PROPERTIES OF PARALLELOGRAMS
THEOREM : Opposite sides of a parallelogram are congruent.
THEOREM : Opposite angles of a parallelogram are congruent.
THEOREM : Diagonals of a parallelogram bisect each other.
SPECIAL PARALLELOGRAMS
THREE SPECIAL TYPES OF PARALLELOGRAMS ARE RECTANGLES, RHOMBUSES,
AND SQUARES.
SPECIAL PARALLELOGRAMS
THEOREM : The diagonals of a rectangle are congruent.
THEOREM : The diagonals of a rhombus are perpendicular.
THEOREM : Each diagonal of a rhombus bisects two angles of the rhombus.
DEFINITIONS: PARALLEL LINES AND PLANES; SKEW LINES
THEOREM If two parallel planes are cut by a third plane, then the
lines of intersection are parallel.
TRANSVERSALS
A ''transversal" is a line that intersects 2 or more lines
in different points.
PROPERTIES OF PARALLEL LINES
IF TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL THEN:
POSTULATE : . . . corresponding angles are congruent.
THEOREM : . . . alternate interior angles are congruent.
THEOREM : . . . same-side interior angles are supplementary.
PROPERTIES OF PARALLEL LINES
THEOREM : If a transversal is perpendicular to one of two parallel
lines, then it is perpendicular to the other one also.
PROVING LINES PARALLEL
POSTULATE : If two parallel lines are cut by a transversal, then
corresponding angles are congruent.
POSTULATE : If two lines are cut by a transversal so that corresponding angles
are congruent, then the lines are parallel.
THEOREM : If two lines are cut by a transversal so that alternate interior
angles are congruent, then the lines are parallel.
THEOREM : If two lines are cut by a transversal and same-side interior angles
are supplementary, then the lines are parallel.
THEOREM : In a plane, two lines perpendicular to the same line are parallel.
PROVING LINES PARALLEL
THEOREM : Through a point outside a line, there is exactly one
line parallel to the given line.
THEOREM : Through a point outside a line, there is exactly one line perpendicular
to the given line.
THEOREM : Two lines parallel to a third line are parallel to each other.
WAYS TO PROVE TWO LINES PARALLEL
TRIANGLES: DEFINITION
TRIANGLE: the figure formed by three segments joining three
noncollinear points.
TRIANGLES: CLASSIFICATION
I.Classification by number of congruent sides.
SCALENE: no sides concgruent
ISOSCELES: at least 2 sides congruent.
EQUILATERAL: all 3 sides congruent.
.
II. Classification by angles.
ACUTE: 3 acute angles
OBTUSE: 1 obtuse angle
RIGHT: 1 right angle
EQUIANGULAR: all 3 angles congruent
ANGLES OF A TRIANGLE
THEOREM : TRIANGLE SUM THEOREM: The sum of the measures of the
three angles of a triangle is 180.
ANGLES OF A TRIANGLE: COROLLARIES
COROLLARY : If two angles of one triangle are congruent to two
angles of another triangle, then the third angles are ___________________
.
COROLLARY : Each angle of an equiangular triangle measures __________ .
COROLLARY : In a triangle, there can be at most one ___________ angle or one
____________ angle.
COROLLARY : The acute angles of a right triangle are _________________ .
ANGLES OF A TRIANGLE
THEOREM : EXTERIOR ANGLE SUM THEOREM: The measure of an exterior
angle of a triangle equals the sum of the measures of the remote interior
angles.
ANGLES OF A POLYGON
POLYGON--closed plane figure with segments as sides.
TRIANGLE
QUADRILATERAL
PENTAGON
ANGLES OF A POLYGON
HEXAGON
OCTAGON
ANGLES OF A POLYGON
CONVEX POLYGON
non-convex polygon
ANGLES OF A POLYGON
THEOREM : The sum of the measures of the angles of a convex polygon
with n sides is:
THEOREM: The sum of the measures of the exterior angles of a convex polygon
is always 360.
THEOREMS INVOLVING PARALELL LINES
THEOREM : The segment that joins the midpoint of two sides of a
triangle:
1) Is parallel to the third side.
2) is half as long as the third side
TRAPEZOIDS
DEF. OF TRAPEZOID: A quadrilateral with exactly one pair of parallel
lines.
ISOSCELES TRAPEZOID: A trapeziod with congruent legs.
TRAPEZOIDS
THEOREM : The base angles of an isosceles trapezoid are congruent.
TRAPEZOIDS
THEOREM : The median of a trapezoid:
1) is parallel to the bases.
2) has a length equal to the average of the base lengths
SLOPE OF A LINE
SLOPE OF A LINE
SLOPE FACTS.
* Lines with positive slope rise to the right.
* Lines with negative slope fall to the right.
* The greater the absolute value of the slope, the steeper the line.
* Horizontal lines have slopes of zero.
* Vertical lines have slopes that are undefined.
PARALLEL AND PERPENDICULAR LINES
THEOREM : Two nonvertical lines are parallel if and only if their
slopes are equal.
THEOREM : Two nonvertical lines are perpendicular if and only if the product
of their slopes is -1.
THEOREM : THE MIDPOINT FORMULA
