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NOTECARDS > GEOMETRY THEOREMS > Flashcards

GEOMETRY THEOREMS Flashcards

1
Q

Parts Whole Theorem for Segments

A

1) If AB = CD and AE = CF, then FD = EB
2) If AE = CF and FD = EB, then AB = CD
C_____F________D
A_____E________B

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2
Q

Parts Whole Theorem for Angles

A

2 on the theorem list - same as parts whole for segments

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3
Q

Midpoint Theorem

A

If M is the midpoint of AB them AM = 1/2AB

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4
Q

Angle Bisector Theorem

A

If ray BD bisects angle ABC then angle ABD = 1/2 angle ABC

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5
Q

Halves Whole Theorem for Segments

A 
Let M be the midpoint of AB and N be the midpoint of CD;
1) If AB=CD, them AM = CM
2) If AM = CN, then MB=ND and AB = CD
A\_\_\_\_\_\_\_\_\_\_M\_\_\_\_\_\_\_\_\_\_B
C\_\_\_\_\_\_\_\_\_\_N\_\_\_\_\_\_\_\_\_\_D
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6
Q

Halves Whole Theorem for Angles

A

6 on the theorem list - same as halves whole theorem for segments

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7
Q

Common Segment Theorem

A

1) If AB = CD then AC = BD
2) If AC = BD then AB = CD
A_____B__________C_____D

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8
Q

Common Angle Theorem

A

8 on the theorem list - same as common segment theorem

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9
Q

Vertical Angle Theorem (VAT)

A

Verticle Angles are Congruent

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10
Q

Perpendicular Line Theorem

A 
If any one of the following statements about two intersecting lines m and n is true, then all the statements are true 
       -
   1   - 2
 ----------
    4  - 3
        -

1) m is perpendicular to n
2) angle 1 = angle 2 (adjacent angles are congruent)
3) angle 1 is a right angle (any angle is right)
4) angle 1 is 90 degrees ( any angle has 90 degrees

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11
Q

Congruent Complements Theorem

A

If two angles are complements of congruent angles (or the same angle), then those angles are congruent

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12
Q

Congruent Supplements Theorem

A

If two angles are supplements of congruent angles ( or the same angle), then those two angles are congruent

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13
Q

Parallel Lines Imply Corresponding Angles congruent postulate
Abbreviation: CAPP or // –> corr angles congruent

A

If two parallel lines are cut by a transversal, then corresponding angles are congruent

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14
Q

Parallel Lines Imply Alternate Interior Angles Congruent

Abbreviation: // –> alt int angles congruent

A

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

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15
Q

Parallel Lines Imply Alternate Exterior Angles Congruent

Abbreviation: // –> alt ext angles congruent

A

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

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16
Q

Parallel Lines Imply Same Side Interior Angles Supplementary

Abbreviation: Same Side Int Sup

A

If two parallel lines are cut by a transversal, then same side interior angles are supplementary

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17
Q

Parallel Lines Imply Alternate Same Side Exterior Angles Supplementary
Abbreviation: // –> same side ext. sup

A

If two parallel lines are cut by a transversal, them same side exterior angles are supplementary

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18
Q

Corresponding Angles are Congruent Implies Lines Parallel Postulate
Abbreviation: CCAP or corr angles are congruent –> //

A

If two lines are cut by a transversal and corresponding angles are congruent then the lines are parallel.

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19
Q

Alternate Interior Angles (congruent) Implies Lines Parallel

A

If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel

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20
Q

Same Side Interior Angles Supplementary Implies Lines Parallel
Abbreviation: Same Side Sup –> //

A

If two lines are cut by a transversal and same side interior angles are supplementary, then the lines are parallel

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21
Q

If two lines are perpendicular to the same line

A

they are parallel

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22
Q

Transitivity with parallel lines:

A

If line 1// line 2 and line 2 // line 3 then line 1 // line 3

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23
Q

Triangle Sum Theorem

A

The sum of the measures of the angles of a triangle is 180 degrees

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24
Q

Corollaries to the Triangle Sum Theorem:

1) Remaining Angle in a Triangle Theorem

A

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

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25
Q

Corollaries to the Triangle Sum Theorem

A

2) Each angle of an equiangular triangle has measure 60°.

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26
Q

Corollaries to the Triangle Sum Theorem

A

3) In a triangle, there can be at most one right or obtuse angle.

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27
Q

Corollaries to the Triangle Sum Theorem

A

4) The acute angles of a right triangle are complementary.

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28
Q

Exterior Angle of a Triangle Theorem

A

The measure of an exterior angle of a triangle

equals the sum of the measures of the two remote interior angles.

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29
Q

Polygon Angle Sum Theorem

A

The sum of the measures of the interior angles of a

convex (or concave!) polygon with n sides is: (n − 2)180.

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30
Q

Polygon Exterior Angle Sum Theorem

A

The sum of the measures of the exterior

angles of any convex polygon, one angle at each vertex, is 360.

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31
Q

Definition of and ideas about congruent triangles from pages 117-8

A

Corresponding Parts of Congruent Triangles are Congruent (i.e. Corr. parts of ≅ Δ’s are ≅ )

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32
Q

SSS Postulate

A

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

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33
Q

SAS Postulate

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 triangles are congruent

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34
Q

ASA Postulate

A

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

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35
Q

Isosceles Triangle Theorem

A

Two sides of a triangle are congruent if and only if the

angles opposite those sides (the base angles) are congruent.

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36
Q

Corollaries of the Isosceles Triangle Theorem

A

1) A triangle is equilateral if and only if it is equiangular.
2) An equilateral triangle has three 60° angles.
3) The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint.

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37
Q

AAS

A

If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.

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38
Q

HL

A

If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent.

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39
Q

Magic Line Theorem or Isosceles-Median Theorem

A 
If any two of the following statements about ΔABC with point M on BC are true, then all four statements are true
1) AB = AC
2) Angle BAM = Angle CAM
3) MB = MC
4) AM is perpendicular to BC
(Diagram on Theorem List - #35)
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40
Q

Parallelogram ⇒ opp. sides ≅

A

The opposite sides of a parallelogram are congruent.

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41
Q

Parallelogram ⇒ opp. ∠’s ≅

A

The opposite angles of a parallelogram are congruent.

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42
Q

Parallelogram ⇒ diagonals bisect each other

A

The diagonals of a parallelogram bisect each other.

43
Q

Parallelogram ⇒ consec∠’s supplementary

A

Consecutive angles in a parallelogram are supplementary.

44
Q

Opp. sides quad. ≅ ⇒ Parallelogram

A

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

45
Q

Opp.∠’s ≅ ⇒ Parallelogram

A

If the opposite angles of a quadrilateral are congruent,

then the quadrilateral is a parallelogram.

46
Q

Opp. sides quad. ≅ and ⇒ Parallelogram

A

If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram.

47
Q

Diagonals bisect each other ⇒ Parallelogram

A

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

48
Q

Consec∠’s supp. ⇒ Parallelogram

A

If consecutive angles of a quadrilateral are supplementary, then the quadrilateral is a parallelogram.

49
Q

Midsegment Theorem

A

The segment that joins the midpoints of two sides of a

triangle is parallel to the third side and is half as long as the third side (Theorem List #45)

50
Q

Corollary to ≅ Cutter Theorem

A

A line that contains the midpoint of one side of a

triangle and is parallel to another side passes through the midpoint of the third side

51
Q

Inner Alien or (Midpoint Quadrilateral Theorem)

A

The quadrilateral formed by joining the midpoints of the sides of any quadrilateral is a parallelogram.

52
Q

rectangle → ≅ diagonals

A

The diagonals of a rectangle are congruent.

53
Q

parallelogram + ≅ diagonals → rectangle

A

If a parallelogram has congruent diagonals, then it is a rectangle.

54
Q

parallelogram + right ∠ → rectangle

A

If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle.

55
Q

2 consec. sides ≅ + parallelogram → rhombus

A

If two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.

56
Q

rhombus → ⊥ diagonals

A

The diagonals of a rhombus are perpendicular.

57
Q

parallelogram +⊥ diagonals → rhombus

A

If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

58
Q

rhombus → diagonals bisect each angle

A

Each diagonal of a rhombus bisects two angles of the rhombus.

59
Q

diagonals of a parallelogram bisect 1∠→ rhombus

A

If a diagonal bisects at least one angle of a parallelogram, then the parallelogram is a rhombus.

60
Q

Isosceles Trapezoid Theorem

A

1) If a trapezoid is isosceles, then its base angles are congruent.
2) If the base angles of a trapezoid are congruent, then the trapezoid is isosceles.

61
Q

The diagonals of a trapezoid are congruent if and only if

A

the trapezoid is isosceles

62
Q

Trapezoid Median Theorem

A

The median of a trapezoid is parallel to both bases and
has a length equal to the average of the base lengths. (i.e in Trapezoid ABCD with median MN, AB // MN // CD and MN = 1/2 (AB + CD)

63
Q

Whole > Part Theorem

A

If a > 0, b > 0, and a + b = c, then c > a and c > b.

64
Q

Mo mo le le

A

If a > b and c > d then a + c > b + d

65
Q

Exterior Angle Inequality Theorem

A

The measure of an exterior angle of a triangle is greater than the measure or either remote interior angle.

66
Q

Alligator Theorem (longer side ó bigger opp. angle)

A

If one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle
opposite the second side.

67
Q

Triangle Inequality Theorem

A

The sum of the lengths of any two sides of a triangle is

greater than the length of the third side.

68
Q

AA ~ Post

A

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

69
Q

3 in 1 Right Triangles

A

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.

70
Q

The Pythagorean Theorem

A

In a right triangle, the square of the hypotenuse is equal

to the sum of the squares of the legs.

71
Q

Converse of the Pythagorean Theorem

A

If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

72
Q

45-45-90 Triangle Theorem

A

the hypotenuse is 2 times as long as a leg

73
Q

30-60-90 Triangle Theorem

A

the hypotenuse is twice as long as the shorter leg, and

the longer leg is 3 times as long as the shorter leg

74
Q

Sine, Cosine, and Tangent Ratios

A 
Let x be an acute (not the right) angle in a right
triangle. Then,
sin(x) = opposite side/hypotenuse
cos(x) = adjacent side/hypotenuse
tan(x) = opposite side/adjacent side
Remember: Soh Cah Toa
75
Q

Tangent ↔ ⊥ to radius (Lauren’s Theorem)

A

A line is tangent to a circle, if and only if the line is perpendicular to the radius drawn to the point of tangency.

76
Q

≅ tangents (Watermelon Graduation Cap)

A

Tangents from the same point to a circle are congruent.

77
Q

Arc add. post.

A

The measure of the arc formed by two adjacent arcs is the sum of the measures of these two arcs.

78
Q

arcs are ≅ ⟺ central ∠s are ≅ (Defn of ≅ arcs)

A

In the same circle or in congruent circles, two arcs are congruent if and only if their central angles are congruent.

79
Q

Arcs ≅ ⟺ chords ≅

A

In the same circle or in congruent circles, two arcs are

congruent if and only if they have congruent chords.

80
Q

⊥ diameter bisects the chord and arc

A

A diameter that is perpendicular to a chord bisects the chord and its arc.

81
Q

diameter bisects the chord and arc ⇒ ⊥

A

if a diameter bisects the chord and arc then it is perpendicular to the chord

82
Q

In the same circle or congruent circles, chords equidistant from the center (or centers) are ≅.

A

Also, ≅ chords are equidistant from the center (or centers).

83
Q

Inscribed Angle Theorem

A

The measure of an inscribed angle is equal to half the measure of its intercepted arc.

84
Q

Corollaries of the Inscribed Angle Theorem

A

1) If two inscribed angles intercept the same arc, then the angles are congruent.
2) An angle inscribed in a semicircle is a right angle.
3) If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

85
Q

Extension of Inscribed Angle Theorem

A

The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc.

86
Q

Area of a Square

A

s⋅s (s squared)

87
Q

Area congruence postulate

A

If two figures are congruent, then they have the same

area.

88
Q

Area Addition postulate

A

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

89
Q

Area of a rectangle

A

l ⋅w

90
Q

Area of a parallelogram

A

b⋅ h where h is the height of the parallelogram

91
Q

Area of a Triangle

A

0.5 (b⋅ h)

92
Q

Area of a Rhombus

A

A = 0.5 (d1 ⋅ d2) where d1 and d2 are the diagonals of the rhombus

93
Q

Area of a Trapezoid

A

A = 0.5h⋅(b1 + b2 ) or A = h ⋅median

94
Q

Area of a Circle

A

A = πr(squared)

95
Q

Circumference of a Circle

A

C = 2πr or C = πd

96
Q

Area of a Sector

A

= x/360⋅(πrsquared) where x is the arc measure of the arc that forms the sector.

97
Q

Length of a Sector

A

= x/360⋅(2πr) where x is the arc measure of the arc that forms the sector.

98
Q

The distance, d, between the two points x1, y1 and x2, y2 can be found using the following distance formula:

A

d = radical (x2 - x1)squared + (y2 - y1) squared

99
Q

The slope of a line between the points x1, y1 and x2, y2

A

rise over run
delta y over delta x
y2 - y1 over x2 - x1
The slope of a horizontal line is zero, and the slope of a vertical line is undefined.

100
Q

The Equation of a Circle

A

(x-a)squared + (y-b) squared = r squared
Center: (a, b)
Radius: r

101
Q

Parallel lines

A

:) have equal slopes.

102
Q

Perpendicular lines have slopes that are

A

negative reciprocals of one another

103
Q

The midpoint between the points is the average of the x-coordinates and y coordinates. Given (x1, y1) and (x2, y2) the midpoint is

A

(x1, y1)/2 + (x2, y2)/2

(x1, y1) divided by two plus (x2, y2) divided two

NOTECARDS (25 decks)

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