Teaching Textbook Geometry Definitions, Theorems, Properties, and Postulates

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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

...

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


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