Download Theorem and postulate list for Scholarship Geometry Chapter 1

Theorem and postulate list for Scholarship Geometry
Chapter 1

Through any two points there is exactly one line.
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Through any three noncollinear points there is exactly one plane containing
them

If two points lie in a plane, then the line containing those points lies in the
plane.

If two lines intersect, then they intersect in exactly one point.

If two planes intersect, then they intersect in exactly one line.
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Segment Addition Postulate
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Angle Addition Postulate

Midpoint Formula: The midpoint M of AB with endpoints A(x1, y1) and B(x2, y2)
x  x2 y1  y2
is:
M( 1
,
)
2
2

Distance Formula: d  (x2  x1 )2  (y2  y1 )2
Chapter 2
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Reflexive Property
Symmetric Property
Transitive Property
Addition Property
Subtraction Property
Multiplication Property
Division Property
Distributive Property

Substitution Property

All right angles are congruent.

The sum of the measures of the angles of a linear pair is 180.

Two angles that form a linear pair are supplementary.

Vertical angles are congruent.
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If two angles are supplementary to the same angle (or to two congruent angles),
then the two angles are congruent.

If two angles are complementary to the same angle (or to two congruent
angles), then the two angles are congruent.
Chapter 3

If two parallel lines are cut by a transversal, then corresponding angles are
congruent.

If two parallel lines are cut by a transversal then alternate interior angles are
congruent.
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If two parallel lines are cut by a transversal, then alternate exterior angles are
congruent.
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If two parallel lines are cut by a transversal, then same side interior angles are
supplementary.

If two lines are cut by a transversal so that corresponding angles are
congruent, then the lines are parallel.

If two lines are cut by a transversal so that alternate interior angles are
congruent, then the lines are parallel.
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If two lines are cut by a transversal so that alternate exterior angles are
congruent, then the lines are parallel.

If two lines are cut by a transversal so that same side interior angles are
supplementary, then the lines are parallel
Chapter 4
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The sum of the measures of the angles of a triangle is 180.

The measure of each angle of an equiangular triangle is 60.

The acute angles of a right triangle are complementary.

The measure of an exterior angle of a triangle is equal to the sum of the
measures of its two remote interior angles.

If two angles of one triangle are congruent to two angles of another triangle,
then the third angles are congruent.

SAS (Side-Angle-Side)

ASA (Angle-Side-Angle)

AAS (Angle-Side-Side)
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SSS (Side-Side-Side)
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HL (Hypotenuse-Leg)
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CPCTC

If two sides of a triangle are congruent, then the angles opposite those sides
are congruent.

If three sides of a triangle are congruent, then the three angles are also
congruent.

If a triangle is equilateral, then it is equiangular.

If two angles of a triangle are congruent, then the sides opposite those angles
are congruent.

If three angles of a triangle are congruent, then the three sides are also
congruent.

If a triangle is equiangular, then it is equilateral.
Chapter 5

If a point is on the perpendicular bisector of a segment, then it is equidistant
from the endpoints of the segment.

If a point is equidistant from the endpoints of a segment, then it is on a
perpendicular bisector of the segment.

If a point is on the bisector of an angle, then it is equidistant from the sides of
the angle.

If a point in the interior of an angle is equidistant from the sides of the angle,
then it is on the bisector of the angle.

The circumcenter of a triangle (where perpendicular bisectors of the sides
meet) is equidistant from the vertices of the triangle.

The incenter of a triangle (where angle bisectors meet) is equidistant from the
sides of the triangle.
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The centroid of a triangle is located 2/3 the distance from the vertex to the
midpoint of the opposite side.

The midsegment of a triangle is parallel to the third side of the triangle and its
length is half the length of the third side.
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If two sides of a triangle are not congruent, then the largest angle is opposite
the longest side.
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If two angles of a triangle are not congruent, then the longest side is opposite
the largest angle.

The sum of the lengths of any two sides of a triangle is greater than the length
of the third side.

THE Pythagorean Theorem: The sum of the squares of the lengths of the legs
of a right triangles is equal to the square of the length of the hypotenuse.
Better remembered as a2 + b2 = c2.

Converse of the Pythagorean Theorem and its Inequalities
If a ≤ b ≤ c are the lengths of sides of a triangle and:
1) c2 = a2 + b2, then the triangle is a right triangle;
2) c2 < a2 + b2, then the triangle is an acute triangle;
3) c2 > a2 + b2, then the triangle is an obtuse triangle.

In a 45-45-90 triangle, both legs are congruent, and hypotenuse is a leg times
2.
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In a 30-60-90 triangle, the hypotenuse is two times the shorter leg, and the
longer leg is the shorter leg times 3 .
Chapter 7
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AA Similarity Postulate
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SSS Similarity Postulate
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SAS Similarity Postulate
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If a line parallel to one side of a triangle intersects the other two sides, then it
divides those sides proportionally.

If a line divides two sides of a triangle proportionally, then it is parallel to the
third side.

The bisector of an angle of a triangle divides the opposite side into two
segments whose lengths are proportional to the lengths of the other two sides.
Chapter 6

The sum of the interior angle measures of a convex polygon with n sides is:
S = (n – 2) 180.

Each angle of a regular polygon of n sides is: a = (n – 2)180
n

The sum of the measures of the exterior angles of a polygon, one angle at each
vertex, is 360

The measure of each exterior angle of a regular polygon of n sides is 360 .
n

If a quadrilateral is a parallelogram, then opposite sides are congruent.
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If a quadrilateral is a parallelogram, then opposite angles are congruent.
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If a quadrilateral is a parallelogram, then consecutive angles are supplementary.
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If a quadrilateral is a parallelogram, then its diagonals bisect each other.
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If both pairs of opposite sides of a quadrilateral are congruent, then the
quadrilateral is a parallelogram.
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If both pairs of opposite angles of a quadrilateral are congruent, then the
quadrilateral is a parallelogram.
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If an angle of a quadrilateral is supplementary to both of its consecutive angles,
then the quadrilateral is a parallelogram.
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If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a
parallelogram.
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If one pair of opposite sides of a quadrilateral is both parallel and congruent,
then the quadrilateral is a parallelogram.
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If a parallelogram is a rectangle, then its diagonals are congruent.
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If a parallelogram is a rhombus, then its diagonals are perpendicular.
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If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite
angles.
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If one pair of consecutive sides of a parallelogram is congruent, then the
parallelogram is a rhombus.
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If the diagonals of a parallelogram are perpendicular, then the parallelogram is
a rhombus.
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If one diagonal of a parallelogram bisects a pair of opposite angles, then the
parallelogram is a rhombus.
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If one angle of a parallelogram is a right angle, then the parallelogram is a
rectangle.
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If the diagonals of a parallelogram are congruent, then the parallelogram is a
rectangle.
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If a quadrilateral is an isosceles trapezoid, then its base angles are congruent.
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A trapezoid is isosceles if and only if its diagonals are congruent.

If a quadrilateral is a kite, then one pair of opposite angles are congruent and
one diagonal bisects a pair of opposite angles.

If a quadrilateral is a kite, then its diagonals are perpendicular and one diagonal
bisects the other (but not both).
Chapter 9
~ Prect = 2l + 2w
~ Arect = lw
(or bh)
~ Psq = 4s
~ Asq = s2
~ Apar = bh
~ Atri = ½bh
~ Atrap = ½h(b1 + b2)
~ Arhom = ½d1d2
~ Akite = ½d1d2
~ C = 2πr (or C = dπ)
~ Acir = πr2
~ Areg polygon = ½ap
~ Lateral Area of a prism
LAprism = Ph P = perimeter of the base and h = height of the prism
~ Surface Area of a prism
SAprism = LA + 2B LA = lateral area and B = area of the base
~ Volume of a rectangular prism
Vrect prism = lwh l = length of base, w = width of base, h = height of prism
~ Volume of any prism
Vprism = Bh B = area of base, h = height of prism
Chapter 11
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If a line is tangent to a circle, then it is perpendicular to eh radius drawn to the
point of tangency.
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If a line is perpendicular to a radius of a circle at a point on the circle, then the
line is tangent to the circle.
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If two segments are tangent to a circle from the same exterior point, then the
segments are congruent.
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The measure of an inscribed angle is half the measure of its intercepted arc.

If inscribed angles of a circle intercept the same arc, then the angles are
congruent.

If a tangent and a chord (or secant) intersect at a point on a circle, then the
measure of the angle formed is half the measure of the intercepted arc.

If two chords intersect in the interior of a circle, then the measure of each
angle formed is half the sum of the measures of its intercepted arcs.

If a tangent and a secant, two tangents or two secants intersect at a point in
the exterior of a circle, then the measure of the angle formed is half the
difference of the measures of the intercepted arcs.
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~ Arc length =
m
2 r
360
1
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m = arc measure (degrees)
r = radius
~ Area of a sector =
m
360

r 2
1
m = arc measure (degrees)
r = radius