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Math
Geometry
OHS H Geometry Postulates 2014-15 Dec. Final
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Terms in this set (82)
Postulate 101
Through any 2 points there is exactly one line.
Postulate 102
If two distinct lines intersect, they intersect at exactly one point.
Postulate 103
If two distinct planes intersect, they intersect at exactly one line.
Postulate 104
Through any noncollinear 3 points there is exactly one plane.
Ruler Postulate
Every point on a line can be paired with a real number. This makes a one‐to‐one correspondence between the points on the line and the real numbers. The real number that corresponds to a point is called the coordinate point.
Segment Addition Postulate
If any three points A, B, and C are collinear and B is between A and C, then AB+BC=AC.
Angle Addition Postulate
If point B lies in the interior of <AOC , then
m<AOB + m<BOC = m<AOC.
Linear Pair Postulate
If <AOC is a straight angle and B is any point not
on it, then <AOB and <BOC form a linear pair, and are supplementary.
Area Addition Postulate
The area of a region is the sum of the areas of its non overlapping parts.
Law of Detachment
If the hypothesis of a true conditional statement is true, then the conclusion is true.
Law of Syllogism
If p→q is true, and q →r is true then p →r is true.
Addition Property
If a = b and c = d, then a + c=b + d OR if a = b, then a + c = b + d.
Subtraction Property
If a = b and c = d, then a ‐ c=b ‐ d OR if a = b, then a - c = b - d.
Multiplication Property
If a = b, then ca = cb.
Division Property
if a = b and c ≠ 0, then a/c = b/c.
Substitution Property
If a = b, then either a or b may be substituted for the other in any equation (or inequality).
Reflexive Property
a = a.
Symmetric Property
If a = b, then b = a.
Transitive Property
If a = b, and b = c, then a = c.
Congruent Supplements Theorem
If two angles are supplements of the same angle (or congruent angles), then the two angles are congruent.
Congruent Complements Theorem
If two angles are complements of the same angle (or congruent angles), then the two angles are congruent.
Congruent Linear Pair Angles (not a real theorem)
If two angles are congruent and supplementary, then each is a right angle.
Alternate Interior Angles
Nonadjacent interior angles that lie on opposite sides of the transversal.
Alternate Exterior Angles
Nonadjacent exterior angles that lie on opposite sides
Same Side Interior Angles
Interior angles that lie on the same side of the transversal.
Corresponding Angles
Angles that lie on the same side of the transversal and in corresponding positions.
Same Side Interior Angles Postulate
If a transversal intersects two parallel lines, then same‐side interior angles are supplementary.
Alternate Interior Angles Theorem
If a transversal intersects two parallel lines, then alternate interior angles are congruent.
Alternate Exterior Angles Theorem
If a transversal intersects two parallel lines, then alternate exterior angles are congruent.
Corresponding Angles Theorem
If a transversal intersects two parallel lines, then corresponding angles are congruent.
Converse of the Same Side Interior Angles Postulate
If two lines and a transversal form same‐side interior angles that are supplementary, then the two lines are parallel
Converse of the Alternate Interior Angles Theorem
If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel.
Converse of the Corresponding Angles Theorem
If two lines cut by a transversal form corresponding angles that are congruent, then the lines are parallel.
Converse of the Alternate Exterior Angles Theorem
If two lines and transversal form alternate exterior angles that are congruent, then the two lines are parallel.
Theorem 101
In the same plane, f two lines are parallel to the same line, then they are parallel to each other.
Theorem 102
In the same plane, if two lines are perpendicular to the same line, then they are parallel to each other.
Theorem 103
In the same plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other.
Parallel Postulate
Through a point not on a line, there is one and only one line parallel to the given line.
Equilateral
Having sides of all the same length.
Equiangular
Having angles of all the same measure.
Exterior Angle
An angle formed by extending the side of a polygon.
Triangle Angle Sum Theorem
The sum of the measures of the angles of a triangle is 180.
Triangle Exterior Angle Theorem
The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.
Corollary to the Triangle Exterior Angle Theorem
The measure of an exterior angle of a triangle is greater than the measures of either remote interior angle
Congruent Polygons
Congruent polygons have congruent corresponding parts—their marching sides and angles. When you name congruent polygons, you must list corresponding vertices in the same order.
Third Angle Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
SSS Postulate
If three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent.
SAS Postulate
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
AAS Theorem
If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of the other triangle, then the triangles are congruent.
ASA Theorem
If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
Hypotenuse - Leg Theorem (HL)
If the hypotenuse and a leg of one right triangle is congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
CPCTC
Corresponding Parts of Congruent Triangles
Isosceles Triangle Theorem
If two sides of one triangle are congruent, then the angles opposite those sides are congruent.
Converse of the Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
Corollary to the Isosceles Triangle Theorem
If a triangle is equilateral, then it is also equiangular.
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Hinge Theorem (SAS Inequality)
If two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger included angle.
Converse of the Hinge Theorem
If two sides of one triangle are congruent to two sides of another triangle, and the third sides are not congruent, then the larger included angle is opposite the longer third side.
Theorem 104
If a line bisects the vertex angle(non base angle) of an isosceles triangle, then the line is also the perpendicular bisector of the base.
Theorem 105
If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and is half as long.
Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
Converse of the Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of the segment.
Angle Bisector Theorem
If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle.
Converse of the Angle Bisector Theorem
If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector.
Concurrency in Perpendicular Bisectors Theorem
The perpendicular bisectors of the sides of a triangle are concurrent (meet, share a point) at a point equidistant from the vertices, called the circumcenter
Concurrency in Angle Bisectors Theorem
The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle, called the centroid.
Circumscribe
Since the circumcenter is equidistant from the vertices, you can use the circumcenter as the center of a circle that contains each vertex of the triangle. You say the circle is circumscribed
about the triangle.
Theorem 106
If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side.
Converse of Theorem 106
If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle.
Polygon Angle - Sum Theorem
The sum of the measures of the interior angles of a polygon with n sides is 180(n - 2).
Regular polygon
A polygon that is both equilateral and equiangular.
Corollary to the Polygon Angle - Sum Theorem
The measure of each interior angle of a regular polygon with n sides is 180(n-2)/n.
Polygon Exterior Angle Theorem
The sum of the measures of the exterior angles of a polygon is 360.
Parallelogram Definition
If a quadrilateral is a parallelogram, then its opposite sides are congruent/then its consecutive angles are supplementary/then its opposite angles are congruent/then its diagonals bisect each other.
(Converse is also true).
Theorem 107
If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on any transversal.
Rhombus Definition
If a parallelogram is a rhombus, then all of its sides are congruent/then its diagonals are perpendicular/then its diagonals bisect a pair of opposite angles.
Rectangle Definition
If a parallelogram is a rectangle, then its angles are all 90 degrees/then its diagonals are congruent.
Square
A parallelogram that is a rhombus and a rectangle.
Trapezoid
A quadrilateral with exactly one pair of parallel lines.
Isosceles Trapezoid Theorem/Definition
If a trapezoid is an isosceles trapezoid, then each pair of its base angles are congruent/then its diagonals are congruent.
Trapezoid Midsegment Theorem
The midsegment of a trapezoid is parallel to the bases; connects the midpoints of the non - base lines/is halfway between the bases; is the average of the length of the bases.
Kite Definition
If a quadrilateral is a kite, then its diagonals are perpendicular/then each pair of distinct adjacent sides are congruent.
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