Chapter 5 – Theorems, Definitions and Postulates
Parallel Lines:
Definition of Parallel
Two lines are parallel if they lie on the same plane and never intersect.
Parallel Postulate
There is only one line through a given point, parallel to a given line.
Proving lines parallel:
AIC  P
If two lines are cut by a transversal such that the alternate interior angles are congruent, then the
lines are parallel.
AEC  P
If two lines are cut by a transversal such that the alternate exterior angles are congruent, then the
lines are parallel.
CAC  P
If two lines are cut by a transversal such that the corresponding angles are congruent, then the
lines are parallel.
SSIS  P
If two lines are cut by a transversal such that the same side interior angles are supplementary,
then the lines are parallel.
SSES  P
If two lines are cut by a transversal such that the same side exterior angles are supplementary,
then the lines are parallel.
Perpendicular  Parallel Postulate
If two coplanar lines are perpendicular to a third line, then they are parallel.
Angles Associated with Parallel Lines:
P  AIC
If two parallel lines are cut by a transversal, then the alternate interior angles formed by that
transversal are congruent.
P  AEC
If two parallel lines are cut by a transversal, then the alternate exterior angles formed by that
transversal are congruent.
P  CAC
If two parallel lines are cut by a transversal, then the corresponding angles formed by that
transversal are congruent.
P  SSIS
If two parallel lines are cut by a transversal, then the same side interior angles formed by that
transversal are supplementary.
P  SSES
If two parallel lines are cut by a transversal, then the same side exterior angles formed by that
transversal are supplementary.
Parallel  Perpendicular Postulate
In a plane if a lines is perpendicular to one of two parallel lines, it is perpendicular to the other.
Transitive Property of Parallel Lines
If two lines are parallel to a third line, they are parallel to each other.
Angle Theorems:
Exterior Angle Inequality
The measure of an exterior angle of a triangle is greater than the measure of either remote
interior angle.
Sum of the Measures of the Interior Angles of a Triangle
The measures of the interior angles of a triangle sum to 180 degrees.
No Choice Theorem
If two angles of one triangle are congruent to two angles of a second triangle, then the third
angles are congruent.
AAS
If two angles and a non-included side of one triangle are congruent to two angles and the
corresponding non-included side of a second triangle, then the triangles are congruent.
Definitions and Properties of Quadrilaterals and Polygons
Definition of a Polygon
A polygon is a closed, rectilinear, planar figure where any two consecutive sides intersect at their
endpoints and non-consecutive sides do not intersect.
Diagonal of a Polygon
A segment connecting any two non-consecutive vertices of a polynomial is a diagonal of that
polynomial.
Convex Polygon
A polygon in which all interior angles are less than 180 degrees or a polygon in which all
diagonals are in the interior of the polygon.
Definition of a Quadrilateral
A quadrilateral is a four-sided polygon.
Parallelograms
Definition of a Parallelogram
A parallelogram is a quadrilateral with opposite sides parallel.
Properties of a Parallelogram:
(i.e. if a shape is a parallelogram, then: )
1. The opposite sides are congruent
2. The opposite sides are parallel (by definition)
3. The opposite angles are congruent
4. The diagonals bisect each other
5. Any pair of consecutive angles are supplementary
Proving that a quadrilateral is a parallelogram
If a quadrilateral:
1. has both sets of opposite sides parallel, OR
2. has both sets of opposite sides congruent, OR
3. has one set of opposite sides parallel and congruent, OR
4. diagonals bisect each other, OR
5. has both sets of of opposite angles congruent,
Then it is a parallelogram.
Rectangle
Definition of a Rectangle:
A rectangle is a parallelogram in which at least one angle is a right angle.
Properties of a Rectangle:
(i.e. if a shape is a parallelogram, then: )
1. All angles of a rectangle are right angles.
2. The diagonals are congruent.
3. All properties of a parallelogram apply by definition
Proving that a quadrilateral is a rectangle
1. If a parallelogram contains at least one right angle, then it is a rectangle.
2. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
3. If all four angles of a quadrilateral are right angles, then it is a rectangle.
(For the first two you must first prove that the shape is a parallelogram)
Kite
Definition of a Kite:
A kite is a quadrilateral with two sets of consecutive sides congruent. (those two sets cannot
share a side, they must be disjoint).
Properties of a Kite:
(i.e. if a shape is a Kite, then: )
1. The diagonals are perpendicular
2. One diagonal bisects the other.
3. One diagonal bisects a pair of opposite angles
4. One pair of opposite angles is congruent
5. Two disjoint pairs of consecutive sides are congruent by definition.
Proving that a quadrilateral is a kite
1. If two disjoint pairs of consecutive sides of a quadrilateral are congruent, then it is a kite.
2. If one of the diagonals of a quadrilateral is the perpendicular bisector of the other
diagonal, then the quadrilateral is a kite.
Rhombus
Definition of a Rhombus:
A rhombus is a parallelogram with at least one set of consecutive sides congruent.
Properties of a Rhombus:
(i.e. If a shape is a Rhombus, then: )
1. All four sides of a rhombus are congruent
2. The diagonals are perpendicular bisectors of each other
3. The diagonals bisect opposite angles
4. The diagonals create right isosceles triangles
5. All properties of a parallelogram apply.
6. All properties of a kite apply.
Proving that a quadrilateral is a rhombus
1. If a parallelogram contains a pair of consecutive sides congruent, then it is a rhombus.
2. If either diagonal of a parallelogram bisects two angles of the parallelogram, then it is a
rhombus.
3. If the diagonals of a quadrilateral are perpendicular bisectors of each other, then it is a
rhombus.
Square
Definition of a Square:
A square is a parallelogram that is both a rectangle and a rhombus.
Properties of a Square:
(i.e. If a shape is a square, then: )
1. All properties of a rectangle apply by definition
2. All properties of a rhombus apply by definition
3. The diagonals form four isosceles right triangles.
Proving that a quadrilateral is a square
1. If a quadrilateral is both a rectangle and a rhombus, then it is a square.
(so you have to prove it is first a rectangle and a rhombus)
Trapezoid
Definition of a Trapezoid:
A trapezoid is a quadrilateral with one set of opposite sides (called bases) parallel.
Isosceles Trapezoid
Definition of an Isosceles Trapezoid:
An isosceles trapezoid is a quadrilateral with one set of parallel sides, with the non-parallel sides
congruent.
Properties of an Isosceles Trapezoid:
1. The lower base angles are congruent
2. The upper base angles are congruent
3. The diagonals are congruent
4. Any lower base angle and upper base angle are supplementary
Proving that a trapezoid is isosceles
1. If the non-parallel sides of a trapezoid are congruent, then it is isosceles.
Proving that the non-parallel sides of a trapezoid are congruent
1. If the lower or upper base angles of a trapezoid are congruent, then the sides opposite
those angles are congruent.
2. If the diagonals of a trapezoid are congruent, then the non-parallel sides are congruent.