MATH

KINDS OF QUADRILATERALS

Quadrilaterals

• can be grouped according to the number of parallel sides.

Convex Quadrilateral

• Diagonals intersect

Non-Convex Quadrilateral.

• Diagonals do not intersect.

Trapezium

• is a quadrilateral with no parallel sides.

Kite

• A special kind of trapezium. It is a quadrilateral with exactly two pairs of distinct congruent consecutive sides.

Trapezoid

• is a quadrilateral with exactly one pair of parallel sides.

Altitude

• ```any segment from a point on one base perpendicular to the line contains the other base.

Median or Midline

• is the segment that joins the midpoints of the legs.

Bases

• the parallel sides in a trapezoid.

Legs

• The nonparallel sides.

There are two kinds of trapezoids:

• isosceles trapezoid

• non-isosceles trapezoids.

Isosceles trapezoid

• is a trapezoid with two congruent legs

Non-isosceles trapezoid

• has no congruent legs.

Parallelogram

• is a quadrilateral where two pairs of sides are parallel and congruent.

Rectangle

• A parallelogram with four right angles.

Rhombus

• is a parallelogram with four congruent sides.

Square

• is a special kind of parallelogram because it is considered as a rectangle. It has four right angles and four equal sides.

CHARACTERISTICS OF A PARALLELOGRAM

• Opposite sides of a parallelogram are parallel.

• Opposite sides of a parallelogram are congruent.

• Opposite angles of a parallelogram are congruent.

• Any two consecutive angles of a parallelogram are supplementary.

PROPERTIES OF PARALLELOGRAMS INVOLVING DIAGONALS

1. The diagonals of parallelogram bisect each other. Each diagonal divides a parallelogram into two congruent triangles.

2. The diagonals of a rectangle are congruent and they bisect each other.

3. In a rhombus, the diagonals are perpendicular and they bisect each other.

4. The diagonals of a square bisect each other and are congruent and perpendicular.

THEOREMS ON DIFFERENT KINDS OF PARALLELOGRAMS

1. In a parallelogram, opposite sides are congruent.

2. in a parallelogram, opposite angles are congruent.

3. Any two consecutive angles in a parallelogram are supplementary.

4. The diagonals of a parallelogram bisect each other.

5. A diagonal of a parallelogram divides the parallelogram into two congruent triangles.

6. If a parallelogram has a right angle, then it has four right angles and the parallelogram is a rectangle.

7. The diagonals of a rectangle are congruent.

8. Each diagonal of a rhombus bisects the angles of a rhombus.

9. In a rhombus, the diagonals are perpendicular to each other

The Midline Theorem

• The segment that joins the midpoints of a triangle is parallel to the third side and half as long.

Midline = Base

THEOREMS ON TRAPEZOIDS

1. The median of a trapezoid is parallel to its bases.

2. The median of a trapezoid is equal to half of the lengths of the bases.

THEOREMS ON ISOSCELES TRAPEZOIDS

1. The base angles of an isosceles trapezoid are congruent.

2. Opposite angles of an isosceles trapezoid are supplementary.

3. The diagonals of an isosceles trapezoid are congruent.

THEOREMS ON KITES

1. In a kite, the perpendicular bisector of one diagonal is the other diagonal.

2. The area of a kite is half the product of the lengths of its diagonals.

PROPORTION

• is simply two or more ratios that are equated with each other.

Similar to ratios, proportions can be written as:

a:b=c:d

"a is to b as c is to d'

We call the outer terms as extremes, while the inner terms called means

FUNDAMENTAL THEOREMS OF PROPORTIONALITY

Cross Products Property

• The first one is called cross products property. This states that the product of the means is equal to product of the extremes:

Ex:

(4)(10) = (5)(8)

40 = 40

Alternation Property

• The next is called alternation property, or the switching of the means or the extremes;

to

Ex:

,

(14)(4) = (7)(8)

56 = 56

Inverse Property

• We also have what we call the inverse property, or the switching of the numerator and the denominator:

to

Ex:

,

(100)(3) = (60)(5)

300 = 300

Addition Property

• Lastly, we have the additional property. This is when we add the denominators to the numerators and we still get the same proportion:

to

Ex:

,

(9)(10) = (5)(18)

90 = 90

SAS Similarity Theorem (Side-Angle-Side)

• 2 sides and an included angle.

ASA Postulate (Angle-SideAngle)

• 2 angles and an included side.

SAA Postulate (Side-Angle-Side)

• Non included side with two consecutive angles.

SSS Similarity Theorem (Side-Side-Side)

• If the sides of one triangle are proportional to the corresponding sides of another triangle, then the triangles are similar.

AA Similarity Theorem (Angle-Angle)

• If two angles of one triangle are congruent to the two angles of another triangle, then the two triangles are congruent.

Pythagorean Theorem

a² + b² = c²

a and b are the legs of the triangle and c is the hypotenuse.