Midpoint Theorem
If M is the midpoint of line segment AB, then line segment AM is congruent to line segment MB
What defines a line?
Through any two points, there is exactly one line. (p. 105), A line contains at least two points. (p. 106)
What defines a plane?
Through any three non-collinear points, there is exactly one plane. (p. 105) A plane contains at least three non-collinear points. (p. 106)
How can you tell if an entire line lies in a plane?
If two points lie in a plane, then the entire line containing those points lies in that plane. (p. 106)
How do two lines intersect?
If two lines intersect, then their intersection is exactly one point. (p. 106)
How do two planes intersect?
If two planes intersect, then their intersection is a line. (p. 106)
Midpoint Theorem
If M is the midpoint of line segment AB, then line segment AM is congruent to line segment MB (p. 107)
Supplement Theorem
If two angles form a linear pair, then they are supplementary angles. (p. 125)
Complement Theorem
If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles. (p. 125)
Vertical Angle Theorem
If two angles are vertical angles, then they are congruent. (p. 127)
Corresponding Angles Postulate
If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent. (p. 149). If two lines in a plane are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. (p. 172)
Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent. (p. 150) If two lines in a plane are cut by a transversal so that a pair of alternate interiorangles is congruent, then the lines are parallel. (p. 1
Alternate Exterior Angles Theorem
If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent. (p. 150) If two lines in a plane are cut by a transversal so that a pair of alternate exterior angles is congruent, then the two lines are parallel.
Consecutive Interior Angles Theorem
If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary. (p. 150) If two lines in a plane are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the lines are pa
Right Angle Theorems
1) Perpendicular lines intersect to form four right angles. 2) Perpendicular lines form congruent adjacent angles. 3) All right angles are congruent. 4) If two angles are congruent and supplementary, then each angle is a right angle. 5) If two congruent a
Perpendicular Transversal Theorem
In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other. (p. 150). In a plane, if two lines are perpendicular to the same line, then they are parallel.
Parallel and Perpendicular Lines
Two nonvertical lines have the same slope if and only if they are parallel. Two nonvertical lines are perpendicular if and only if the product of their slopes is -1. (p. 158)
Shortest Distance
The perpendicular segment from a point to a line is the shortest segment from the point to the line. (p. 298)
Triangle Angle Sum Theorem
The sum of the measures of the angles of a triangle is 180.(p. 210)
Third Angle Theorem
If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent (p. 211)
Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. (p. 212)
CPCTC
Two triangles are congruent if and only if their corresponding parts are congruent.
Side-Side-Side Congruence (SSS)
If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. (p. 226)
Side-Angle-Side Congruence (SAS)
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. (p. 227)
Angle-Side-Angle Congruence (ASA)
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. (p. 235)
Angle-Angle-Side Congruence (AAS)
If two angles and a nonincluded side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent. (p. 236)
Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent. (p. 245)
Equilateral Triangle Corollaries
A triangle is equilateral if and only if it is equiangular. Each angle of an equilateral triangle measures 60�. (p. 247)
Perpendicular bisector
A perpendicular bisector of a side of a triangle is a line, segment, or ray that passes through the midpoint of the side and is perpendicular to that side.
Median
A median is a segment whose endpoints are a vertex of a triangle and the midpoint of the side opposite the vertex.
Altitude
An altitude of a triangle is a segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side.
Centroid Theorem
The centroid of a triangle is located two-thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median. (p. 271) In other words, it is where the medians intersect.
Relationship of Triangle sides and angles
If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. (p. 282) Therefore, if one angle of a triangle has a greater measure than another angle, then th
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side. (p. 296) Therefore, just any three numbers can not form a triangle.
Interior Angle Sum Theorem
If a convex polygon has n sides and S is the sum of the measures of its interior angles, then S = 180(n - 2). (p. 318)
Exterior Angle Sum Theorem
If a polygon is convex, then the sum of the measures of the exterior angles, one at each vertex, is 360. (p. 320)
Parallelogram Theorem - Sides
Opposite sides of a parallelogram are congruent and parallel. (p. 326)
Parallelogram Theorem - Angles
Opposite angles of a parallelogram are congruent and consecutive angles in a parallelogram are supplementary. If a parallelogram has one right angle, then it has 4 right angles. (p. 326)
Parallelogram Theorem - Diagonals
The diagonals of a parallelogram bisect each other and each diagonal of a parallelogram separates the parallelogram into two congruent triangles. (p. 328)
Proving Parallelograms
The quadrilateral is a parallelogram if 1) both pairs of opposite sides are parallel, 2) both pairs of opposite sides are congruent, 3) both pairs of opposite angles are congruent, 4) the diagonals bisect each other, 5) one pair of opposite sides of is bo
Rectangle Theorem
If a parallelogram is a rectangle, then the diagonals are congruent. (p. 340) If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. (p. 342)
Rectangle Definition
A rectangle is a parallelogram with 4 right angles. (p. 340)
Rhombus Theorems
1) The sides of a rhombus are congruent, 2) the diagonals are perpendicular, 3) each diagonal bisects a pair of opposite angles. (p. 348)
Square Properties
A square has all of the properties of both the rectangle and the rhombus; therefore, it has 4 right angles and 4 congruent sides, diagonals bisect the angles, are perpendicular, and congruent.
Trapezoid Definition
A trapezoid is a quadrilateral with exactly one pair of parallel sides called bases. (p. 356)
Isosceles Trapezoid Theorems
Each pair of base angles are congruent, the diagonals are congruent, and the legs are congruent. (p. 356)
Trapezoid Median Theorem
The median of a trapezoid connects the midpoints of the legs, is parallel to the bases, and its measure is one-half of the measures of the bases. (p. 359)
Area Postulate
Congruent figures have equal areas. (p. 642)
Area of a Regular Polygon
The area of a regular polygon is one-half the product of the perimeter and the apothem (p. 650)
Area of a Circle
The area of a circle is the product of pi and the square of the radius. (p. 651)
Area of a Region
The area of a region is the sum of the areas of all of its non-overlapping parts. (p. 658)
Dimension Properties
1) The smallest unit of area is the square in whatever unit is being used. 2) Area is found for two-dimensional figures only and is always computed by multiplying the two dimensions. 3) All two-dimensional figures have two, and only two, dimensions.
Congruence Transformations
The congruence transformations are those transformations that result in a congruent image: translation, reflection, and rotation.
Reflection as Rotation
Reflecting an image successively in two perpendicular lines results in a 180� rotation. (p. 512)
Transitive Property
If a=b and b=c, then a=c
Substitution Property
If a = b, then either a or b may be substituted for the other in any equation
Angle of Rotation
In a given rotation, if A is the preimage, A' is the image, and P is the center of rotation, then the measure of the angle of rotation APA' is twice the measure of the acute or right angle formed by the intersecting lines of reflection. (p. 512)
Translation
A transformation that "slides" each point of a figure the same distance in the same direction.
Reflection
A transformation that "flips" a figure over a mirror or reflection line.
Rotation
A transformation in which the coordinate axes are rotated ("turned") by a fixed angle about the origin.
Line of Reflection
The line over which a figure is reflected. The vertices of the original and new figure are the same distance from this. The line of reflection can be the x-axis, the y-axis, or any line on the coordinate plane.
Reflexive Property
A quantity is congruent (equal) to itself.
a = a
Likewise, a line segment is congruent to itself and any object is congruent to itself.
Deductive Argument
An argument that uses rules, laws, theorems, definitions, and properties to guarantee that if the premises are true, the conclusion must be true.
Inductive Argument
Intend to provide probable, not conclusive, support for its conclusion by using examples or experience
Slopes of Parallel Lines
If lines are parallel, then their slopes are exactly the same.
Slopes of Perpendicular Lines
If two lines are perpendicular, then the product of their slopes is -1. One slope is the negative reciprocal of the other slope.
Examples: slope 1/3 so perpendicular slope is -3. slope -2/3 so perpendicular slope is 3/2. slope 4 so perpendicular slope is
Got coordinates?
Graph them!
Distance "formula
Graph the points and connect.
Draw a right triangle
Use the Pythagorean Theorem to find the hypotenuse (distance between the points)
Congruent right triangles
If the legs are congruent, the hypotenuse of each triangle must be congruent.
Got a geometric shape?
Label it with all the givens.
As you work, label some more
Got info about a shape but no shape?
Sketch the shape!
Triangle inequality - longest side
In a triangle, the longest side is across from the largest angle.
Triangle inequality - largest angle
In a triangle, the largest angle is across from the longest side.
Kite theorem
A quadrilateral is a kite if it has two pairs of adjacent sides congruent and no opposite sides congruent