GEOMETRY 1



dimensions

Point, Line, Plane and Solid

A Point has no dimensions, only position
A Line is one-dimensional
A Plane is two dimensional (2D)
A Solid is three-dimensional (3D)









Acute angle:


An angle whose measure is less than 90 degrees. The following is an acute angle.

Acute-angle-image

Right angle:

An angle whose measure is 90 degrees. The following is a right angle.

Right-angle-image

Obtuse angle:

An angle whose measure is bigger than 90 degrees but less than 180 degrees. Thus, it is between 90 degrees and 180 degrees. The following is an obtuse angle.

Obtuse-angle-image

Straight angle

An angle whose measure is 180 degrees.Thus, a straight angle look like a straight line. The following is a straight angle.

Straight-angle-image

Reflex angle:

An angle whose measure is bigger than 180 degrees but less than 360 degrees.The following is a reflex angle.Reflex-angle-image

Adjacent angles:

Angle with a common vertex and one common side. <1 and <2, are adjacent angles. Adjacent-angle-image

Complementary angles:

Two angles whose measures add to 90 degrees. Angle 1 and angle 2 are complementary angles because together they form a right angle.

Note that angle 1 and angle 2 do not have to be adjacent to be complementary as long as they add up to 90 degrees

complemenetary-angle-image




These two angles (40° and 50°) areComplementary Angles, because they add up to 90°.

Notice that together they make a right angle.



But the angles don't have to be together.

These two are complementary because 27° + 63° = 90°



Supplementary angles:

Two angles whose measures add to 180 degrees. The following are supplementary angles.

Supplementary-angle-image

Supplementary Angles

Two Angles are Supplementary if they add up to 180 degrees.


These two angles (140° and 40°) are Supplementary Angles, because they add up to 180°.

Notice that together they make a straight angle.



But the angles don't have to be together.

These two are supplementary because 60° + 120° = 180°


Vertical angles:

Angles that have a common vertex and whose sides are formed by the same lines. The following(angle 1 and angle 2) are vertical angles.

Verical-angle-image

When two parallel lines are crossed by a third line(Transversal), 8 angles are formed. Take a look at the following figure

Transverlines-image

Angles 3,4,5,8 are interior angles

Angles 1,2,6,7 are exterior angles

Alternate interior angles:

Pairs of interior angles on opposite sides of the transversal.

For instance, angle 3 and angle 5 are alternate interior angles. Angle 4 and angle 8 are also alternate interior angles.

Alternate exterior angles:

Pairs of exterior angles on opposite sides of the transversal.

Angle 2 and angle 7 are alternate exterior angles.

Corresponding angles:

Pairs of angles that are in similar positions.

Angle 3 and angle 2 are corresponding angles.

Angle 5 and angle 7 are corresponding angles


 types of angle



Reflexive PropertyA quantity is congruent (equal) to itself.  a = a 
Symmetric PropertyIf a = b, then b = a.
Transitive PropertyIf a = b and b = c, then a = c.
Addition PostulateIf equal quantities are added to equal quantities, the sums are equal.
Subtraction PostulateIf equal quantities are subtracted from equal quantities, the differences are equal.
Multiplication PostulateIf equal quantities are multiplied by equal quantities, the products are equal.  (also Doubles of equal quantities are equal.)
Division PostulateIf equal quantities are divided by equal nonzero quantities, the quotients are equal. (also Halves of equal quantities are equal.)
Substitution PostulateA quantity may be substituted for its equal in any expression.
Partition PostulateThe whole is equal to the sum of its parts.
Also:  Betweeness of Points:  AB + BC = AC
Angle Addition Postulate:  m
ConstructionTwo points determine a straight line.
 
ConstructionFrom a given point on (or not on) a line, one and only one perpendicular can be drawn to the line.

Angles:
 

Right AnglesAll right angles are congruent.
 
Straight AnglesAll straight angles are congruent.
 
Congruent SupplementsSupplements of the same angle, or congruent angles, are congruent.
Congruent ComplementsComplements of the same angle, or congruent angles, are congruent. 
Linear PairIf two angles form a linear pair, they are supplementary.
 
Vertical AnglesVertical angles are congruent.
 
Triangle SumThe sum of the interior angles of a triangle is 180º.
 
Exterior AngleThe measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.
The measure of an exterior angle of a triangle is greater than either non-adjacent interior angle.
Base Angle Theorem
(Isosceles Triangle)
If two sides of a triangle are congruent, the angles opposite these sides are congruent.
Base Angle Converse
(Isosceles Triangle)
If two angles of a triangle are congruent, the sides opposite these angles are congruent.

Triangles:
 

Side-Side-Side (SSS) CongruenceIf three sides of one triangle are congruent to three sides of  another triangle, then the triangles are congruent.
Side-Angle-Side (SAS) CongruenceIf two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
Angle-Side-Angle (ASA) CongruenceIf two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
Angle-Angle-Side (AAS) CongruenceIf two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
Hypotenuse-Leg (HL) Congruence (right triangle)If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the two right triangles are congruent.
CPCTCCorresponding parts of congruent triangles are congruent.
Angle-Angle (AA) SimilarityIf two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
SSS for SimilarityIf the three sets of corresponding sides of two triangles are in proportion, the triangles are similar.
SAS for SimilarityIf an angle of one triangle is congruent to the corresponding angle of another triangle and the lengths of the sides including these angles are in proportion, the triangles are similar.
Side ProportionalityIf two triangles are similar, the corresponding sides are in proportion.
Mid-segment Theorem
(also called mid-line)
The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long.
Sum of Two Sides

The sum of the lengths of any two sides of a triangle must be greater than the third side

Longest SideIn a triangle, the longest side is across from the largest angle.
In a triangle, the largest angle is across from the longest side.
Altitude RuleThe altitude to the hypotenuse of a right triangle is the mean proportional between the segments into which it divides the hypotenuse. 
Leg RuleEach leg of a right triangle is the mean proportional between the hypotenuse and the projection of the leg on the hypotenuse.

Parallels:

Corresponding AnglesIf two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
Corresponding Angles ConverseIf two lines are cut by a transversal and the corresponding angles are congruent, the lines are parallel.
Alternate Interior Angles
 
If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.
Alternate Exterior AnglesIf two parallel lines are cut by a transversal, then the alternate exterior angles are congruent.
Interiors on Same SideIf two parallel lines are cut by a transversal, the interior angles on the same side of the transversal are supplementary.
Alternate Interior Angles 
Converse
If two lines are cut by a transversal and the alternate interior angles are congruent, the lines are parallel.
Alternate Exterior Angles
Converse
If two lines are cut by a transversal and the alternate exterior angles are congruent, the lines are parallel.
Interiors on Same Side ConverseIf two lines are cut by a transversal and the interior angles on the same side of the transversal are supplementary, the lines are parallel.
 

Quadrilaterals:

Parallelograms

 

 

 


About Sides
 
* If a quadrilateral is a parallelogram, the opposite
   sides are parallel.
* If a quadrilateral is a parallelogram, the opposite
   sides are congruent.
About Angles* If a quadrilateral is a parallelogram, the opposite
   angles are congruent.
If a quadrilateral is a parallelogram, the
   consecutive angles are supplementary.
About DiagonalsIf a quadrilateral is a parallelogram, the diagonals
   bisect each other.
* If a quadrilateral is a parallelogram, the diagonals
   form two congruent triangles.
Parallelogram Converses

 

 

 

 



About Sides
 
If both pairs of opposite sides of a quadrilateral
   are parallel, the quadrilateral is a parallelogram.
* If both pairs of opposite sides of a quadrilateral
   are congruent, the quadrilateral is a
   parallelogram.
About Angles* If both pairs of opposite angles of a quadrilateral
   are congruent, the quadrilateral is a 
   parallelogram.
* If the consecutive angles of a quadrilateral are
 supplementary, the quadrilateral is a parallelogram.
About Diagonals

 

If the diagonals of a quadrilateral bisect each
   other, the quadrilateral is a 
   parallelogram.
* If the diagonals of a quadrilateral form two
   congruent triangles, the quadrilateral is a
   parallelogram.
ParallelogramIf one pair of sides of a quadrilateral is BOTH parallel and congruent, the quadrilateral is a parallelogram.
RectangleIf a parallelogram has one right angle it is a rectangle
A parallelogram is a rectangle if and only if its diagonals are congruent.
A rectangle is a parallelogram with four right angles.
RhombusA rhombus is a parallelogram with four congruent sides.
If a parallelogram has two consecutive sides congruent, it is a rhombus.
A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.
A parallelogram is a rhombus if and only if the diagonals are perpendicular.
SquareA square is a parallelogram with four congruent sides and four right angles.
A quadrilateral is a square if and only if it is a rhombus and a rectangle.
TrapezoidA trapezoid is a quadrilateral with exactly one pair of parallel sides.
Isosceles TrapezoidAn isosceles trapezoid is a trapezoid with congruent legs.
A trapezoid is isosceles if and only if the base angles are congruent
A trapezoid is isosceles if and only if the diagonals are congruent
If a trapezoid is isosceles, the opposite angles are supplementary.

Circles:

RadiusIn a circle, a radius perpendicular to a chord bisects the chord and the arc.
In a circle, a radius that bisects a chord is perpendicular to the chord.
In a circle, the perpendicular bisector of a chord passes through the center of the circle.
If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency.
Chords

In a circle, or congruent circles, congruent chords are equidistant from the center. (and converse)

In a circle, or congruent circles, congruent chords have congruent arcs. (and converse0
In a circle, parallel chords intercept congruent arcs
In the same circle, or congruent circles, congruent central angles have congruent chords (and converse)
TangentsTangent segments to a circle from the same external point are congruent
ArcsIn the same circle, or congruent circles, congruent central angles have congruent arcs. (and converse)
AnglesAn angle inscribed in a semi-circle is a right angle.

In a circle, inscribed angles that intercept the same arc are congruent.

The opposite angles in a cyclic quadrilateral are supplementary
In a circle, or congruent circles, congruent central angles have congruent arcs.