Download Undefined terms: Point, line, set, between ness

MG21 Geometry
10/6/2009
Undefined terms:
Set is a collection of distinct objects
Point P indicates location, has no length, width or depth
Line l is set of continuous points that extend indefinitely in either direction
Plane R is a set of points that forms a flat surface that has length and width but no depth and that
extends infinitely in either direction
Definitions:
1. Collinear points are points that lie on the same straight line
2. Line segment is part of line consisting of two endpoints and the set of all points between
them
3. Ray is part of line consisting of a given endpoint and the set of all points on one side.
Opposite rays have the same endpoint and form a line
4. Betweenness. Point P is between points A and B if point A,P,B are three different
collinear points and AB = AP +PB
5. Angle is the union of two rays called sides having the same endpoint called vertex
6. Midpoint of a line segment is a point on that segment such that the two segments formed
are congruent. Divides a segment into two segments of equal length
Point M is the midpoint of AB if M is between A and B and AM=MB
7. The bisector of a line segment is a line that intersects the line segment at the midpoint of
the line segment
8. Bisector of an angle is a ray such that its endpoint is the vertex of the angle and it forms
two angles of equal measure with the sides of the angle.
BM is the bisector of ABC if M lies in the interior of ABC and ABM  CBM
9. Congruent angles(segments) are angles(segments) having equal measures
10. A right angle is an angle of 90 degrees. Oblique angle is angle not a right angle
11. A straight angle is an angle of 180 degrees
12. An acute angle is an angle whose measure is greater than 0  and less than 90
13. An obtuse angle is an angle whose measure is greater than 90 and less than 180
14. Two angles are supplementary if the sum of their measure is 180.
15. Two angles are complementary if the sum of their measures is 90
16. Two angles are adjacent pair if have the same vertex, share a common side and have no
interior points in common. The side not shared are called exterior sides
17. Vertical angles are two angles such that the sides of one are rays that are opposite to
those of the sides of the other. Nonadjacent (Opposite) angles formed by two intersecting
lines
18. Perpendicular lines are two lines that intersect and form right angles. Two segments or a
line and a segment that intersect to form a right angle are perpendicular
19. A perpendicular bisector of a line segment is a line or segment that is perpendicular to the
given segment at its midpoint.
20. Distance between two points is the length of the segments joining the points
21. Distance between a line and a point not on the line is the length of the perpendicular
segment drawn from the point to the line
22. An exterior angle of a triangle is adjacent and supplementary to an angle of the triangle
23. Transversal is a line that intersects r lines at r distinct points( the lines need not be
parallel)
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POSTULATES IN GEOMETRY
1.
2.
3.
4.
5.
6.
7.
8.
A line may be extended as far as desired in either directions
For any two points on a line, there exists a third point that is between them.
There exists a one to one correspondence between the points on a line and the real numbers
There exists one and only one line through two points
The Addition Postulate
The Subtraction Postulate
The Multiplication Postulate
The Division postulate
Corollary. Half of congruent wholes are congruent.
Halves of equals are equal
9.
Reflexive property of equality/congruence: AB  AB , ABC  ABC
10.
Symmetric property of equality/congruence: If AB  CD then CD  AB
11. Transitive property of equality/congruence. If a  b and b  c then a  c
12. Substitution property. An equivalent number/quantity may be substituted/replaced in place
of the numerical expression on the right side of the equation
13. Law of Detachment(Modus Ponens)
pq
p
q
14. Angle Addition Postulate
If ray OP is in the interior of AOB , then mAOB  mAOP  mPOB
15. Segment Addition Postulate. If collinear points B is between A and C, then AB  BC  AC
16. Ruler Postulate: The points on a line can be paired with the real numbers. The distance
between any two points equals the absolute value of the difference of their numbers.
17. Protractor Postulate: On AB in a given plane, choose any point O between A and B.
Consider OA and OB and all the rays that can be drawn from O to one side of AB . These
rays can be paired with the real numbers from 0 to 180.
“Simple” Theorems:
1. If two angles are right angles, then they are congruent
2. If two angles are straight angles, then they are congruent
3. If two angles are complementary to the same angle, then they are congruent
4. If two angles are supplementary to the same angle, then they are congruent
5. If two angles are complementary to two congruent angles, then they are congruent
6. If two angles are supplementary to two congruent angles, then they are congruent
7. If two angles are congruent to two congruent angles, then they are congruent
8. If two angles are vertical angles, then they are congruent
9. If the exterior sides of a pair of adjacent angles form a straight line, then the angles
are supplementary. Linear pair
10. If two angles are congruent and supplementary, then each is a right angle
11. If the exterior sides of a pair of adjacent angles are perpendicular, then the angles are
complementary
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12. Perpendicular lines intersect to form right angles
13. Through a given point on a line, there exists exactly one perpendicular to the given
line
14. Through a given point not on the line there exists exactly one perpendicular to the
line
15. If two lines intersect to form congruent adjacent angles, then the lines are
perpendicular
16. If two lines intersect, then they intersect in exactly one point
Basic Postulates:
1. Every angle has one and only one bisector
2. Every line segment has one and only one midpoint
3. Partition Postulate : The whole is the sum of its parts
4. The whole is greater than any of its parts
5. A line that intersects one side of a triangle and enters the interior of the triangle must
intersect a second side of the triangle. (Pasch’s Axiom)
Congruent Triangles Postulates and Theorems
1. SAS Postulate (two sides and included angle)
2. ASA Postulate (two angles and included side)
3. SSS Postulate
4. If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
5. If two angles of a triangle are congruent, then the sides opposite those angles are
congruent.
6. All radii of same circle are congruent
7. If two points are each equidistant from the endpoints of a line segment then the
line joining them will be the perpendicular bisector of the line segment
8. If a point is on the perpendicular bisector of a line segment then it is equidistant
from the endpoints of the line segment
9 If a point is equidistant from the endpoints of a line segment then it lies on the
perpendicular bisector of the line segment.
10. Exterior Angle Inequality Theorem. The measure of an exterior angle of a triangle is
greater than the measure of either of the remote interior angles
11. The sum of the measures of the angles of a triangle is equal to 180
12. If two angles of one triangle are congruent respectively to two angles of another triangle
then the third pair of angles is congruent
13. Two triangles are congruent if there exists a correspondence between the vertices in
which two angles and a side opposite one of them in one triangle are congruent to the
corresponding parts in another triangle. (AAS Theorem)
14. The measure of an exterior angle of triangle is equal to the sum of the measures of its
remote interior angles.
15. Two right triangles are congruent if there exists a correspondence between the vertices in
which the hypotenuse and leg of one are congruent to those corresponding parts in the other
(HL Theorem)
16. If two triangles are congruent to the same triangle then they are congruent to each other
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Parallel lines are two coplanar lines that do not intersect.
A transversal is a line that intersects two other lines in two distinct points.
Parallel Postulate Through a given point not on a given line there exists one and only one
line that is parallel to the given line
Perpendicularity and Parallelism
1. If two lines are cut by a transversal such that the alternate interior (corresponding,
alternate exterior) angles are congruent, then the lines are parallel.
2. If two lines are perpendicular to the same line then they are parallel.
3. If two lines are parallel to the same line, then they are parallel to each other.
4. If two lines are cut by a transversal such that the same side interior angles are
supplementary, then the lines are parallel.
5. If a line is perpendicular to one of two parallel lines, it is  to the other.
6. If two lines are parallel and cut by a transversal, then the alternate (corresponding,
alternate exterior) angles are congruent.
7. If two parallel lines are cut by a transversal, the bisectors of the congruent interior angles
are parallel.
8. The bisectors of congruent interior (corresponding) angles are parallel
9. The bisectors of same side supplementary interior angles are perpendicular
10. Parallel lines are everywhere equidistant
Polygons
Definition. A polygon is a closed figure in a plane that is the union of line segments such that
the segments intersect only at their endpoints and no segments sharing a common endpoint
are collinear
A convex polygon is a polygon in which the measure of each angle is less than 180 .
A polygon is convex if it contains all the line segments connecting any pair of its points
Polygonal diagonal is a line segment connecting two nonadjacent vertices
An equiangular polygon is a polygon in which each angle has the same measure
An equilateral polygon is a polygon in which each side has the same length
A regular polygon is a polygon that is both equiangular and equilateral
Two polygons are similar if their vertices can be paired so that corresponding angles are
congruent and the ratios of the lengths of all corresponding sides are equal
Theorem
1. If a polygon has n sides then n(n-3)/2 non-adjacent diagonals can be drawn
2. The sum of the measures of the angles of a polygon of n sides is 180(n-2)
3. The sum of the measures of the exterior angles of a polygon formed by extending the
sides in the same order is equal to 360
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Triangle Inequalities
1. If equals are added to (subtracted from) unequals the sums (differences) will be unequal
in the same order.
2. If unequals are divided (multiplied) by positive equals, the quotients (products) will be
unequal in the same order. Halves of unequal quantities are unequal in the same order.
3. If the first of the three quantities is greater than the second and the second is greater than
the third, then the first is greater than the third. Transitive property of inequality.
Transitive Postulate of Order
4. If unequal quantities are added to unequal quantities of the same order, the sums are
unequal in the same order
5. The whole is greater than any of its parts
6. The sum of the lengths of two sides of a triangle is greater than the length of the third
side
7. If two sides of a triangle are unequal, then the angles opposite them are unequal and the
measure of larger angle is opposite the greater side.
8. If two angles of a triangle are unequal, then the sides opposite them are unequal and the
greater side is opposite the larger angle measure.
9. Hinge Theorem. (SAS Theorem of Inequality)
If two sides of one triangle are congruent to two sides of another triangle and the included
angle of the first triangle is greater than the included angle in the second triangle then the
third side of the first triangle is greater than the third side of the second triangle.
Indirect Proof Postulates
1. Either p or ~ p is true. No other possibilities exist. Law of the Excluded Middle
2. Both p and ~p cannot be true at the same time. Law of Contradiction
3. Law of Elimination
Definition of Quadrilaterals
A parallelogram is a quadrilateral in which the opposite sides are parallel
A rectangle is a parallelogram with one right angle
A square is a rectangle with two adjacent sides congruent
A rhombus is a parallelogram with two adjacent sides congruent
A trapezoid is a quadrilateral that has one and only one pair of sides parallel
An isosceles trapezoid is a trapezoid in which the legs (nonparallel sides) are congruent
Quadrilaterals Theorems
The opposite sides of a parallelogram are congruent
The opposite angles of a parallelogram are congruent
The diagonals of a parallelogram bisect each other
All four sides of a square are congruent
All sides of a rhombus are congruent
The lower base angles of an isosceles trapezoid are congruent
Two consecutive angles of parallelogram are supplementary
If the diagonals of a parallelogram are congruent then the parallelogram is a rectangle
The diagonals of a rhombus are perpendicular
The diagonals of a rhombus bisects its angles
A square is a rhombus
If a trapezoid is isosceles then the diagonals are congruent
If a trapezoid is isosceles then the base angles are congruent
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If both pairs of opposite sides of a quadrilateral are parallel then the quadrilateral is a
parallelogram.
If the opposite sides of a quadrilateral are congruent then the quadrilateral is a parallelogram
If the diagonals of a quadrilateral bisect each other then the quadrilateral is a parallelogram
If a quadrilateral has one pair of sides congruent and parallel then the quadrilateral is a
parallelogram
If both pairs of opposite angles of a quadrilateral are congruent then the quadrilateral is a
parallelogram.
If a quadrilateral is equilateral then it is a rhombus
If the diagonals of a parallelogram are perpendicular to each other then the parallelogram is a
rhombus
If one of the angles of a rhombus is a right angle then the rhombus is a square
The median to the hypotenuse of right triangle is congruent to one half the hypotenuse.
In a quadrilateral if two angles are adjacent and congruent they are right angles
The line segment joining the midpoints of two sides of triangle is parallel to the third side and
one half measure of the third side.
The line segment joining the midpoints of the nonparallel side of trapezoid is parallel to the bases
and has measure equal to one half of the sum of the lengths of the bases.
If three or more parallel lines intercept congruent segments on one transversal, they intercept
congruent segments on every transversal.
Proportions
In a proportion product of the means equals the product of the extremes
In a proportion, the means may be interchanged.
In a proportion the extremes may be interchanged
If three parallel lines intersect two transversals, then they divide the transversals proportionally.
Ratios and Proportions
In a proportion product of the means equals the product of the extremes
In a proportion, the means may be interchanged.
In a proportion the extremes may be interchanged
If three or more parallel lines intercept congruent segments on one transversal, they intercept
congruent segments on every transversal. (Lewis p. 324)
If three parallel lines intersect two transversals, then they divide the transversals proportionally.
Similar Triangles
The line segment joining the midpoints of two sides of triangle is parallel to the third side and
one half measure of the third side.
If the three angles of one triangle are congruent to corresponding angles of another triangle then
the two triangles are similar.
Corresponding angles of similar triangles are congruent
Corresponding sides of similar triangles are in proportion
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AA Theorem of Similar Triangles
Two triangles are similar if two angles of one triangle are  to two corresponding angles of the
other. (Dressler p. 294)
If the ratios of the measures of two pairs of corresponding sides of two triangles are equal and
the included angles are congruent, then the two triangles are similar. (SAS Theorem of ~
triangles)
If the ratios of the measures of corresponding sides of two triangles are equal, then the triangles
are similar. (SSS Theorem of ~ triangles)
Medians
The median to the hypotenuse of a right triangle is congruent to one half the hypotenuse.
A segment joining the midpoints of two sides of a triangle is parallel to the third side and its
measure is one-half the measure of the third side.
The median of a trapezoid is parallel to the bases and its length equals the average of the
measures of the bases.
Medians Concurrency Theorem
The medians of a triangle are concurrent at a point that is two-thirds of the way from any vertex
to the midpoint of the opposite side.
Side Splitter Theorem or Triangle Proportionality Theorem
If a line parallel to one side of a triangle intersects the other two sides in different points, it
divides those sides in the same ratio (proportionally). (part/part = part/part)
Postulate. If a line intersects two sides of a triangle so that the ratios of the measures of the
corresponding segments are equal, then the line is parallel to the third side
Corollary to the Side Splitter Theorem
If a line parallel to one side of a triangle intersects the other two sides in different points, it cuts
off segments proportional to the sides. (part /whole = part/whole)
Corollary. A line that is parallel to one side of a triangle and intersects the other two sides in
different points cuts off a triangle similar to the given triangle. (Dressler p. 296)
If a line divides two sides of a triangle proportionally, the line is parallel to the third side
Angle Bisector Theorem
An angle bisector in a triangle divides the opposite sides into segments that have the same ratio
as the other two sides
The perimeter of two similar triangles has the same ratio as the lengths of any pair of
corresponding sides.
If two triangles are similar then the ratio of their areas equals the square of the ratio of the
lengths of any two corresponding sides
If two triangles are similar then the ratio of their corresponding altitudes (medians, angle
bisectors) equal the ratio of the lengths of any two corresponding sides
Postulate.
Two triangles that are similar to the same triangle are similar to each other.
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Right Triangle
The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices
In a right triangle with an altitude to the hypotenuse the measure of the altitude is the mean
proportional between the measures of the segments of the hypotenuse.
In a right triangle with an altitude to the hypotenuse the measure of either leg is the mean
proportional between the measure of the entire hypotenuse and the measure of the hypotenuse
adjacent to the leg (projection)
The product of the measures of the legs of right triangle is equal to the product of the measures
of the hypotenuse and the altitude to the hypotenuse.
Pythagorean Theorem
The sum of the measure of the hypotenuse of a right triangle is equal to the sum of the squares of
the measure of the legs.
Converse
If the square of the length of longest side of a triangle is equal to the sum of the squares of the
lengths of the other two sides, then the triangle is a right triangle.
Pythagorean Triples: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25)
If a 2  b 2  c 2 then the triangle is obtuse.
If a 2  b 2  c 2 then the triangle is acute
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