Teaching Textbook Geometry Definitions, Theorems, Properties, and Postulates
division property of inequality
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vertical angles are
A pair of non-adjacent angles formed by the intersection of two straight lines
multiplication property of inequality
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subtraction property of inequality
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the sum of the measures of the angles of a triangle is
180 degrees
the measure of each angle of an equiangular triangle is
60 degrees
a line segment is
A PART OF A LINE CONSISTING OF TWO POINTS, CALLED END POINTS, AND THE SET OF ALL POINTS BETWEEN THEM
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
an obtuse angle is
An angle greater than 90 degrees and less than 180 degrees
the protractor postulate d
And the measure of the angle is the absolute value of the difference between the coordinates of its rays.
the protractor postulate b
And to every number from 0-180, there corresponds exactly one ray.
reflexive property
Any quantity is equal to itself: a = a.
betweenness of points
If F, G, and H are collinear, and if FG + GH = FH, then G is between F and H
substitution property
If a = b, then either a or b may be substituted for the other in any equation.
addition property
If equals are added to equals, the results are equal. If a=b, then a + c = b + c.
division property
If equals are divided by nonzero equals, their quotients 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.
subtraction property
If equals are subtracted from equals, the results are equal. If a=b, then a - c = b - c.
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.
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.
converse of the base angles theorem
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
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.
base angles theorem
If two sides of a triangle are congruent, then the angles opposite those sides 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.
perpendicular lines are
Lines which intersect to form right angles
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.
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.
symmetric property
The positions of the expressions on either side of an equals sign may be reversed. If a = b, then b = a.
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 c
To every pair of rays there corresponds exactly one real number called the measure of the angle that they determine.
the ruler postulate a
To every point of the line there corresponds exactly one real number called its coordinate
complementary angles are
Two angles whose measures have the sum of 90 degrees.
a polygon is
a geometric figure whose sides are line segments.
a transversal is
a line that intersects two or more lines in different points.
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
a triangle is
a polygon that has three sides.
a median of a triangle is
a segment drawn from any vertex of the triangle to the midpoint of the opposite side.
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.
Three noncollinear points determine
a unique plane (postulate 2)
two points determine
a unique straight line (postulate 1)
an exterior angle of a polygon is
an angle that forms a linear pair with one of the interior angles of the polygon.
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.
congruent angles are
angles that have equal measures.
supplementary angles are
angles with measures that add to 180 degrees
adjacent angles are
are a pair of angles with a common vertex and a common side, but no common interior points
noncollinear points
are points that do not lie on the same line
collinear points
are points that lie on the same line
if the exterior sides of a pair of adjacent angles are perpendicular the angles are
complementary
the acute angles of a right triangle are
complementary
If two parallel lines are cut (crossed) by a transversal, then their alternate exterior angles are
congruent
all right angles are
congruent
if two angle of a triangle are congruent to two angles of another triangle then the remaining pair of angles is
congruent
pairs of vertical angles are
congruent
the altitudes extending to the legs of an isosceles triangle are
congruent
the medians extending to the legs of an isosceles triangle are
congruent
if two triangles are congruent to the same triangle then they are
congruent to each other
lines, segments, rays, or points which lie in the same plane are said to be
coplanar
If two angles in a linear pair have equal measures (are congruent) then
each is a right angle
through a given point not on a line there exists
exactly one perpendicular to the given line
through a given point on a line there exists
exactly one perpendicular to the given line
perpendicular lines intersect to form
four right angles
an equiangular triangle
has all three angles with equal measures.
an acute triangle
has all three angles with measure of less than 90°.
an equilateral triangle
has all three congruent (equal) sides.
an isosceles triangle
has at least two congruent (equal sides).
a scalene triangle
has no congruent (equal) sides.
a right triangle
has one angle with a measure of 90°.
an obtuse triangle
has one angle with a measure of greater than 90°.
addition property of inequality
if a>b then a+c > b+c
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.
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.
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 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.
if a triangle is equilateral then
it is also equiangular
if a triangle is equiangular then
it is also equilateral
congruent line segments are
line segments that have the same length
parallel lines are
lines that lie in the same plane and that never intersect.
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
if two lines form supplementary interior angles on the same side of a transversal then the lines are
parallel
ir two lines form congruent alternate exterior angles with a transversal then the lines are
parallel
through a given point not on a line exactly one line may be drawn
parallel to the line
if a point lies on the perpendicular bisector of a segment, the
point is equidistant from the endpoints of the segment
If two parallel lines are cut (crossed) by a transversal, the interior angles on the same side of the transversal are
supplementary
the ruler postulate d
the difference between tow points is the absolute value of the difference between their coordinates
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
if two lines are parallel to a third line then
the lines are parallel to each other
if a point is equidistant from the endpoints of a segment then
the point lies on the perpendicular bisector of the segment
the midpoint of a line segment is
the point that divides the line segment into two congruent line segments.
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.
if two parallel lines are cut (crossed) by a transversal then
their corresponding angles are congruent
If two angles are supplementary to the same angle or equal (congruent) angles then
they are congruent
If two angles are complementary to the same angle or equal angles then
they are congruent (equal)
if two angles are a linear pair then
they are supplementary
the ruler postulate c
to every pair of points there corresponds exactly oone real number called the distance between the points
the ruler postulate b
to every real number there corresponds exactly one point of the line
a linear pair is
two adjacent angles, whose exterior sides form a straight line