collinear points
are points that lie on the same line
noncollinear points
are points that do not lie on the same line
two points determine
a unique straight line (postulate 1)
Three noncollinear points determine
a unique plane (postulate 2)
a line segment is
A PART OF A LINE CONSISTING OF TWO POINTS, CALLED END POINTS, AND THE SET OF ALL POINTS BETWEEN THEM
congruent line segments are
line segments that have the same length
the ruler postulate a
To every point of the line there corresponds exactly one real number called its coordinate
the ruler postulate b
to every real number there corresponds exactly one point of the line
the ruler postulate c
to every pair of points there corresponds exactly oone real number called the distance between the points
the ruler postulate d
the difference between tow points is the absolute value of the difference between their coordinates
betweenness of points
If F, G, and H are collinear, and if FG + GH = FH, then G is between F and H
A ray is
a part of a line consisting of a given point called the end point, and the set of all points on one side of the end point
an angle is
the union of two rays having the same end point. The end point is called the vertex of the angle; the rays are called the sides of the angle.
congruent angles are
angles that have equal measures.
the protractor postulate a
The rays in a half rotation (180⁰) can be numbered so that to every ray there corresponds exactly one real number called its coordinate.
the protractor postulate b
And to every number from 0-180, there corresponds exactly one ray.
the protractor postulate c
To every pair of rays there corresponds exactly one real number called the measure of the angle that they determine.
the protractor postulate d
And the measure of the angle is the absolute value of the difference between the coordinates of its rays.
betweenness of rays
Ray PS is between ray PQ and ray PR, if point S lies in the interior of ∠QPR and m∠SPR + m∠QPS = ∠QPR.
a right angle is
an angle that has a measure of 90 degrees
an acute angle is
an angle with a measure of less than 90⁰.
an obtuse angle is
An angle greater than 90 degrees and less than 180 degrees
the midpoint of a line segment is
the point that divides the line segment into two congruent line segments.
a segment bisector is
A line, ray, or segment that passes through the midpoint of a segment
angle bisector
A ray that divides an angle into two congruent adjacent angles
addition property
If equals are added to equals, the results are equal. If a=b, then a + c = b + c.
subtraction property
If equals are subtracted from equals, the results are equal. If a=b, then a - c = b - c.
multiplication property
If equals are multiplied by equals, their products are equal. If a=b, then ac = bc.
division property
If equals are divided by nonzero equals, their quotients are equal. If a=b, then a ÷ c = b ÷ c.
substitution property
If a = b, then either a or b may be substituted for the other in any equation.
transitive property
If two quantities are equal to the same quantity, then they are equal to each other: If a = b and b =c, then a = c.
reflexive property
Any quantity is equal to itself: a = a.
symmetric property
The positions of the expressions on either side of an equals sign may be reversed. If a = b, then b = a.
complementary angles are
Two angles whose measures have the sum of 90 degrees.
If two angles are complementary to the same angle or equal angles then
they are congruent (equal)
supplementary angles are
angles with measures that add to 180 degrees
If two angles are supplementary to the same angle or equal (congruent) angles then
they are congruent
adjacent angles are
are a pair of angles with a common vertex and a common side, but no common interior points
a linear pair is
two adjacent angles, whose exterior sides form a straight line
if two angles are a linear pair then
they are supplementary
vertical angles are
A pair of non-adjacent angles formed by the intersection of two straight lines
pairs of vertical angles are
congruent
perpendicular lines are
Lines which intersect to form right angles
perpendicular lines intersect to form
four right angles
all right angles are
congruent
through a given point on a line there exists
exactly one perpendicular to the given line
through a given point not on a line there exists
exactly one perpendicular to the given line
if the exterior sides of a pair of adjacent angles are perpendicular the angles are
complementary
the distance between two points is
the length of the line segment joining the points
the distance between a line and a point not on the line is
the length of the perpendicular segment drawn form the point to the line
parallel lines are
lines that lie in the same plane and that never intersect.
lines, segments, rays, or points which lie in the same plane are said to be
coplanar
a transversal is
a line that intersects two or more lines in different points.
if two parallel lines are cut (crossed) by a transversal then
their corresponding angles are congruent
If two angles in a linear pair have equal measures (are congruent) then
each is a right angle
If two parallel lines are cut (crossed) by a transversal, then their alternate exterior angles are
congruent
If two parallel lines are cut (crossed) by a transversal, the interior angles on the same side of the transversal are
supplementary
If two lines form congruent alternate interior angles with a transversal the the lines are
parallel
if two lines form congruent alternate interior angles with a transversal the the lines are
parallel
ir two lines form congruent alternate exterior angles with a transversal then the lines are
parallel
if two lines form supplementary interior angles on the same side of a transversal then the lines are
parallel
a polygon is
a geometric figure whose sides are line segments.
a triangle is
a polygon that has three sides.
a scalene triangle
has no congruent (equal) sides.
an isosceles triangle
has at least two congruent (equal sides).
an equilateral triangle
has all three congruent (equal) sides.
an acute triangle
has all three angles with measure of less than 90°.
a right triangle
has one angle with a measure of 90°.
an obtuse triangle
has one angle with a measure of greater than 90°.
an equiangular triangle
has all three angles with equal measures.
if two lines are parallel to a third line then
the lines are parallel to each other
through a given point not on a line exactly one line may be drawn
parallel to the line
the sum of the measures of the angles of a triangle is
180 degrees
the acute angles of a right triangle are
complementary
the measure of each angle of an equiangular triangle is
60 degrees
if two angle of a triangle are congruent to two angles of another triangle then the remaining pair of angles is
congruent
an exterior angle of a polygon is
an angle that forms a linear pair with one of the interior angles of the polygon.
Exterior angle of a triangle theorem
The measure of an exterior angle of another triangle is equal to the sum of the measures of the two remote interior angles.
definition of congruent triangles
if the vertices of two triangles can be paired in a correspondence so that all pairs of corresponding angles are congruent and all pairs of corresponding sides are congruent, then the triangles are congruent.
side angle side postulate
if the vertices of two trangles can be paired so that two sides and the included angle of one triangle are congruent to the corresponding parts of the second triangle, then the two triangles are congruent.
angle side angle postulate
If the vertices of two triangles can be paired so that two angles and the included side of one triangle are congruent to the corresponding parts of the second triangle, then the two triangles are congruent.
angle angle side theorem
if the verices of two triangles can be paired so that two angles and the side opposite one of them in one triangle are congruent to the corresponding parts of the second triangles, then the two triangles are congruent.
hypotenuse-leg postulate
If the vertices of two right triangles can be paired so that the hypotenuse and leg of one of them are congruent to the corresponding parts of the second right triangle, then the two right triangles are congruent.
side side side postulate
if the vertices of two triangles can be paired so that three sides of one triangles are congruent to the corresponding sides of the second triangle, then the two triangles are congruent.
corresponding parts of congruent triangles are congruent
If two triangles are congruent, then their vertices can be paired in a correspondence so that all pairs of corresponding angles are congruent and all pairs of corresponding sides are congruent.
if two triangles are congruent to the same triangle then they are
congruent to each other
an altitude of a triangle is
a segment drawn from any vertex of the triangle, perpendicular to the opposite side, extended outside the triangle if necessary.
a median of a triangle is
a segment drawn from any vertex of the triangle to the midpoint of the opposite side.
if a point lies on the perpendicular bisector of a segment, the
point is equidistant from the endpoints of the segment
if a point is equidistant from the endpoints of a segment then
the point lies on the perpendicular bisector of the segment
base angles theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
if a triangle is equilateral then
it is also equiangular
the altitudes extending to the legs of an isosceles triangle are
congruent
converse of the base angles theorem
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
if a triangle is equiangular then
it is also equilateral
the medians extending to the legs of an isosceles triangle are
congruent
addition property of inequality
if a>b then a+c > b+c
subtraction property of inequality
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multiplication property of inequality
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division property of inequality
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substitution property of inequality
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transitive property of inequality
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whole greater than its part property
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triangle inequality postulate
The sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
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.
if unequal sides then unequal angles
If two sides of a triangle are not congruent, then the angles opposite those sides are not congruent, and the greater angle is opposite the longer side.
if unequal angles then unequal sides
If two angles of a triangle are not congruent, then the sides opposite those angles are not congruent, and the longer side is opposite the greater angle.
a quadrilateral is
a polygon that has four sides.
the sum of the measures of the angles of a quadrilateral is
360 degrees
the length of a line segment drawn from any vertex of an equilateral triangle to a point on the opposite side is
less than the length of any side of the triangle
a trapezoid is
a quadrilateral which has exactly one pair of parallel sides. The parallel sides are called bases and the nonparallel sides are called legs.
an isosceles trapezoid is
a trapezoid that has both legs congruent.
the lower (and upper) base angles of an isosceles trapezoid are
congruent
the diagonals of an isosceles trapezoid are
congruent
a parallelogram is
a quadrilateral that has both pairs of opposite sides parallel.
if a quadrilateral is a parallelogram then both pairs of consecutive angles are
supplementary
if a quadrilateral is a parallelogram then both pairs of opposite angles are
congruent
if a quadrilateral is a parallelogram then both pairs of opposite sides are
congruent
if a quadrilateral is a parallelogram then the diagonals
bisect each other
if both pairs of opposite angles are congruent then a quadrilateral is a
parallelogram
if both pairs of opposite sides are congruent then a quadrilateral is a
parallelogram
if one pair of opposite sides is both parallel and congruent then a quadrilateral is a
parallelogram
if the diagonals bisect each other than a quadrilateral is a
parallelogram
the line segment joining the midpoints of two sides of a triangle is parallel to the
third side and is one half its length
a rectangle is
a parallelogram with four right angles.
a rhombus is
parallelogram with four congruent sides.
a square is
parallelogram with four congruent sides and four congruent and right angles.
the diagonals of a rectangle are
congruent
the diagonals of a rhombus are
perpendicular to each other
the diagonals of a rhombus
bisect the angles at the vertices which they join
the sum of the measures of the interior angles of a polygon with n sides is
180(n-2)
the sum of the measures of the exterior angles of any polygon (one exterior angle per vertex) is
360 degrees
a regular polygon is
a convex polygon that is both equilateral and equiangular.
The measure of each interior angle of a regular polygon is
180(n-2) / n
where n is the number of sides of the regular polygon
the perimeter of a polygon is
the sum of the lengths of its sides.
the perimeter of a regular polygon with n sides and side length s is equal to
ns
