Points, Lines, and Planes

Undefined Terms in Geometry

Point – a location that has neither size nor shape

Line- Made up of points and has no thickness or width. There is exactly

one line through any two points.

Plane – a flat surface made up of points that extend infinitely in all

directions. There is exactly one plane through any three noncolinear

points.

Coplaner – points on the same plane

Colinear – points on the same line

Two lines intersect in exactly one point.

Two planes intersect in exactly one line.

Distance between two points on the number line |𝑥1 − 𝑥2|

Distance Formula – d = √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2

Midpoint Formula (𝑋1+𝑥2

2,

𝑦1+𝑦2

2)

Right Angle – an angle whose measure is 90

Acute Angle – an angle whose measure is greater than 0 but less than

90

Obtuse Angle – an angle whose measure is greater than 90 but less

than 180

Adjacent angles are two angles that lie in the same plane and have a

common side but no common interior points

A linear pair is a pair of adjacent angles with noncommon sides that are

opposite rays.

Vertical angles are two angles formed by two intersecting lines.

Complementary angles are two angles whose sum is 90

Supplementary angles are two angles whose sum is1 80

Reasoning and Proof

Inductive reasoning is reasoning that uses a number of specific

examples to arrive at a conclusion. When you assume that an observed

pattern will continue, you are applying inductive reasoning. A

concluding statement reached using inductive reasoning is called

conjecture. To show that a conjecture is true for all cases, you must

prove it. It only takes one false example, however, to show that a

conjecture is not true. This false statement is called a counterexample,

and it can be a number, a drawing, or a statement.

A statement is a sentence that is either true (T) or false (F) is called the

truth value often represented using a letter such as p or q. The negation

of a statement has the opposite meaning, and well as the opposite truth

value which is represented by ~p (not p). A conjunction is a compound

statement by joining two or more statements with the word and (⋀). A

disjunction is a compound statement by joining two or more statements

with the word or (∨).

An if-then statement is of the form if p, then q. (p→q). The hypothesis

of a conditional statement immediately following the word if (p). The

conclusion of a conditional statement immediately following the word

then (q). The converse is formed by exchanging the hypothesis and the

conclusion q→p. The inverse is formed by negating both the hypothesis

and the conclusion ~p→ ~𝑞. The contrapositive is formed by negating

both the hypothesis and conclusion of the converse. A conditional and

its contrapositive are logically equivalent. The converse and the inverse

are logically equivalent.

Deductive reasoning uses facts, rules, definitions, or properties to reach

a logical conclusion from the given statements. Law of Detachment

p→q, p is true, q is true. Law of Syllogism p→ 𝑞, 𝑞 → 𝑟, 𝑝 → 𝑟.

Through any two points, there is exactly one line.

Through any three noncollinear points, there is exactly one plane

A plane contains at least three noncollinear points.

If two points lie in a plane, then the entire line containing those

points lies in that plane.

If two lines intersect, their intersection is exactly one point.

If two planes intersect, their intersection is a line.

If M is the midpoint of AB, then AM ≅MB

Reflexive Property of Congruence AB ≅ AB

Symmetric Property of Congruence If AB≅ 𝐶𝐷, 𝑡ℎ𝑒𝑛 𝐶𝐷 ≅ 𝐴𝐵.

Transitive Property of Congruence If AB ≅ 𝐶𝐷, 𝑎𝑛𝑑 𝐶𝐷 ≅

𝐸𝐹, 𝑡ℎ𝑒𝑛 𝐴𝐵 ≅ 𝐸𝐹

Angles supplementary to the same angle or to congruent angles

are congruent.

Angles complementary to the same angle or to congruent angles

are congruent.

If two angles are vertical angles, then they are congruent

Perpendicular lines that intersect to form four right angles

All right angles are congruent

Perpendicular lines form congruent adjacent angles

If two angles are congruent and supplementary, then each angle is

a right angles

If two congruent angles form a linear pair, then they are right

angles.

Parallel Lines and Transversals

Parallel lines are coplanar lines that do not intersect. Skew lines

are lines that do not intersect and are not coplanar. Parallel

planes are planes that do not intersect.

Interior angles ∠3, ∠4, ∠5, ∠6 .

Exterior angles ∠1, ∠2, ∠7, ∠7

Consecutive interior angles are interior angles on the same side of the

transversal ∠3 𝑎𝑛𝑑 ∠5, ∠4 𝑎𝑛𝑑 ∠6

87

65

43

21

Alternative interior angles are nonadjacent interior angles that lie on

opposite sides of the transversal ∠ 3 𝑎𝑛𝑑 ∠6, 𝑎𝑛𝑑 ∠4 𝑎𝑛𝑑 ∠5

Alternate exterior angles are nonadjacent exterior angles that lie on

opposite sides of the transversal ∠ 1 𝑎𝑛𝑑 ∠8, 𝑎𝑛𝑑 ∠2 𝑎𝑛𝑑 ∠7

Corresponding angle lie on the same side of the transversal and on the

same side of the parallel lines. ∠ 1 𝑎𝑛𝑑 ∠5, 𝑎𝑛𝑑 ∠2 𝑎𝑛𝑑 ∠6, ∠ 3 𝑎𝑛𝑑 ∠7,

𝑎𝑛𝑑 ∠4 𝑎𝑛𝑑 ∠8

If two parallel lines are cut by a transversal, then each pair of

corresponding angles are congruent

If two parallel lines are cut by a transversal, then each pair of

consecutive interior angles are supplementary

If two parallel lines are cut by a transversal, then each pair of

alternate exterior angles is congruent

If the corresponding angles are congruent, then the lines are

parallel

If the consecutive interior angles are supplementary, then the

lines are parallel

If the alternate exterior angles are congruent, then the lines are

parallel

Slope intercept form y = mx + b

Point slope form y = 𝑦1 + 𝑚(𝑥 − 𝑥1)

If the slope of two lines are congruent, then the lines are parallel

If the slopes of two lines are the negative reciprocals of each other,

then the lines are perpendicular.

Congruent Triangle

Acute triangles have three acute angles

Equiangular triangles have three congruent acute angles

Obtuse triangles have one obtuse angle

Right triangles have one right triangle

Equilateral triangles have three congruent sides

Isosceles triangles have at least two congruent sides

Scalene triangles have no congruent sides

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

The measure of an exterior angle of a triangle is equal to the sum

of the measure of the two remote interior angles

The acute angles of a right triangle are complementary

There can be at most one right or obtuse angle in a triangle.

Two polygons are congruent if and only if their corresponding

parts are congruent

If two angles of one triangle are congruent to two angles of a

second triangle, then the third angles of the triangles are

congruent

If three sides of one triangle are congruent to three sides of a

second triangle, then the triangles are congruent (SSS)

If two sides and the included angle of one triangle are congruent

to two sides and the included angle of a second triangle, then the

triangles are congruent (SAS)

If two angles and the included side of one triangle are congruent

to two angles and the included side of a second triangle, then the

triangles are congruent (ASA)

If two angles and the nonincluded side of one triangle are

congruent to the corresponding angles and side of a second

triangle, then the two triangles are congruent (AAS).

In and isosceles triangle the two congruent sides are called the

legs of the triangle, and the angle with sides that are the legs is

called the vertex angle. The side opposite the vertex angle is

called the base. The two angles formed by the base and the

congruent sides are called the base angles.

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

those sides are congruent.

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

those angles are congruent.

Relationships in Triangles

Perpendicular bisector-circuumcenter. The circumcenter P 0f

∆ABC is equidistant from each vertex.

Angle Bisector – Incenter- The incenter Q of ∆ABC is equidistance from

each side of the triangle.

Median – centroid – The center R of ∆ABC is two thirds of the distance

from each vertex to the midpoint of the opposite side

Altitude – orthocenter- The lines of containing the altitudes of ∆ABC

are concurrent at the orthocenter S.

P

C

B

A

Q

C

B

A

R

D

C

BA

S

C

B

A

If one side of a triangle is longer than another side then the angle

opposite the longer side has a greater measure than the angle

opposite the shorter side.

If one angle of a triangle has a greater measure than another,

then the side opposite the greater angle is longer than the side

opposite the lesser angle.

The sum of the lengths of any two sides of a triangle must be

greater than the length of the third side

If two sides of a triangle are congruent to two sides of another

triangle and the included angle of the first is larger than the angle

of the second triangle, then the third side of the first triangle is

longer than the third side of the second triangle.

If two sides of a triangle are congruent to two sides of another

triangle, and the third side in the first triangle is longer than the

third side in the second triangle, then the included angle measure

of the first triangle is greater than the included angle measure in

the second triangle.

Quadrilaterals

The sum of the interior angles of an n-sided convex polygon is

180(n – 2) where n is the number of sides.

The sum of the exterior angle measures of a convex polygon, one

at each vertex, is 360

Parallelograms

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

congruent

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

congruent

If a quadrilateral is a parallelogram, then its consecutive angles

are supplementary

If a parallelogram has one right triangle, then it has four right

triangles.

If a quadrilateral is a parallelogram, then its diagonals bisect each

other

If a quadrilateral is a parallelogram, then each diagonal separates

the parallelogram into two congruent triangles

If both pairs of opposite sides of a quadrilateral are congruent,

then the quadrilateral is a parallelogram.

If both pairs of opposite angles of a quadrilateral are congruent,

then the quadrilateral is a parallelogram.

If the diagonals of a parallelogram bisect each other, then the

quadrilateral is a parallelogram.

If one pair of opposite sides or a quadrilateral is both parallel and

congruent, then the quadrilateral is a parallelogram.

Rectangles

All four angles are right angles

Opposite sides are parallel and congruent

Opposite angles are congruent

Consecutive angles are supplementary

Diagonals bisect each other

Diagonals are congruent

Rectangles are parallelograms

Rhombi

Diagonals are perpendicular

Diagonals bisect a pair of opposite angles

All sides are congruent

Rhombi are parallelograms

Square

Four congruent sides

Four congruent angles

Squares are parallelograms

Trapezoids

Bases are parallel

nonparallel sides are legs

If the base angles are congruent, then the trapezoid is an isosceles

trapezoid

If a trapezoid is isosceles, then each pair of base angles is

congruent

If a trapezoid has one pair of congruent base angles, then it is an

isosceles trapezoid

A trapezoid is isosceles if and the diagonals are congruent

The midsegment of a trapezoid is the segment that connects the

midpoints of the legs of a trapezoid and it is equal to one-half the

sums of the lengths of the bases.

Kite

If a quadrilateral is a kite, then its diagonals are perpendicular

If a quadrilateral is a kite, then exactly one pair of opposite angles

is congruent

Proportions and Similarity

If 𝑎

𝑏=

𝑐

𝑑, then ad = bc

Similar polygons have the same size and shape

The ratio of the lengths of the corresponding sides of two similar

polygons is called the scale factor

If two angles of one triangle are congruent to two angles of

another triangle, then the triangles are similar (AA Similarity)

If the corresponding side length of two triangles are proportional,

then the triangles are similar (SSS Similarity)

If the lengths of two sides of one triangle are proportional to the

lengths of two corresponding sides of another triangle and the

included angles are congruent, then the triangles are similar (SAS

Similarity)

If a line is parallel to one side of a triangle and intersects the

other two sides, then it divides the sides into segments of

proportional lengths

If a line intersects two sides of a triangle and separates the sides

into proportional corresponding segments, then the line is parallel

to the third side of the triangle.

A midsegment of a triangle is parallel to one side of the triangle,

and its length is one half the sum of the lengths of the bases

If three or more parallel lines intersect two transversals, then

they cut off the transversals proportionally

If three or more parallel lines cut off congruent segments on one

transversal, then they cut off congruent segments on every

transversal.

If two triangles are similar, the lengths of corresponding altitudes

are proportional to the lengths of corresponding sides.

If two triangles are similar, the lengths of corresponding angle

bisectors are proportional to the lengths of corresponding sides.

If two triangles are similar, the lengths of corresponding medians

are proportional to the lengths of corresponding sides.

Right Triangles and Trigonometry

𝐴𝐷

𝐵𝐷=

𝐵𝐷

𝐷𝐶

𝐴𝐵

𝐴𝐷=

𝐴𝐶

𝐴𝐵

𝐵𝐶

𝐶𝐷=

𝐴𝐶

𝐵𝐶

a2 + b2 = c2 where a and b are the legs of a right triangle and c is

the hypotenuse

a2 + b2 = c2 then the triangle is a right triangle

If c2 < a2 + b2, then the triangle is an acute triangle

If c2 > a2 + b2, then the triangle is an obtuse triangle

Sine = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

Cosine = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

DC

B

A

2

1

45

45

2

1

45

45

Tangent = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

An angle of elevation is the angle formed by a horizontal line and

an observer’s line of sight to an object above the horizontal line

An angle of depression is the angle formed by a horizontal line and

an observer’s line of sight to an object below the horizontal line

Transformations

A reflection is a flip over a line called the line of reflection.

A reflection in y = x is (x, y)→(y, x)

A reflection in y = -x is (x, y)→ (-y, -x)

A translation or slide that moves all points of the original figure

the same distance in the same direction (x, y)→(x + h, y + k)

A rotation or turn is a transformation around a fixed point called

the center of rotation, through a specific angle, and in a specific

direction. Each point of the original figure and its image are the

same distance from the center.

Rotating 90 counterclockwise (x, y) → (−𝑦, 𝑥)

Rotating 180 counterclockwise (x, y) →(-x, -y)

Rotating 270 counterclockwise (x, y)→ (𝑦, − 𝑥)

Line symmetry is a reflection in a line

Point symmetry is if an object rotates on itself 180

Rotational symmetry is if you rotate an object and it rotates on

itself in a 360 turn

Order is the number of times you rotate on object on itself in a 360

turn

Magnitude =360

𝑜𝑟𝑑𝑒𝑟

If k>1 the dilation is an enlargement

if k < 1 the dilation is a reduction

Circles

C = 2𝜋𝑟 or C = 𝜋𝑑

Arc length = 𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑎𝑛𝑔𝑙𝑒

360∙ 𝜋𝑑

If a diameter of a circle is perpendicular to a chord, then it bisects

the chord and its arc

The perpendicular bisector of a chord is a diameter of the circle

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

congruent if and only if they are equidistant from each other

If and angle is inscribed in a circle, then the measure of the angle

equals one half the measure of its intercepted arc

An inscribed angle of a triangle intercepts a diameter or semicircle

if and only if the angle is a right angle

If a quadrilateral is inscribes in a circle, then its opposite angles

are supplementary

In a plane, a line is tangent to a circle if and only if it is

perpendicular to a radius drawn to the point of tangency

If two segments from the same exterior point are tangent to a

circle, then they are congruent

If two secants or chords intersect in the interior of a circle, then

the measure of an angle formed is one half the sum of the

measures of the arcs intercepted by the angle and its vertical

angle

If a secant and a tangent intersect at the point of tangency, then

the measures of each angle formed is one half the measures of its

intercepted arc.

If two secants, a secant and a tangent, or two tangents intersect in

the exterior of a circle, then the measure of the angles formed is

one half the difference of the measures of the intercepted arcs.

If two chords intersect in a circle, then the products of the lengths

of the chord segments are equal

If two secants intersect in the exterior of a circle, then the product

of the measures of one secant segment and its external secant

segment is equal to the product of the measures of the other

secant and its external secant segment

If a tangent and a secant intersect in the exterior of a circle, then

the produce of the measures of one secant segment and its

external secant segment is equal to the tangent squared

The standard form of a the equation of a circle at (h, k) and radius

r is (x – h)2 + (x – k)2 = r2

Area of polygons and circles, surface area and volume

Area of a parallelogram = base times height

Area of a triangle = ½ base times the height

Area of a trapezoid = ½ height(base1 + base2)

Area of a kite = ½ diagonal1 times diagonal2

Area of a rhombus = ½ diagonal1 times diagonal2

Area of a circle = 𝜋𝑟2

Area of a circle = 𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑎𝑛𝑔𝑙𝑒

360∙ 𝜋𝑟2

Area of a regular polygon = ½ times the perimeter times the

apothem

Lateral area of a prism = perimeter times the height

Surface area of a prism = perimeter times the height + 2 times the

area of the base

Lateral area of a cylinder = 2𝜋𝑟ℎ

Surface area of a cylinder = 2𝜋𝑟ℎ + 2𝜋𝑟2

Lateral area of a regular pyramid = ½ times the perimeter times

the slant

Surface area of a regular pyramid =½ times the perimeter times

the slant plus the area of the base

Lateral area of a cone = 𝜋 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑟𝑎𝑑𝑖𝑢𝑠 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑠𝑙𝑎𝑛𝑡

Surface area of a cone = 𝜋 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑟𝑎𝑑𝑖𝑢𝑠 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑠𝑙𝑎𝑛𝑡 + 𝜋𝑟2

Volume of a prism = Area of the base times the height

Volume of a cylinder = ℎ

Volume of a pyramid = 1/3 times the area of the base times the

height

Volume of a come = 1/3 𝜋𝑟2h

Surface area of a sphere = 4 𝜋𝑟2

Volume of a sphere 4/3 𝜋𝑟3

Surface area of a hemisphere = 3 𝜋𝑟2

Volume of a hemisphere = 2/3 𝜋𝑟3

Lengths of a side of solid = a to b, surface area of a solid =

𝑎2 𝑡𝑜 𝑏2 and volume of a solid = 𝑎3 𝑡𝑜 𝑏3