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Geometry Terms Commonly Used in Proofs
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Gravity
Definitions, Theorems, and Postulates
Key Concepts:
Terms in this set (66)
congruent segments
line segments with the same length
Segment Addition Postulate
If B is between A and C, then AB + BC=AC.
Angle Addition Postulate
if P is in the interior of angle RST, then measure of angle RST= measure of angle RSP + measure of angle PST
right angle
an angle that measures exactly 90 degrees
straight angle
an angle that measures exactly 180 degrees
complementary angles
the sum of two angles' measures that equals 90 degrees
supplementary angles
the sum of two angles' measures that equals 180 degrees
linear pair
two adjacent angles that are supplementary (the sides they don't share form a line)
vertical angles
angles where their sides form two pairs of opposite rays and they share a common vertex
Addition Property of Equality
If a=b, then a + c=b + c.
Subtraction Property of Equality
If a=b, then a - c=b - c.
Multiplication Property of Equality
If a=b, then ac=bc.
Division Property of Equality
If a b and c doesn't =0, then a/c=b/c.
Substitution Property of Equality
If a=b, then a can be substituted for b in any equation or expression.
Distributive Property
a(b + c)=ab + ac, where a, b, and c are real numbers.
Reflexive Property of Equality or Congruence
Real Numbers: For any real number a, a=a.
Segment Length: For any segment AB, AB=AB.
Angle Measure: For any angle A, measure of angle A= measure of angle A.
Triangle: For any triangle ABC, triangle ABC=triangle ABC.
Symmetric Property of Equality or Congruence
Real Numbers: For any real numbers a and b, if a=b, then b=a.
Segment Length: For any segments AB and CD, if AB=CD, then CD=AB.
Angle Measure: For any angles A and B, if measure of angle A= measure of angle B, then measure of angle B= measure of angle A.
Triangle: For any triangles ABC and DEF, if triangle ABC= triangle DEF, then triangle DEF=triangle ABC.
Transitive Property of Equality or Congruence
Real Numbers: For any real numbers a, b, and c, if a=b and b=c, then a=c.
Segment Length: For any segments AB, CD, and EF, if AB=CD and CD=EF, then AB=EF.
Angle Measure: For any angles A,B, and C, if measure of angle A= measure of angle B and measure of angle B= measure of angle C, then measure of angle A= measure of angle C.
Triangle: For any triangles ABC, DEF, and GHI, if triangle ABC=triangle DEF and triangle DEF= triangle GHI, then triangle ABC= triangle GHI.
Right Angles Congruence Theorem
all right angles are congruent
Congruent Supplements Theorem
if two angles are supplementary to the same angle, then they are congruent
Congruent Complements Theorem
if two angles are complementary to the same angle, then they are congruent
Linear Pair Postulate
if two angles form a linear pair, then they are supplementary
Vertical Angles Congruence Theorem
any pair of vertical angles are congruent
Corresponding Angles Postulate
if two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent
Alternate Interior Angles Theorem
if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent
Alternate Exterior Angles Theorem
if two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent
Consecutive Interior Angles Theorem
if two parallel lines are cut by at transversal, then the pairs of consecutive interior angles are supplementary
Corresponding Angles Converse
if two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel
Alternate Interior Angles Converse
if two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel
Alternate Exterior Angles Converse
if two lines are cut by a transversal so the alternate exterior angles are congruent, then the lines are parallel
Consecutive Interior Angles Converse
if two lines are cut by a transversal so the consecutive interior angles are supplementary, then the lines are parallel
Transitive Property of Parallel Lines
if two lines are parallel to the same line, then they are parallel to each other
Theorem 3.8
if two lines intersect to form a pair of congruent angles, then the lines are perpendicular
Theorem 3.9
if two lines are perpendicular, then they intersect to form four right angles
Theorem 3.10
if two sides of two adjacent angles are perpendicular, then the angles are complementary
Perpendicular Transversals Theorem
if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other
Lines Perpendicular to a Transversal Theorem
in a plane, if two lines are perpendicular to the same line, then they are parallel to each other
Triangle Sum Theorem
the sum of the measures of the interior angles of a triangle is 180 degrees
Exterior Angle Theorem
the measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles
Corollary to the Triangle Sum Theorem
the acute angles of a right triangle are complementary
Base Angles Theorem
if two sides of a triangle are congruent, then the angles opposite to the sides are congruent
Base Angles Theorem Converse
if two angles of a triangle are congruent, then the sides opposite those angles are congruent
Corollary to the Base Angles Theorem
if a triangle is equilateral, then it is equiangular
Corollary to the Base Angles Theorem Converse
if a triangle is equilateral, then it is equiangular
Definition of Congruent Triangles
two triangles are congruent if and only if their vertices can be matched up so that the corresponding parts of the triangle are congruent
Third Angles Theorem
if two angles of one triangle are congruent to two angles of another, then the third angles are also congruent
midpoint
a point that divides a segment into two congruent segments
SSS Congruence Postulate
if three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent
SAS Congruence Postulate
if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent
HL Congruence Theorem
if the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent
AAS Congruence Postulate
if two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent
ASA Congruence Theorem
if two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent
Midsegment Theorem
the midsegment of a triangle is parallel to the third side and is half as long as that side
Perpendicular Bisector Theorem
in a plane, if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment
Perpendicular Bisector Theorem Converse
in a plane, if a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment
Angle Bisector Theorem
if a point is on the bisector of an angle, then it is equidistant from the two sides of the angle
Angle Bisector Theorem Converse
if a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle
segment bisector
a segment, ray, line, or plane that intersects a segment at its midpoint
perpendicular bisector
a segment bisector that is perpendicular to the segment
angle bisector
a ray that divides an angle into two congruent angles
Theorem 5.10
if one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side
Theorem 5.11
if one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle
Triangle Inequality Theorem
the sum of the lengths of any two sides of a triangle is greater than the length of the third side
Exterior Angle Inequality Theorem
the measure of an exterior angle of a triangle is greater than the measure of either of the nonadjacent angles
Hinge Theorem
if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first triangle 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
Hinge Theorem Converse
if two sides of one triangle are congruent to two sides of another triangle, and 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 triangle
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