopes is -1. p.162
- angle
- the union of two rays that have the same endpoint. p. 124
- bisector
- VR is the bisector of <PVQ if and only if VR (except for point V) is in the interior of <PVQ and m<PVR = m<RVQ.
- adjacent angles
- Two non-straight and non zero angles are adjacent angles if and only if a common side (OB in the figure) is interior to the angle formed by the non-common sides (<AOC). p. 139
- Linear pair
- Two adjacent angles form a linear pair if and only if their non-common sides are opposite rays.
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Complimentary
Supplementary -
If the measures of two angles are m1 and m2, then the angles are
a. complementary if and only if m1 + m2 = 90.
b. supplementary if and only if m1 + m2 = 180 - parallel lines
- two coplanar lines m and n are parallel lines, written m n, if and only if they have no points in common or they are identical.
- segment
- The segment (or line segment) with endpoints A and B, denoted AB, is the set consisting of the distinct points A and B and all points between A and B.
- ray
- The ray with endpoint A and containing a second point B, denoted AB consists of the points on AB and all points for which B is between each of them and A.
- opposite rays
- AB and AC are opposite rays if and only if A is between B and C.
- Angle Addition Property
- If VC (except for point V) is in the interior of <AVB, then m<AVC + m<CVB = m<AVB.
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Angles:
zero
acute
right
obtuse
straight -
zero if m = 0
acute if and only if 0 < m < 90
right if and only if m = 90
obtuse if and only if 90 < m < 180
straight if and only if m = 180 - vertical angles
- Two non-straight angles are vertical angles if and only if the union of their sides is two lines.
- Reflexive property of equality
- a = a
- Symmetric property of equality
- If a = b, then b = a.
- Transitive Property of Equality
- If a = b and b = c, then a = c.
- Multiplication Property of Equality
- If a = b, then ac = bc.
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Corresponding Angles Postulate
Suppose two coplanar lines are cut by a transversal... -
a. If two corresponding angles have the same measure, then the lines are parallel.
b. If the lines are parallel, then corresponding angles have the same measure. - perpendicular
- Two segments, rays, or lines are perpendicular if and only if the lines containing them form a 90 angle.
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Point Line Postulate:
a. Unique Line Assumption - Through any two points, there is exactly one line. p.42
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Point Line Postulate:
b. Number Line Assumption - Every line is a set of points that can be put into a one-to-one correspondence with the real numbers, with any point corresponding to 0 and any other point corresponding to 1. p.42
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Point Line Postulate:
c. Dimension Assumption -
1. Given a line in a plane, there is at least one point in the plane that is not on the line.
2. Given a plane in space, there is at least one point in space that is not in the plane. p.42 -
Distance Postulate:
a. Uniqueness property - On a line, there is a unique distance between two points.
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Distance Postulate:
b. Distance formula - If two points on a line have coordinates x and y the distance between them is X - Y .
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Distance Postulate:
c. Additive property- - If B is on AC, then AB = BC = AC.
- Convex-
- A convex set is a set in which every segment that connects points of the set lies entirely in the set.
- Instance of a conditional-
- An instance of a conditional is a specific case in which both the antecedent (if part) and the consequent (then part) of the conditional are true.
- Counterexample to a conditional
- A counterexample to a conditional is a specific case for which the antecedent (if part) of the conditional is true and its consequent (then part) is false.
- Converse
- The converse of P q is q P.
- Midpoint-
- The midpoint of a segment AB is the point M on AB with AM = MB.
- Union of two sets-
- The union of two sets A and B, written A B, is the set of elements which are in A, in B, or in both A and B.
- Intersection of two sets
- The intersection of two sets A and B, written A B, is the set of elements which are in both A and B.
- Polygon-
- A polygon is the union of segments in the same plane such that each segment intersects exactly two others, one at each of its endpoints.
- Triange Inequality Postulate-
- The sum of the lenghts of any two sides of a triange is greater than the length of the third side.
- vertices- (vertex)
- The endpoints of the sides of a polygon. Singular is vertex.
- Consecutive (or adjacent)-
- Consecutive (or adjacent) sides are sides which share an endpoint. p.96
- Diagonal-
- A diagonal is a segment connecting nonadjacent vertices. p. 96
- equilateral triangle
- all three sides are equal
- isosceles triangle-
- has at least two sides of equal length. p. 97
- scalene triangle-
- a triangle with no sides of the same length. p. 96
- Triangle Inequality Postulate-
- The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
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Postulates of Inequality and Operations:
Transitive
Addition
Multiplication -
Transitive-
If a<b and b<c, then a<c.
Addition-
If a<b, then a+c<b+c.
Multiplication-
If a < b and c > 0, then ac < bc.
If a < b and c < 0, then ac > bc. - slope
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The slope of the line through
(X1, Y1) and (X2, Y2),
with X1 not equal to X2 is
Y2 - Y1
X2 - X1 - Circle-
- A circle is the set of all points in a plane at a certain distance, its radius, from a certain point, its center.