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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 has 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). 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: 2 m ABX = m ABC and m ABX = m ABC 2 m 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 non collinear 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. 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 non collinear 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 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...
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