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Unit 1 math: Points, Lines, and Planes; 9/15/24 > Postulates: All the units > Flashcards

Postulates: All the units Flashcards

1
Q

Postulate 1: Unit 1

A

Through any 2 points there is exactly 1 line.

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2
Q

Postulate 2: Unit 1

A

If 2 lines intersect, then they intersect in exactly 1 point (systems of equations.)

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3
Q

Postulate 3: Unit 1

A

If 2 planes intersect , they intersect in exactly 1 line.

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4
Q

Postulate 4: Unit 1

A

For any 3 noncollinear points there is exactly 1 plane containing them.

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5
Q

Ruler Postulate (Postulate 5): Unit 1

A

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

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6
Q

Segment Addition Postulate (Postulate 6): Unit 1

A

If 3 points A, B, C are collinear and B is between A
and C, then AB + BC= AC.

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7
Q

Protractor Postulate (Postulate 7): Unit 1

A

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

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8
Q

Angle Addition Postulate (Postulate 8) : Unit 1

A

If B is in the interior of <AOC, then
m <AOB + m<BOC = m<AOC

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9
Q

Postulate 9: Unit 1

A

If 2 figures are congruent, then their areas are equal.

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10
Q

Postulate 10: Unit 1

A

The area of a region is the sum of the areas of its nonoverlapping parts.

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11
Q

Angle Bisector Theorem (Theorem 11): Unit 2

A

If ray BD is the bisector of < ABC then
< ABD = 1/2 < ABC

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12
Q

Vertical Angle Theorem; VAT
(Theorem 12): Unit 2

A

Vertical angles are congruent.

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13
Q

Linear Pair Postulate (Postulate 13): Unit 2

A

2 angles that form a linear pair are supplementary.

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14
Q

Linear Pair Theorem (Theorem 14): Unit 2

A

The sum of the measures of a linear pair is 180 degrees.

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15
Q

Congruent Complement Theorem (Theorem 15): Unit 2

A

If 2 angles that are complementary to the same angle or to congruent angles, then they are congruent.

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16
Q

Congruent Supplements Theorem (Theorem 16): Unit: 2

A

If 2 angles that are supplementary to the same angle or to congruent angles, then they are congruent.

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17
Q

Common Segment Theorem (Theorem 17): Unit: 2

A

If segment AB is congruent to segment CD, then segment AC is congruent to segment BD.

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18
Q

Converse of the Common Segment Theorem (Theorem 18): Unit 2

A

If AC is congruent to segment BD, then segment AB is congruent to segment CD.

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19
Q

Common Angle Theorem (Theorem 19): Unit 2

A

If angle ABC is congruent to angle DBE, then angle ABD is congruent to angle CBE.

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20
Q

Converse of the Common Angle Theorem (Theorem 20): Unit 2

A

If angle ABD is congruent to angle CBE, then angle ABC is congruent to angle DBE

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21
Q

Corresponding Angles Postulate: (Postulate 21): Unit 3

A

If a transversal intersects two parallel lines, then the corresponding angles are congruent.

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22
Q

Alternate Interior Angles Theorem (Theorem 22): Unit 3

A

If a transversal intersects two parallel lines, then alternate interior angles are congruent.

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23
Q

Alternate Exterior Angles Theorem (Theorem 23): Unit 3

A

If a transversal intersects two parallel lines, then alternate exterior angles are congruent.

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24
Q

Same-side Interior Angles Theorem (Theorem 24): Unit 3

A

If a transversal intersects two parallel lines, then same-side interior angles are supplementary.

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25
Q

Same-side Exterior Angles Theorem (Theorem 25): Unit 3

A

If a transversal intersects two parallel lines, then same-side exterior angles are supplementary.

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26
Q

Converse of the Corresponding Angles Postulate: (Postulate 26): Unit 3

A

If the corresponding angles are congruent, then a transversal intersects two parallel lines.

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27
Q

Converse of the Alternate Interior Angles Theorem (Theorem 22): Unit 3

A

If alternate interior angles are congruent ,then a transversal intersects two parallel lines

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28
Q

Converse of the Alternate Exterior Angles Theorem (Theorem 28): Unit 3

A

If alternate exterior angles are congruent, then a transversal intersects two parallel lines

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29
Q

Converse of the Same-Side Interior Angles Theorem (Theorem 29): Unit 3

A

If same-side interior angles are supplementary, then a transversal intersects two parallel lines.

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30
Q

Converse of the Same-Side Exterior Angles Theorem (Theorem 30): Unit 3

A

If same-side exterior angles are supplementary, then a transversal intersects two parallel lines

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31
Q

Theorem 31: Unit 3

A

All right angles are congruent.

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32
Q

Theorem 34: Unit 3

A

If a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.

32
Q

Theorem 32: Unit 3

A

If two lines are parallel to the same line, then they are parallel to each other.

33
Q

Theorem 33: Unit 3

A

If two lines are perpendicular to the same line, then they are parallel to each other.

34
Q

Theorem 35: Unit 3

A

Given line, l, and a point P not on l, there exists one and only one line through P, parallel to l.

35
Q

Triangle Sum Theorem (Theorem 36): Unit 3

A

The sum of the measures of the angles of a triangle is 180°.

36
Q

Angle Sum Theorem (Theorem 37): Unit 3

A

The measure of an exterior angle of a triangle equals the sum of it’s two remote interior angles.

37
Q

Polygon Angle Sum Theorem (Theorem 38):
Unit 3

A

The sum of the measures of the angles of a convex polygon of n sides is: (n-2)(180)

38
Q

Polygon Exterior Angle-Sum Theorem
(Theorem 39): Unit 3

A

The sum of the measures of the exterior angles of a convex polygon is 360.

39
Q

Theorem 40: Unit 3

A

If two non-vertical lines are parallel, then their slopes are equal

40
Q

Theorem 41: Unit 3

A

Any two vertical lines are parallel

41
Q

Theorem 42: Unit 3

A

If the slopes of two distinct non-vertical lines are equal, the lines are parallel

42
Q

Theorem 43: Unit 3

A

If two non-vertical lines are perpendicular, the product of their slopes is -1

43
Q

Theorem 44: Unit 3

A

If the slopes of two non-vertical lines have a product of -1, the lines are parallel

44
Q

Theorem 45: Unit 3

A

Any horizontal line and vertical line are perpendicular

45
Q

Theorem 46: Unit 4

A

If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.

46
Q

SSS postulate (Postulate 47): Unit 4

A

If the thee sides of one triangle are congruent to the three sides of another, then the two triangles, are congruent.

47
Q

SAS postulate (Postulate 48): Unit 4

A

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.

48
Q

ASA postulate ( Postulate 49): Unit 4

A

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

49
Q

AAS Theorem (Theorem 50): Unit 4

A

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

50
Q

CPCTC (Entry 51): Unit 4

A

Corresponding parts of congruent triangles are congruent.

51
Q

Isosceles Triangle Theorem (Theorem 52): Unit 4

A

If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

52
Q

Corollary 53: Unit 4

A

If a triangle is equilateral, then it is equiangular

53
Q

Converse of the Isosceles Triangle Theorem (Theorem 54): Unit 4

A

If two angles of a triangle are congruent then the sides opposite those angles are congruent.

54
Q

Corollary 55: Unit 4

A

If the triangle is equiangular, then the triangle is equilateral.

55
Q

Theorem 56: Unit 4

A

The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.

56
Q

HL Theorem (Theorem 57): Unit 4

A

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.

57
Q

HA Theorem (Theorem 58): Unit 4

A

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.

58
Q

Triangle Midsegment Theorem (Theorem 59): Unit 5

A

If a segment joins the midpoints of a triangle, then the segment is parallel to the third side and half its length.

59
Q

Perpendicular Bisector Theorem (Theorem 60): Unit 5

A

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment

60
Q

Converse of the Perpendicular Bisector Theorem (Theorem 61): Unit 5

A

If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

61
Q

Angle Bisector Theorem (Theorem 62): Unit 5

A

If a point is on the bisector of an angle, then the point is equidistant from the sides of an angle.

62
Q

Converse of the Angle Bisector Theorem (Theorem 63): Unit 5

A

If a point is equidistant form the sides of an angle, then is on the bisector of the angle.

63
Q

Theorem 64: Unit 4

A

The perpendicular bisector of the sides of a triangle are concurrent at a point equidistant from the vertices.

64
Q

Theorem 65: Unit 4

A

The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.

65
Q

Theorem 66: Unit 5

A

The lines that contain the altitudes of a triangle are concurrent.

66
Q

Theorem 67: Unit 5

A

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.

67
Q

Corollary to the Triangle Exterior Angle Theorem (Corollary 68): Unit 5

A

The measure of an exterior angle of a triangle is greater than the measure of each of the remote interior angles.

68
Q

Theorem 69: Unit 5

A

If two sides of a triangle are not congruent, then the larger angle lies opposite the larger side.

69
Q

Theorem 70: Unit 5

A

If two angle are not congruent, then the larger side lies opposite the larger angle.

70
Q

Triangle Inequality Theorem (Theorem 71):
Unit 5

A

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

71
Q

SAS Inequality Theorem (Theorem 72): Unit 5

A

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.

72
Q

SSS Inequality Theorem (Theorem 73): Unit 5

A

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.

73
Q

Theorem 74: Unit 6

A

If a quadrilateral is a parallelogram, then opposite sides are congruent

74
Q

Theorem 75: Unit 6

A

If a quadrilateral is a parallelogram, then opposite angles are congruent

75
Q

Corollary 76; Corollary to Theorem 75: Unit 6

A

If a quadrilateral is a parallelogram, then consecutive angles are supplementary

76
Q

Theorem 77: Unit 6

A

If a quadrilateral is a parallelogram, then the diagonals bisect each other.

77
Q

Theorem 78: Unit 6

A

If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segment on every transversal.

78
Q

Theorem 79: Unit 6

A

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Unit 1 math: Points, Lines, and Planes; 9/15/24 (12 decks)

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