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Definitions, Postulates, Theorems for chapters 1-7 for Geometry
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u can't really test urself on this........ its just kinda like familiarizing urself with the stuff
Terms in this set (154)
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
Ruler Postulate
if B is between A and C, then: AB+BC=AC
Segment Addition Postulate
on line AB in a given plane, choose any point O between A and B. Consider ray OA and ray 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: ray OA is paired with 0, and ray OB with 180, or, if ray OP is paired with X, and ray OQ with y, then angle POQ = l x-y l.
Protractor Postulate
If point B lies in the Interior of angle AOC, then angle AOB + angle BOC = angle AOC
Angle Addition Postulate
if angle AOC is a straight and and B is any point not on line AC, then angle AOB + angle BOC = 180
Angle Addition Postulate
a line contains at least two points; a plane contains at least three points not all in one line; space contains at least four points not all in one plane
Postulate 5
through any two points there is exactly one line
Postulate 6
through any three points there is at least one plane, and through any three non collinear points there is exactly one plane.
postulate 7
if two points are in a plane, then the line that contains the points is in that plane
Postulate 8
if two planes intersect, then their intersection is a line.
Postulate 9
If two lines intersect, then exactly one plane contains the lines
Theorem 1-1
through a line and a point not in the line there is exactly one plane
Theorem 1-2
If two lines intersect, then exactly one plane contains the lines
Theorem 1-3
if M is the midpoint of segment AB, then AM=1/2AB and MB=1/2AB
Midpoint Theorem
If ray BX is the bisector of angle ABC, then angle ABX = 1/2angleABC and angle XBC = 1/2 angle ABC
Angle Bisector Theorem
Reasons used in Proofs
Given info, Definitions, Postulates (and properties from algebra), Proven Theorems
Vertical angles are congruent
Theorem 2-3
Complementary angles
two angles whose measures have the sum 90
Supplementary angles
two angles whose measures have the sum 180
if two lines are perpendicular, then they from congruent adjacent angles
Theorem 2-4
if two lines form congruent adjacent angles, then the lines are perpindicular
Theorem 2-5
if the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary
Theorem 2-6
if two angles are supplements of congruent angles (or of the same angle) then the two angles are congruent
Theorem 2-7
if two angles are complements of congruent angles (or of the same angle), then the two angles are congruent
Theorem 2-8
Parallel lines
coplanar lines that do not intersect
Skew lines
noncoplanar lines
if two parallel planes are cut by a third plane, then the lines of intersection are parallel
Theorem 3-1
two nonadjacent interior angles on opposite sides of the transversal
Alternate interior angles
Same-Side Interior Angles
two interior angles on the same side of the transversal
two angles in corresponding positions relative to the two lines
Corresponding angles
if two parallel lines are cut by a transversal, then corresponding angles are congruent
Postulate 10
if two parallel lines are cut by a transversal, then alternate interior angles are congruent
Theorem 3-2
if two parallel lines are cut by a transversal, then same-side interior angles are supplementary
Theorem 3-3
if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other ones also
Theorem 3-4
if two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel
Postulate 11
if two lines are cut by a transversal and alternate interior angles are congruent, then the liens are parallel
Theorem 3-5
if two lines are cut by a transversal and same-side interior angles are supplementary, then the lines are parallel
Theorem 3-6
in a plane two lines perpendicular to the same lines are parallel
Theorem 3-7
through a point outside a line, there is exactly one line parallel to the given line
Theorem 3-8
through a point outside a line, there is exactly one line perpendicular to the given line
Theorem 3-9
two lines parallel to a third line are parallel to each other
Theorem 3-10
Ways to prove two lines parallel
show that a pair of corresponding angles are congruent
Ways to prove two lines parallel
show that a pair of alternate interior angles are congruent
Ways to prove two lines parallel
show that a pair of same-side interior angles are supplementary
Ways to prove two lines parallel
in a plane show that both lines are perpendicular to a third line
Ways to prove two lines parallel
show that both lines are parallel to a third line
Triangle
figure formed by three segments joining three noncollinear points
Scalene Triangle
no sides congruent
Isosceles Triangle
at least two sides congruent
Equilateral triangle
all sides congruent
Acute triangle
three acute angles
Obtuse triangle
one obtuse angle
Right triangle
one right angle
Equiangular triangle
all angles congruent
the sum of the measures of the angles of a triangle is 180
Theorem 3-11
if two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent
Corollary 3-1
each angle of an equiangular triangle has measure of 60
Corollary 3-2
in a triangle, there can be at most one right angle or obtuse angle
Corollary 3-3
the acute angles of a right triangle are complementary
Corollary 3-4
the measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles
Theorem 3-12
Polygon
many angles
Convex Polygon
polygon such that no line containing a side of the polygon contains a point in the interior of the polygon
the sum of the measures of the angles of a convex polygon with n sides is (n-2)180
Theorem 3-13
the sum of the the measures of the exterior angles of any convex polygon, one angle at each vertex, is 360
Theorem 3-14
Inductive Reasoning
reasoning that is widely used in science and everyday life
Deductive Reasoning
conclusion based on accepted statements (definitions, postulates, previous theorems, corollaries, and given information) conclusion MUST be true if hypotheses are true
Inductive Reasoning
Conclusion based on several past observations. conclusion is PROBABLY true, but not necessarily true
Congruent
having the same size and shape
SSS postulate
if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent
SAS postulate
if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent
ASA postulate
if two angles and the included sides of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent
A way to prove two segments or two angles are congruent
identify two triangles in which the two segments or angles are corresponding parts
A way to prove two segments or two angles are congruent
prove that the triangles are congruent
A way to prove two segments or two angles are congruent
state that the two parts are congruent, using the reason CPCTC
Isosceles Triangle Theorem
if two sides of a triangle are congruent, then the angles opposite those sides are congruent
an equilateral triangle is also equiangular
Corollary 4-1
an equilateral triangle has three 60° angles
Corollary 4-2
the bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint
Corollary 4-3
if two angles of a triangle are congruent, then the sides opposite those angles are congruent
Theorem 4-2
an equiangular triangle is also equilateral
Corollary 4-4
AAS theorem
if two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the triangls are congruent
HL theorem
if the hypotenuse and a leg of one right triangle
Ways to prove triangles congruent
SSS, SAS, ASA, AAS
Ways to prove right triangles congruent
HL
Median
segment from a vertex to the midpoint of the opposite side in a triangle
Altitude
perpendicular segment from a vertex to the line that contains the opposite side in a triangle
Perpendicular bisector
line (or ray or segment) that is perpendicular to the segment at its midpoint
if a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment
Theorem 4-5
if a point is equidistant from the endpoints of a segment, then the point is on the perpendicular bisector of the segment
Theorem 4-6
if a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle
Theorem 4-7
if a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle
Theorem 4-8
opposite sides of a parallelogram are congruent
Theorem 5-1
opposite angles of a parallelogram are congruent
Theorem 5-2
diagonals of a parallelogram bisect each other
Theorem 5-3
if both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram
Theorem 5-4
if one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram
Theorem 5-5
if both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram
Theorem 5-6
if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram
Theorem 5-7
Ways to prove that a quadrilateral is a parallelogram
show that BOTH pairs of opposite sides are parallel
Ways to prove that a quadrilateral is a parallelogram
show that BOTH pairs of opposite sides are congruent
Ways to prove that a quadrilateral is a parallelogram
show that ONE pair of opposite sides are both congruent and parallel
Ways to prove that a quadrilateral is a parallelogram
show that both pairs of opposite angles are congruent
Ways to prove that a quadrilateral is a parallelogram
show that the diagonals bisect each other
if two lines are parallel, then all points on one line are equidistant from the other line
Theorem 5-8
if three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal
Theorem 5-9
a line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side
Theorem 5-10
the segment that joins the midpoints of two sides of a triangle is parallel to the third side and is half as long as the third side
Theorem 5-11
Rectangle
quadrilateral with four right angles
Rhombus
quadrilateral with four congruent sides
Square
quadrilateral with four right angles and four congruent sides
the diagonals of a rectangle are congruent
Theorem 5-12
the diagonals of a rhombus are perpendicular
Theorem 5-13
each diagonal of a rhombus bisects two angles of the rhombus
Theorem 5-14
the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices
Theorem 5-15
if an angle of a parallelogram is a right angle, then the parallelogram is a rectangle
Theorem 5-16
if two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus
Theorem 5-17
base angles of an isosceles trapezoid are congruent
Theorem 5-18
the median of a trapezoid is parallel to the bases and has a length equal to the average of the base lengths
Theorem 5-19
Exterior Angle Inequality Theorem
the measure of an exterior angle of a triangle is greater than the measure of either remote interior angle
if not p, then not q
Inverse
if not q, then not p
Contrapositive
if p, then q
Given statement
if q, then p
Converse
How to write an Indirect Proof
Assume temporarily that the conclusion is not true
How to write an Indirect Proof
reason logically until you reach a contradiction of a known fact
How to write an Indirect Proof
point out that the temporary assumption must be false, and that the conclusion must then be true
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 6-2
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
Theorem 6-3
the perpendicular segment from a point to a line is the shortest segment from the point to the line
Corollary 6-1
the perpendicular segment from a point to a plane is the shortest segment form the point to the plane
Corollary 6-2
the sum of the lengths of any two sides of a triangle is greater than the length of the third side
Triangle Inequality
SAS Inequality Theorem
if two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first trangle 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
SSS Inequality Theorem
if two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second, then the included angle of the first triangle is larger than the included angle of the second
Ratio
quotient when the first number is divided by the second
Property of Proportions
a/b = c/d is equivalent to ad = bc
Property of Proportions
a/b = c/d is equivalent to a/c = b/d
Property of Proportions
a/b = c/d is equivalent to b/a = d/c
Property of Proportions
a/b = c/d is equivalent to a+b/b = c+d/d
Property of Proportions
if a/b = c/d = e/f =..., then a + c +e +.../b + d + f + ... = a/b = ...
AA ~
if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar
SAS Similarity Theorem
if an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar
SSS Similarity Theorem
if the sides of two triangles are in proportion, then the triangles are similar
Triangle Proportionality Theorem
if a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally
Triangle Angle-Bisector Theorem
if a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides
set of all points
space
points all in one line
collinear points
points all in one plane
coplanar points
set of points that are in both figures
intersection
two lines that intersect to form right angles
Perpendicular lines
equation stating that two ratios are equal
proportion
corresponding angles are congruent and corresponding sides are in proportion
similar
ratio of similar polygons
scale factor
Corollary 7-1
if three parallel lines intersect two transversals, then they divide the transversals proportionally
auxiliary line
line (or ray or segment) added to a diagram to help in a proof
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