Postulates: All the units Flashcards
Postulate 1: Unit 1
Through any 2 points there is exactly 1 line.
Postulate 2: Unit 1
If 2 lines intersect, then they intersect in exactly 1 point (systems of equations.)
Postulate 3: Unit 1
If 2 planes intersect , they intersect in exactly 1 line.
Postulate 4: Unit 1
For any 3 noncollinear points there is exactly 1 plane containing them.
Ruler Postulate (Postulate 5): Unit 1
The points on a line can be matched one-to-one with real-number coordinates such that:
a. for any 2 points there corresponds a unique positive number called the distance between two points.
b. the distance between any two points is the absolute value of the difference of their coordinates. AB = I a-b I
Segment Addition Postulate (Postulate 6): Unit 1
If 3 points A, B, C are collinear and B is between A
and C, then AB + BC= AC.
Protractor Postulate (Postulate 7): Unit 1
Let Ray OA, and Ray OB be opposite rays in a plane. All the rays from point O on 1 side of Ray OA can be matched one-to-one with real numbers from 0 to 180 so that
a. Ray OA is matched with 0
b. Ray OB is matched with 180
c. is Ray OC matched with x and Ray OD is matched with y, then m<COD = I x-y I
Angle Addition Postulate (Postulate 8) : Unit 1
If B is in the interior of <AOC, then
m <AOB + m<BOC = m<AOC
Postulate 9: Unit 1
If 2 figures are congruent, then their areas are equal.
Postulate 10: Unit 1
The area of a region is the sum of the areas of its nonoverlapping parts.
Angle Bisector Theorem (Theorem 11): Unit 2
If ray BD is the bisector of < ABC then
< ABD = 1/2 < ABC
Vertical Angle Theorem; VAT
(Theorem 12): Unit 2
Vertical angles are congruent.
Linear Pair Postulate (Postulate 13): Unit 2
2 angles that form a linear pair are supplementary.
Linear Pair Theorem (Theorem 14): Unit 2
The sum of the measures of a linear pair is 180 degrees.
Congruent Complement Theorem (Theorem 15): Unit 2
If 2 angles that are complementary to the same angle or to congruent angles, then they are congruent.
Congruent Supplements Theorem (Theorem 16): Unit: 2
If 2 angles that are supplementary to the same angle or to congruent angles, then they are congruent.
Common Segment Theorem (Theorem 17): Unit: 2
If segment AB is congruent to segment CD, then segment AC is congruent to segment BD.
Converse of the Common Segment Theorem (Theorem 18): Unit 2
If AC is congruent to segment BD, then segment AB is congruent to segment CD.
Common Angle Theorem (Theorem 19): Unit 2
If angle ABC is congruent to angle DBE, then angle ABD is congruent to angle CBE.
Converse of the Common Angle Theorem (Theorem 20): Unit 2
If angle ABD is congruent to angle CBE, then angle ABC is congruent to angle DBE
Corresponding Angles Postulate: (Postulate 21): Unit 3
If a transversal intersects two parallel lines, then the corresponding angles are congruent.
Alternate Interior Angles Theorem (Theorem 22): Unit 3
If a transversal intersects two parallel lines, then alternate interior angles are congruent.
Alternate Exterior Angles Theorem (Theorem 23): Unit 3
If a transversal intersects two parallel lines, then alternate exterior angles are congruent.
Same-side Interior Angles Theorem (Theorem 24): Unit 3
If a transversal intersects two parallel lines, then same-side interior angles are supplementary.
Same-side Exterior Angles Theorem (Theorem 25): Unit 3
If a transversal intersects two parallel lines, then same-side exterior angles are supplementary.
Converse of the Corresponding Angles Postulate: (Postulate 26): Unit 3
If the corresponding angles are congruent, then a transversal intersects two parallel lines.
Converse of the Alternate Interior Angles Theorem (Theorem 22): Unit 3
If alternate interior angles are congruent ,then a transversal intersects two parallel lines
Converse of the Alternate Exterior Angles Theorem (Theorem 28): Unit 3
If alternate exterior angles are congruent, then a transversal intersects two parallel lines
Converse of the Same-Side Interior Angles Theorem (Theorem 29): Unit 3
If same-side interior angles are supplementary, then a transversal intersects two parallel lines.
Converse of the Same-Side Exterior Angles Theorem (Theorem 30): Unit 3
If same-side exterior angles are supplementary, then a transversal intersects two parallel lines
Theorem 31: Unit 3
All right angles are congruent.
Theorem 34: Unit 3
If a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.
Theorem 32: Unit 3
If two lines are parallel to the same line, then they are parallel to each other.
Theorem 33: Unit 3
If two lines are perpendicular to the same line, then they are parallel to each other.
Theorem 35: Unit 3
Given line, l, and a point P not on l, there exists one and only one line through P, parallel to l.
Triangle Sum Theorem (Theorem 36): Unit 3
The sum of the measures of the angles of a triangle is 180°.
Angle Sum Theorem (Theorem 37): Unit 3
The measure of an exterior angle of a triangle equals the sum of it’s two remote interior angles.
Polygon Angle Sum Theorem (Theorem 38):
Unit 3
The sum of the measures of the angles of a convex polygon of n sides is: (n-2)(180)
Polygon Exterior Angle-Sum Theorem
(Theorem 39): Unit 3
The sum of the measures of the exterior angles of a convex polygon is 360.
Theorem 40: Unit 3
If two non-vertical lines are parallel, then their slopes are equal
Theorem 41: Unit 3
Any two vertical lines are parallel
Theorem 42: Unit 3
If the slopes of two distinct non-vertical lines are equal, the lines are parallel
Theorem 43: Unit 3
If two non-vertical lines are perpendicular, the product of their slopes is -1
Theorem 44: Unit 3
If the slopes of two non-vertical lines have a product of -1, the lines are parallel
Theorem 45: Unit 3
Any horizontal line and vertical line are perpendicular
Theorem 46: Unit 4
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
SSS postulate (Postulate 47): Unit 4
If the thee sides of one triangle are congruent to the three sides of another, then the two triangles, are congruent.
SAS postulate (Postulate 48): Unit 4
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.
ASA postulate ( Postulate 49): Unit 4
If two angles and the included side of one triangle to two angles and the included side of another triangle, then the triangles are congruent
AAS Theorem (Theorem 50): Unit 4
If two angles and a non included side of one triangle are congruent to two angles and a nonincluded side of another triangle, then the triangles are congruent
CPCTC (Entry 51): Unit 4
Corresponding parts of congruent triangles are congruent.
Isosceles Triangle Theorem (Theorem 52): Unit 4
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
Corollary 53: Unit 4
If a triangle is equilateral, then it is equiangular
Converse of the Isosceles Triangle Theorem (Theorem 54): Unit 4
If two angles of a triangle are congruent then the sides opposite those angles are congruent.
Corollary 55: Unit 4
If the triangle is equiangular, then the triangle is equilateral.
Theorem 56: Unit 4
The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.
HL Theorem (Theorem 57): Unit 4
If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent.
HA Theorem (Theorem 58): Unit 4
If the hypotenuse and one acute angle of one right triangle are congruent to the hypotenuse and one acute angle of another right triangle, then the triangles are congruent.
Triangle Midsegment Theorem (Theorem 59): Unit 5
If a segment joins the midpoints of a triangle, then the segment is parallel to the third side and half its length.
Perpendicular Bisector Theorem (Theorem 60): Unit 5
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 (Theorem 61): Unit 5
If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
Angle Bisector Theorem (Theorem 62): Unit 5
If a point is on the bisector of an angle, then the point is equidistant from the sides of an angle.
Converse of the Angle Bisector Theorem (Theorem 63): Unit 5
If a point is equidistant form the sides of an angle, then is on the bisector of the angle.
Theorem 64: Unit 4
The perpendicular bisector of the sides of a triangle are concurrent at a point equidistant from the vertices.
Theorem 65: Unit 4
The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.
Theorem 66: Unit 5
The lines that contain the altitudes of a triangle are concurrent.
Theorem 67: Unit 5
The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side.
Corollary to the Triangle Exterior Angle Theorem (Corollary 68): Unit 5
The measure of an exterior angle of a triangle is greater than the measure of each of the remote interior angles.
Theorem 69: Unit 5
If two sides of a triangle are not congruent, then the larger angle lies opposite the larger side.
Theorem 70: Unit 5
If two angle are not congruent, then the larger side lies opposite the larger angle.
Triangle Inequality Theorem (Theorem 71):
Unit 5
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
SAS Inequality Theorem (Theorem 72): Unit 5
If two sides of one triangle are congruent to two sides of a second triangle and the included angle of the first triangle if larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.
SSS Inequality Theorem (Theorem 73): Unit 5
If two sides of one triangle are congruent to two sides of a second triangle and the third side of the first triangle is longer than the third side of the second triangle, then the angle opposite the third side of the first triangle is larger than the angle opposite the third side of the second triangle.
Theorem 74: Unit 6
If a quadrilateral is a parallelogram, then opposite sides are congruent
Theorem 75: Unit 6
If a quadrilateral is a parallelogram, then opposite angles are congruent
Corollary 76; Corollary to Theorem 75: Unit 6
If a quadrilateral is a parallelogram, then consecutive angles are supplementary
Theorem 77: Unit 6
If a quadrilateral is a parallelogram, then the diagonals bisect each other.
Theorem 78: Unit 6
If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segment on every transversal.
Theorem 79: Unit 6
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Theorem 80: Unit 6
If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram.
Theorem 81: Unit 6
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Theorem 82: Unit 6
If the consecutive angles of a quadrilateral are supplementary, then the quadrilateral is a parallelogram.
Theorem 83: Unit 6
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
Theorem 84: Unit 6
A parallelogram is a rectangle if and only if it’s diagonals are congruent.
Theorem 85: Unit 6
A parallelogram is a rhombus if and only if it’s diagonals are perpendicular.
Theorem 86: Unit 6
A parallelogram is a rhombus if and only if a diagonal bisects a pair of opposite angles.
Theorem 87: Unit 6
The base angles of an isosceles trapezoid are congruent.
Theorem 88: Unit 6
The diagonals of an isosceles trapezoid are congruent.
Theorem 89: Unit 6
The midsegment of a trapezoid is parallel to the bases and the length of the midsegment of a trapezoid is half the sum of the lengths of the bases.
Theorem 90: Unit 6
The diagonals of a kite are perpendicular.
Cross Product Property (Entry 91): Unit 7
In a proportion, the product of the means equal the product of the extremes.
Angle Angle (AA) Similarity Postulate (Postulate 92): Unit 7
If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.
Side Angle Side (SAS) Similarity Theorem (Theorem 93): Unit 7
If an angle of one triangle are congruent to an angle of a second triangle and the lengths of the sides including the angles are proportional, then the triangles are similar.
Side Side Side (SSS) Similarity Theorem (Theorem 94): Unit 7
If the corresponding side lengths of two triangles are proportional, then the triangles are similar.
Right Triangle Similarity Theorem (Theorem 95): Unit 7
If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.
Corollary to the Right Triangle Similarity Theorem (Corollary 96): Unit 7
The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segment of the hypotenuse.
Corollary to the Right Triangle Similarity Theorem (Corollary 97): Unit 7
The altitude to the hypotenuse of a right triangle separates the hypotenuse such that each leg is the geometric mean between the length of the adjacent hypotenuse segment and the length of the hypotenuse.
Triangle Proportional Segment Theorem (Theorem 98): Unit 7
If a line parallel to one side of a triangle intersect the other two sides, then it divides the two side proportionally.
Converse of the Triangle Proportional Segment Theorem (Theorem 99): Unit 7
If a line divides two sides of a triangle proportionally, then the line is parallel to the third side.
Parallels Proportional Segment Theorem (Theorem 100): Unit 7
If three parallel lines intersect two transversals, then they divide the transversals proportionally.
Triangle Angle Bisector Theorem (Theorem 101): Unit 7
If a ray bisects an angle of a triangle, then it divides the opposite side into two segment whose lengths are proportional to the lengths of the other two sides.
Entry 102: Unit 7
The lengths of bisectors, altitudes, and medians, of corresponding angles of similar triangles are in the same ratio as the lengths of corresponding sides.
Pythagorean Theorem (Theorem 103): Unit 8
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. a^2+b^2=c^2
Converse of the Pythagorean Theorem (Theorem 104): Unit 8
If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the tringle is a right triangle. If c^2=a^2+b^2, then Triangle ABC is a right triangle
Pythagorean Inequalities Theorem (Theorem 105): Unit 8
For any Triangle ABC, where c is the length of the longest side, the following statements are true:
1. If c^2<a^2+b^2, then Triangle ABC is acute.
2. If c^2>a^2+b^2, then Triangle ABC is obtuse.
45-45-90 Triangle Theorem (Theorem 106): Unit 8
In a 45-45-90 triangle, the hypotenuse is root 2 times as long a each leg.
30-60-90 Triangle Theorem (Theorem 107): Unit 8
In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is root 3 times as long as the shorter leg.
