GEOMETRY REVIEW (key terms, postulates, theorems)

1.1: Ruler Postulate

The distance between two points is the absolute value of the difference of their coordinates.

1.2: Segment Addition Postulate

If B is between A and C, AB + BC = AC.

1.3: Protractor Postulate

The measure of an angle is equal to the absolute value of the difference between the real numbers matched with the two sides of the angle on a protractor.

1.4: Angle Addition Postulate

If point P is in the interior of angle RST, then the measure of angle RST is equal to the sum of the measures of angles RSP and PST.

Complementary Angles

2 positive angles (not necessarily adjacent to each other) whose measures have a sum of 90 degrees

Supplementary Angles

2 positive angles (not necessarily adjacent to each other) whose measures have a sum of 180 degrees

Vertical Angles

two angles whose sides form two pairs of opposite rays
- always congruent

Linear Pair

pair of adjacent angles with sum of 180
- supplementary

Properties of Segment Congruence

The Reflexive, Symmetric, and Transitive Properties can be applied to congruent segments

Properties of Angle Congruence

The Reflexive, Symmetric, and Transitive Properties can be applied to congruent angles

Right Angles Congruence Theorem

All right angles are congruent

Congruent Supplements Theorem

If two angles are supplementary to the same angle, then they are congruent

Congruent Complements Theorem

If two angles are complementary to the same angle (or to congruent angles), then they are congruent.

2.1: Two Point Postulate

Through any two points, there exists exactly one line

2.2: Line-Point Postulate

A line contains at least two points. (Converse of the Two Point Postulate)

2.3: Line Intersection Postulate

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

2.4: Three Point Postulate

Through any three noncollinear points, there exists exactly one plane

2.5: Plane-Point Postulate

A plane contains at least three noncollinear points. (Converse of the Three Point Postulate)

2.6: Plane-Line Postulate

If two points lie in a plane, then the line containing them lies in that plane

2.6 Vertical Angles Congruence Theorem

Vertical angles are congruent

2.7 Plane Intersection Postulate

If two planes intersect, then their intersection is a line

2.8: Linear Pair Postulate

If two angles form a linear pair, then they are supplementary

Theorem 3.1: Corresponding Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

3.1: Parallel Postulate

If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line.

Postulate 3.2: Perpendicular Postulate

If there is a line and a point not on the line,
then there is exactly one line through the point
perpendicular to the given line.

Theorem 3.2: Alternate Interior Angles Theorem

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

Theorem 3.3: Alternate Exterior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.

Theorem 3.4: Consecutive Interior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are SUPPLEMENTARY

Theorem 3.5: Corresponding Angles Converse

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

Theorem 3.6: Alternate Interior Angles Converse

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

Theorem 3.7: Alternate Exterior Angles Converse

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

Theorem 3.8: Consecutive Interior Angles Converse

If two lines are cut by a transversal so the
consecutive interior angles are supplementary,
then the lines are parallel.

Theorem 3.9: Transitive Property of Parallel Lines

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

Theorem 3.10: Linear Pair Perpendicular Theorem

If two lines intersect to form a linear pair of
congruent angles, then the lines are perpendicular.

If ∠l ≅ ∠2, then g ⊥ h

Theorem 3.11: Perpendicular Transversal Theorem

In a plane, if a transversal is perpendicular to one
of two parallel lines, then it is perpendicular to the
other line.

If h  k and j ⊥ h, then j ⊥ k.

Theorem 3.12: Lines Perpendicular to a Transversal Theorem

In a plane, if two lines are perpendicular to the
same line, then they are parallel to each other.

If m ⊥ p and n ⊥ p, then m  n.

Theorem 3.13: Slopes of Parallel Lines

In a coordinate plane, two distinct nonvertical lines
are parallel if and only if they have the same slope.
Any two vertical lines are parallel.

Theorem 3.14: Slopes of Perpendicular Lines

In a coordinate plane, two nonvertical lines are
perpendicular if and only if the product of their
slopes is −1 (NEGATIVE RECIPROCOLS)

Horizontal lines are perpendicular to vertical lines.

Vectors

component form:

Translation

moves every point of
a figure in the same direction. (along vector)

Postulate 4.1 Translation Postulate

A translation is a rigid motion. (transformation that preserves length and angle measure)

Theorem 4.1 Composition Theorem

The composition of two (or more) rigid motions is a rigid motion

Reflections

transformation that uses a line like a mirror *line of reflection* to reflect a figure

If (a, b) is reflected in the x axis

then (a, −b)

If (a, b) is reflected in the y-axis

then (−a, b)

If (a, b) is reflected in the line y = x

then (b, a).

If (a, b) is reflected in the line y = −x,

then (−b, −a)

Postulate 4.2 Re flection Postulate

A reflection is a rigid motion (aka congruence
transformation)

glide reflection

a transformation involving a
translation followed by a reflection (a reflection in a line k parallel to the direction of the translation)

Rotation

transformation in which a figure is turned about center of rotation.

Angle of rotation: Rays drawn from the center of rotation to a point and its image

counterclockwise 90
clockwise 270

(-y , x)

counterclockwise/clockwise 180

(-x , -y)

counterclockwise 270
clockwise 90

(y , -x)

Postulate 4.3 Rotation Postulate

A rotation is a rigid motion.

rotational symmetry

when the figure can be mapped
onto itself by a rotation of 180° or less about the center of the figure

Theorem 4.2 Reflections in Parallel Lines Theorem

If lines k and m are parallel, then a refl ection in
line k followed by a refl ection in line m is the
same as a translation.

If A″ is the image of A, then
1. AA—″ is perpendicular to k and m, and
2. AA″ = 2d, where d is the distance
between k and m.

Theorem 4.3 Refl ections in Intersecting Lines Theorem

same as rotation

The angle of rotation is 2x°, where x° is
the measure of the acute or right angle
formed by lines k and m.

dilation

transformation in which a fi gure is enlarged or reduced with respect to center of dilation and a scale factor (ratio)

k > 1, a dilation is an enlargement. When 0 < k < 1, a dilation is a reduction

k = dilated length of point to center/ original length of point to center

Coordinate Rule for Dilations

If P(x, y) is the preimage of a point, then its image
after a dilation centered at the origin (0, 0) with
scale factor k is the point P′(kx, ky).

similarity transformation

dilation OR composition of rigid
motions and dilations.

not always the same side length, but preserves angle measure

distance formula

Theorem 5.1 Triangle Sum Theorem

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

Corollary: The acute angles of a right triangle
are complementary.

Theorem 5.2 Exterior Angle Theorem

measure of an exterior angle = sum of the
measures of the two nonadjacent
interior angles.

Theorem 5.4 Third Angles Theorem

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

SAS congruence theorem

two sides and the included angle of one triangle are congruent to two sides and
the included angle of a second triangle, then the two triangles are congruent.

Theorem 5.6 Base Angles Theorem

two sides of a triangle are congruent = the base angles
are congruent

corollary: If a triangle is equilateral, then it is equiangular.

Theorem 5.7 Converse of the Base Angles Theorem

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

corollary: If a triangle is equiangular, then it is equilateral.

SSS congruence theorem

If three sides of one triangle are congruent to three sides of a second triangle,
then the two triangles are congruent.

Hypotenuse-Leg (HL) Congruence Theorem

If the hypotenuse AND a leg of a RIGHT TRIANGLE (must prove it is a right triangle) = the two triangles are congruent.

ASA CONGRUENCE THEOREM

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

AAS CONGRUENCE THEOREM

If two angles and a non-included side of one triangle are congruent = then the two triangles are congruent

Theorem 6.1 Perpendicular Bisector Theorem

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

Theorem 6.2 Converse of the Perpendicular Bisector Theorem

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

Theorem 6.3 Angle Bisector Theorem

If a point lies on the bisector of an angle, then it is
equidistant from the two sides of the angle (side that forms right angle).

Theorem 6.4 Converse of the Angle Bisector Theorem

If a point is in the interior of an angle and is equidistant
from the two sides of the angle, then it lies on the
bisector of the angle.

circumcenter

The the point of concurrency for perpendicular bisectors

acute= inside
right = midpoint of hypotenuse
obtuse= outside

Theorem 6.5 Circumcenter Theorem

The circumcenter of a triangle is equidistant from
the vertices of the triangle.

Incenter

point of concurrency of angle bisectors
when inscribing, make a circle inside (incircle) using incenter and perpendicular line - point on side

Theorem 6.6 Incenter Theorem

The incenter of a triangle is equidistant
from the sides of the triangle.

centroid

point of concurrency for median (vertex to the midpoint of the opposite side)

Theorem 6.7 Centroid Theorem

The centroid of a triangle is two-thirds of the
distance from each vertex to the midpoint of
the opposite side.

Orthocenter

point of concurrency of altitude (perpendicular segment from a vertex to the opposite side or
to the line that contains the opposite side.)

Theorem 6.8 Triangle Midsegment Theorem

The segment connecting the midpoints of two sides of
a triangle is parallel to the third side and is half as
long as that side

Indirect Proof

Either...or...

Step 1 Identify the statement you want to prove. Assume temporarily that this statement is false by assuming that its opposite is true. (Assume...)

Step 2 Reason logically until you reach a contradiction.

Step 3 Point out that the desired conclusion must be true because the contradiction proves the temporary assumption false.

Theorem 6.9 Triangle Longer Side Theorem

If one side of a triangle is longer than another side,
then the angle opposite the longer side is larger than
the angle opposite the shorter side

Theorem 6.10 Triangle Larger Angle Theorem

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

Theorem 6.11 Triangle Inequality Theorem

The sum of the lengths of ANY two sides of
a triangle IS GREATER than the length of the
third side.

Theorem 6.12 Hinge Theorem

If 2 sides of a triangle is congruent to 2 sides of another..

larger included angle = longer 3rd side
shorter included angle = shorter 3rd side

Theorem 6.13 Converse of the Hinge Theorem

If 2 sides of a triangle is congruent to 2 sides of another..

longer 3rd side = larger included angle
shorter 3rd side = shorter included angle

Theorem 7.1 Polygon Interior Angles Theorem

The sum of the measures of the interior angles
of a convex n-gon is (n − 2) ⋅ 180°.

Corollary: The sum of the measures of the interior angles of a quadrilateral is 360°

Theorem 7.2 Polygon Exterior Angles Theorem

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

Theorem 7.3 Parallelogram Opposite Sides Theorem

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

Theorem 7.4 Parallelogram Opposite Angles Theorem

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

Theorem 7.5 Parallelogram Consecutive Angles Theorem

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

Theorem 7.6 Parallelogram Diagonals Theorem

If a quadrilateral is a parallelogram, then its
diagonals BISECT each other.

Theorem 7.7 Parallelogram Opposite Sides Converse

If BOTH pairs of opposite SIDES of a quadrilateral are
CONGRUENT, then the quadrilateral is a parallelogram.

Theorem 7.8 Parallelogram Opposite Angles Converse

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

Theorem 7.9 Opposite Sides Parallel and Congruent Theorem

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

Theorem 7.10

If the DIAGONALS of a quadrilateral BISECT each other,
then the quadrilateral is a parallelogram.

Corollary 7.2 Rhombus Corollary

A quadrilateral is a rhombus if and only if it has
four congruent SIDES