Geometry Units #1-6

Properties of a Parallelogram

(5 properties)

1. Both pairs of opposite sides are parallel (Definition)

2. Both pairs of opposite sides are congruent

3. Both pairs of opposite angles are congruent

4. Each pair of consecutive angles is supplementary

5. The diagonals bisect each other

A quadrilateral can be proven to be a parallelogram by these theorems

(6 theorems)

1. If both pairs of opposite sides of a quadrilateral are parallel, then it is a parallelogram. (Definition reversed)

2. If both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram.

3. If an angle of a quadrilateral is supplementary to both of its consecutive angles, then it is a parallelogram.

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

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

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

Properties of a Rectangle

(3 properties; 7 in total)

1. All properties of a parallelogram (5)

2. All angles are right angles

3. Diagonals are congruent

A quadrilateral can be proven to be a rectangle by these theorems

(3 theorems)

1. If a quadrilateral is a parallelogram with at least one right angle, then it is a rectangle. (Definition reversed)

2. If a quadrilateral is a parallelogram with congruent diagonals, then it is a rectangle.

3. If a quadrilateral has 4 right angles, then it is a rectangle.

Properties of a Rhombus

(4 properties; 8 in total)

1. All the properties of a parallelogram (5)

2. All sides are congruent

3. Diagonals are perpendicular

4. Each diagonal bisects a pair of opposite angles

A quadrilateral can be proven to be a rhombus by these theorems

(5 theorems)

1. If a quadrilateral is a parallelogram with at least two consecutive sides, then it is a rhombus. (Definition reversed)

2. If a quadrilateral is a parallelogram whose diagonals are perpendicular, then it is a rhombus

3. If a quadrilateral is a parallelogram whose diagonal bisects a pair of opposite angles, then it is a rhombus

4. If a quadrilateral whose diagonals are perpendicular bisectors of each other, then it is a rhombus

5. If a quadrilateral has 4 congruent sides, then it is a rhombus

Properties of a Square

(3 properties)

1. All the properties of a parallelogram

2. All the properties of a rectangle

3. All the properties of a rhombus

A quadrilateral can be proven to be a square by this theorem

(1 theorem)

A quadrilateral is a square if and only if it is both a rhombus and a rectangle

Properties of a Kite

(5 properties)

1. A quadrilateral is a kite if and only if it has two disjoint pairs of consecutive sides that are congruent (Definition)

2. One of the diagonals is the perpendicular bisector of the other

3. The diagonals are perpendicular

4. One of the diagonals bisects a pair of opposite angles

5. One pair of opposite angles is congruent.

A quadrilateral can be proven to be a kite by these theorems

(2 theorems)

1. If a quadrilateral has two disjoint pairs of consecutive sides that are congruent, then it is a kite (Definition reversed)

2. If a quadrilateral has one diagonal that is the perpendicular bisector of the other diagonal, then it is a kite.

Properties of an Isosceles Trapezoid

(4 properties)

1. A quadrilateral is an isosceles trapezoid if and only if it is a trapezoid in which the base angles are congruent.

2. The legs are congruent

3. The diagonals are congruent

4. Both pairs of opposite angles are congruent

A quadrilateral can be proven to be an isosceles trapezoid by these theorems

(3 theorems)

A quadrilateral can be proven to be a trapezoid by this theorem

(1 theorem)

1. If a quadrilateral is a trapezoid with congruent base angles, then it is an isosceles trapezoid. (Definition reversed)

2. If a quadrilateral is a trapezoid with congruent diagonals, then it is an isosceles trapezoid.

3. If a quadrilateral is a trapezoid with one p[air of opposite angles that is supplementary, then it is an isosceles trapezoid.

1. If a quadrilateral has at least one pair of opposite sides that is parallel, then it is a trapezoid.

The formula for the sum of the measures of the interior angles of a n-gon is…

Formula = 180(n-2)

The formula for the sum of the measures of the exterior angles of a n-gon is…

Formula = 360

The formula for the measure of each interior angle of an equiangular n-gon is…

Formula = 180(n-2)/n

The formula for the measure of each exterior angle of an equiangular n-gon is…

Formula = 360/n

Circumcenter is when…

• Location in an acute triangle:

• Location in an obtuse triangle:

• Location in a right triangle:

The perpendicular bisectors of the sides of a triangle are concurrent as a point called the circumcenter, that is equidistant from the vertices of the triangle.

• Inside the triangle

• Outside the triangle

• On the midpoint of the hypotenuse

Incenter is when…

Location in an acute triangle:
Location in an obtuse triangle:
Location in a right triangle:

The bisectors of the angles of a triangle are concurrent as a point called the incenter, that is equidistant from the sides of the triangle.

• Inside the triangle

• Inside the triangle

• Inside the triangle

Orthocenter is when…

Location in an acute triangle:
Location in an obtuse triangle:
Location in a right triangle:

The lines containing the altitudes of a triangle are concurrent as a point called the orthocenter of the triangle.

• Inside the triangle

• Outside the triangle

• On the vertex of the right angle of the triangle

Centroid is when…

Location in an acute triangle:
Location in an obtuse triangle:
Location in a right triangle:

The medians of a triangle are concurrent as a point called the centroid, which is 2/3 of the way from any vertex of the triangle to the midpoint of the opposite side.

1. The centroid is also the center of gravity of the triangle.

2. The centroid of a triangle divides each median into two parts so that the distance from the vertex to the centroid is twice the distance from the centroid to the midpoint of the opposite side (2:1 ratio)

3. The coordinates of the centroid of a triangle are the averages of the coordinates of the 3 vertices of the triangle.

• Inside the triangle

• Inside the triangle

• Inside the triangle