Collinear points are points all in one line. Coplanar points are points all in one plane. The intersection of two figures is the set of points that are in both figures. Statements that are accepted without proof are called postulates or axioms. 1. Any two desired points can have coordinates 0 and 1. 2. The distance between any tow points equals the absolute value of the difference of their coordinates. Congruent segments are segments that have equal lengths. The midpoint of a segment is the point that divides the segment into two congruent segments. A bisector of a segment is a line, segment, ray, or plane that intersects the segment at its midpoint. An angle is a figure formed by two rays that have the same endpoint. The two rays are called the sides of the angle, and their common endpoint is the vertex of the angle. On AB in a given plane, choose any point O between A and B. Consider OA and OB and all the rays that can be drawn from O on one side of AB. Thus rays can be paired with the real numbers from 0 to 180 in such a way that: a. OA is paired with 0, and OB with 180. b. If OP is paired with x, and OQ with y, then m POQ = ?x – y?. Acute angle: Measure between 0 and 90 Obtuse angle: Measure between 90 and 180 If point B lies in the interior of AOC, then If AOC is a straight angle and B is any point not on AC, then Congruent angles are angles that have equal measures. Adjacent angles are two angles in a plane htat have a common vertex and a common side but no common interior points. The bisector of an angle is a ray that divides the angle into two congruent adjacent angles. Addition Property If a = b and c = d, then a + c = b + d. Subtraction Property If a = b and c = d, then a – c = b – d. Multiplication Property If a = b, then ca = cb. Division Property If a = b and c ? 0, then a/c = b/c. Substitution Property If a = b, then either a or b may be substituted for the other in any equation (or inequality).
The Essay on Focal Point Lens Rays Lenses
Light that is emitted from a surface, or reflected from it, leaves the surface in the form of spherical wavefront's. Every point on the surface can be thought of as a source of these wavefront's. Rather than drawing the wavefront's, we customarily illustrate the propagation of light with rays. These are just lines with arrowheads that point in the direction in which the light is traveling. Lenses ...
Symmetric Property If a = b, then b = a. Transitive Property If a = b and b = c, then a = c. Symmetric Property } Same as properties of equality except Distributive Property a(b + c) = ab + ac Statements that are proved are called theorems. If BX is the bisector of ABC, then: 2m ABX = m ABC and m ABX = ?m ABC 2m XBC = m ABC and m XBC = ?m ABC Complementary angles are two angles whose measures have the sum 90. Each angle is called a complement of the other. Supplementary angles are two angles whose measures have the sum 180. Each angle is called a supplement of the other. Vertical angles are two angles whose sides form two pairs of opposite rays. When two lines intersect, they form two pairs of vertical angles. Perpendicular lines (- lines) are two lines that form right angles. Adjacent angles formed by perpendicular lines are congruent. If two lines form congruent adjacent angles, then the lines are perpendicular. If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary. If two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent. If two angles are complements of congruent angles (or of the same angle), then the two angles are congruent.
A line contains at least two points; a plane contains at least three points not all in one line; space contains at least four points not all in one plane. Through any two points there is exactly one line. Through any three points there is at least one plane, and through any three noncollinear points there is exactly one plane. If two points are in a plane, then the line that contains the points is in that plane. If two planes intersect, then their intersection is a line. If two lines intersect, then they intersect in exactly one point. If there is a line and a point not in the line, then exactly one plane contains them. If two lines intersect, then exactly one plane contains them. Parallel lines (|| lines) do not intersect and are coplanar. Skew lines do not intersect and are not coplanar. Parallel planes (|| lines) do not intersect. A line and a plane are parallel if they do not intersect. If two parallel planes are cut by a third plane, then the lines of intersection are parallel. A transversal is a line that intersects two or more coplanar lines in different points. Alternate interior angles are two nonadjacent interior angles on the opposites sides of the transversal. Same-side interior angles are two interior angles on the same side of the transversal.
The Term Paper on Sided Polygon Angle Line Set
... for figures Overlapping triangles - triangles that share a side or angle Parallel lines - two or more coplanar lines that have no points in common ... faces are all parallelograms and congruent (in pairs) Parallel planes - planes that have no points in common Penta decagon - ... two hemispheres; see small circle Grid - a tesselation of congruent squares sometimes used to measure distance Harmonic mean - ...
Corresponding angles are two angles in corresponding positions relative to the two lines. If two parallel lines are cut by a transversal, then corresponding angles are congruent. If two parallel lines are cut by a transversal, then alternate interior angles are congruent. If two parallel lines are cut by a transversal, then same-side interior angles are supplementary. If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other one also. If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel. If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel. If two lines are cut by a transversal and same-side interior angles are supplementary, then the lines are parallel. In a plane, two lines perpendicular to the same line are parallel. Through a point outside a line, there is exactly one line parallel to the given line. Through a point outside a line, there is exactly one line perpendicular to the given line. Two lines parallel to a third line are parallel to each other. A triangle is the figure formed by three segments joining three noncollinear points.
Each of three points is a vertex of the triangle. (The plural of vertex is vertices.) The segments are the sides of the triangle. Scalene triangle Isosceles triangle Equilateral triangle No sides congruent At least two sides congruent All sides congruent Acute triangle Obtuse triangle Right triangle Equiangular triangle Three acute angles One obtuse angle One right angle All angles congruent An auxiliary line is a line (or ray or segment) added to a diagram to help in a proof. The sum of the measures of the angles of a triangle is 180. A statement that can be proved easily by applying a theorem is often called a corollary of the theorem. If two angle of one triangle are congruent to two angles of another triangle, then the third angles are congruent. Each angle of an equiangular triangle has measure 60. In a triangle, there can be at most one right angle or obtuse angle. The acute angles of a right triangle are complementary. When one side of a triangle is extended, an exterior angle is formed. Because an exterior angle of a triangle is always a supplement of the adjacent interior angle of the triangle, its measure is related in a special way to the measure of the other two angle of the triangle, called the remote interior angles.
The Essay on Early Greek Algebra Line Segments
Overview The word algebra came from an Arabic expression, al-jar wa'l muqabala, which was the title of the first Arabic text on algebra. Al-Khwarizmi wrote the book in the ninth century A. D. According to Al-Khwarizmi, algebra was " the art of reducing and solving equations" (van der Waerden 70). From the very beginning of its introduction, algebra was influenced by geometry. For example, the ...
The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles. A polygon is a plane figure formed by coplanar segments (sides) such that (1) each segment intersects exactly two other segments, one at each endpoint; and (2) no two points with a common endpoint are collinear. A convex polygon is a polygon such that no line containing a side of the polygon contains a point in the interior of the polygon. 3 sides: triangle, 4 sides: quadrilateral, 5 sides: pentagon, 6 sides: hexagon, 8 sides: octagon, 10 sides: decagon, n sides: n-gon. A segment joining two nonconsecutive vertices is a diagonal of the polygon. The sum of the measures of the angles of a convex polygon with n sides is (n-2)180. The sum of the measures of the exterior angles of any convex polygon, one angle at each vertex, is 360. If a polygon is both equiangular and equilateral, it is called a regular polygon. Whenever two figures have the same size and shape, they are called congruent. Two triangles are congruent if and only if their vertices can be matched up so that the corresponding parts (angles and sides) of the triangles are congruent. If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. A line and a plane are perpendicular if and only if they intersect and the line is perpendicular to all lines in the plane that pass through the point of intersection. CPCTC – Corresponding parts of congruent triangles are congruent. Recall that an isosceles triangle has two congruent sides. These congruent sides are called are called legs and the third side is called the base. The angles at the base are called base angles and the angle opposite the base is called the vertex angle of the isosceles triangle. Theorem 3-1 The Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those angles are congruent. An equilateral triangle is also equiangular. An equilateral triangle has three 60? angles. The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint. If two angles of a triangle are congruent, then the sides opposite those angles are congruent. An equiangular triangle is also equilateral. If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. In a right triangle the side opposite the right angle is called the hypotenuse (hyp.).
The Essay on The Side Effects Of Breast Implants
In the 1970s women began to look for different options other than padding their bra to get a more voluptuous look. The 70s were an age of a new form of cosmetic surgery called breast implants. These implants became a popular alternative to padding. Serious side effects can result in implants and women should be well aware of these health risks before making a final decision. When you first look ...
The other two sides are called legs. If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent. A median of a triangle is a segment from a vertex to the midpoint of the opposite side. An altitude of a triangle is the perpendicular segment from a vertex to the line that contains the opposite side. A perpendicular bisector of a segment is a line (or ray or segment) that is perpendicular to the segment at its midpoint. If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment. If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment. The distance from a point to a line (or plane) is defined to be the length of the perpendicular segment from the point to the line (or plane).
The Term Paper on The True Side Effects F Sterids part 1
The True Side Effects f Sterids What, exactly, are sterids? Usually called anablic sterids, they are synthetic substances similar in chemical structure t the hrmne teststerne. Teststerne is ften called the male sex hrmne, but females prduce it t. In bth sexes, teststerne prmtes the grwth f skeletal muscles, the muscles that are cnnected t the skeletn and enable a persn t mve. The muscle-grwing ...
If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle. If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle. A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Opposite sides of a parallelogram are congruent. If two lines are parallel, then all points on one line are equidistant from the other line. Opposite angles of a parallelogram are congruent. The diagonals of a parallelogram bisect each other. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. A line that contains the midpoint of one side of a triangle and is parallel to another side bisects the third side.
A quadrilateral with four right angles is a rectangle. Since both pairs of opposite angles are congruent, every rectangle is a parallelogram. A quadrilateral with four congruent sides is a rhombus. Since both pairs of opposite sides are congruent, every rhombus is a parallelogram. A quadrilateral with four right angles and four congruent sides is a square. The diagonals of a rectangle are congruent. The diagonals of a rhombus are perpendicular. Each diagonal of a rhombus bisects two angles of the rhombus. The midpoint of the hypotenuse of a right triangle is equidistant from the three vertices. If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle. If two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. A quadrilateral with exactly one pair of parallel sides is called a trapezoid. The parallel sides are called bases: the other sides are legs. A trapezoid with congruent legs is called isosceles. Base angles of an isosceles trapezoid are congruent. The median of a trapezoid is the segment that joins the midpoints of the legs. (2) has a length equal to half the sum of the lengths of the bases. The segment that joins the midpoints of two sides of a triangle (1) is parallel to the third side;
The Term Paper on Fast Food Nation The Dark Side Of The All American Meal
Fast Food Nation: The Dark Side of the All American Meal The affinity of the American people with fast food can be understood because it enables people to eat on the go and to be able to take out their meals that are set to an affordable price. Fast-food restaurants address a societal need of Americans today which is the lack of time to cook their own food for themselves. If there is such a thing ...
(2) has a length equal to half the length of the third side. In an indirect proof you begin by assuming temporarily that the conclusion is not true. Then you reason logically until you reach a contradiction of the hypothesis or another known fact. If a * b and c * d, then a + c * b + d. If a = b + c, and c * 0, then a * b. If one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side. If one angle of a triangle is larger than a second angle, then the side opposite the first angle is longer than the side opposite the second angle. The perpendicular segment from a point to a line is the shortest segment from the point to the line. The perpendicular segment from a point to a plane is the shortest segment from the point to the plane. Theorem 4-20 The Triangle Inequality The sum of the lengths of any two sides of a triangle is greater than the length of the third side. If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is greater than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle. If two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second, then the included angle of the first triangle is larger than the included angle of the second. The ratio of one number to another is the quotient when the first number is divided by the second.
A proportion is an equation stating that two ratios are equal.
Two polygons are similar if their vertices can be paired so that: (1) Corresponding angles are congruent. (2) Corresponding sides are in proportion. (Their lengths have the same ratio.) The ratio of the lengths of two corresponding sides is called the scale factor of the similarity. Postulate 15 AA Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar. If the sides of two triangles are in proportion, then the two triangles are similar. Points L and M lie on -AB and -CD, respectively. If AL = CM, we say that -AB and -CD are divided proportionally. LB MD Theorem 5-3 Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. If three parallel lines intersect two transversals, then they divide the transversals proportionally. Theorem 5-4 Triangle Angle-Bisector Theorem If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides.
Suppose r, s, and t are positive numbers with r = s. Then s is called the geometric mean between r and t. s t When you write radical expressions you should write them in simplest form. This means writing them so that 1. No radicand has a factor, other than 1, that is a perfect square. 3. No fraction has a denominator that contains a radical. If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the segments of the hypotenuse. When the altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.
