 | Term | Definition |
| |
Ruler Postulate |
the pints on a line can be paired with the real numbers in such a way that any two points can have coordinates 0 and 1 |
| |
Segment Addition Postulate |
if B is between A and C, then: AB+BC=AC |
| |
Protractor Postulate |
on line AB in a given plane, choose any point O between A and B. Consider ray OA and ray OB and all the rays that can be drawn from O on one side of line AB. These rays can be paired with the real numbers from 0 to 180 in such a way that: ray OA is paired with 0, and ray OB with 180, or, if ray OP is paired with X, and ray OQ with y, then angle POQ = l x-y l. |
| |
Ange Addition Postulate |
If point B lies in the Interior of angle AOC, then angle AOB + angle BOC = angle AOC |
| |
Angle Addition Postulate |
if angle AOC is a straight and and B is any point not on line AC, then angle AOB = angle BOC = 180 |
| |
Postulate 5 |
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 |
| |
Postulate 6 |
through any two points there is exactly one line |
| |
Postulate 7 |
through any three points there is at least one plane, and through any three non collinear points there is exactly one plane. |
| |
Postulate 8 |
if two points are in a plane, then the line that contains the points is in that plane |
| |
Postulate 9 |
if two planes intersect, then their intersection is a line. |
| |
Theorem 1-1 |
if two lines intersect, then they intersect in exactly one point |
| |
Theorem 1-2 |
through a line and a point not in the line there is exactly one plane |
| |
Theorem 1-3 |
If two lines intersect, then exactly one plane contains the lines |
| |
Midpoint Theorem |
if M is the midpoint of segment AB, then AM=1/2AB and MB=1/2AB |
| |
Angle Bisector Theorem |
If ray BX is the bisector of angle ABC, then angle ABX = 1/2angleABC and angle XBC = 1/2 angle ABC |
| |
Reasons Used in Proofs |
Given info, Definitions, Postulates (and properties from algebra), Proven Theorems |
| |
Theorem 2-3 |
Vertical angles are congruent |
| |
Theorem 2-4 |
if two lines are perpendicular, then they from congruent adjacent angles |
| |
Theorem 2-5 |
if two lines form congruent adjacent angles, then the lines are perpindicular |
| |
Theorem 2-6 |
if the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary |
| |
Theorem 2-7 |
if two angles are supplements of congruent angles (or of the same angle) then the two angles are congruent |
| |
Theorem 2-8 |
if two angles are complements of congruent angles (or of the same angle), then the two angles are congruent |
| |
Parallel lines |
coplanar lines that do not intersect |
| |
Skew lines |
noncoplanar lines |
| |
Theorem 3-1 |
if two parallel planes are cut by a third plane, then the lines of intersection are parallel |
| |
Transversal |
lines that intersects two or more coplanar lines in different points |
| |
Alternate interior angles |
two nonadjacent interior angles on opposite sides of the transversal |
| |
Same-side interior angles |
two interior angles on the same side of the transversal |
| |
Corresponding angles |
two angles in corresponding positions relative to the two lines |
| |
Postulate 10 |
if two parallel lines are cut by a transversal, then corresponding angles are congruent |
| |
Theorem 3-2 |
if two parallel lines are cut by a transversal, then alternate interior angles are congruent |
| |
Theorem 3-3 |
if two parallel lines are cut by a transversal, then same-side interior angles are supplementary |
| |
Theorem 3-4 |
if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other ones also |
| |
Postulate 10 |
if two parallel lines are cut by a transversal, then corresponding angles are congruent |
| |
Postulate 11 |
if two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel |
| |
Theorem 3-5 |
if two lines are cut by a transversal and alternate interior angles are congruent, then the liens are parallel |
| |
Theorem 3-6 |
if two lines are cut by a transversal and same-side interior angles are supplementary, then the lines are parallel |
| |
Theorem 3-7 |
in a plane two liens perpendicular to the same lines are parallel |
| |
Theorem 3-8 |
through a point outside a line, there is exactly one line parallel to the given line |
| |
Theorem 3-9 |
through a point outside a line, there is exactly one line perpendicular to the given line |
| |
Theorem 3-10 |
two lines parallel to a third line are parallel to each other |
| |
Ways to prove two lines parallel |
show that a pair of corresponding angles are congruent |
| |
Ways to prove two lines parallel |
show that a pair of alternate interior angles are congruent |
| |
Ways to prove two lines parallel |
show that a pair of same-side interior angles are supplementary |
| |
Ways to prove two lines parallel |
in a plane show that both lines are perpendicular to a third line |
| |
Ways to prove two lines parallel |
show that both lines are parallel to a third line |
| |
Scalene Triangle |
no sides congruent |
| |
Isosceles Triangle |
at least two sides congruent |
| |
Equilateral triangle |
all sides congruent |
| |
Acute triangle |
three acute angles |
| |
Obtuse triangle |
one obtuse angle |
| |
Right triangle |
one right angle |
| |
Equiangular |
all angles congruent |
| |
Triangle |
figure formed by three segments joining three noncollinear points |
| |
Theorem 3-11 |
the sum of the measures of the angles of a triangle is 180 |
| |
Corollary 1 |
if two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent |
| |
Corollary 2 |
each angle of an equiangular triangle has measure of 60 |
| |
Corollary 3 |
in a triangle, there can be at most one right angle or obtuse angle |
| |
Corollary 4 |
the acute angles of a right triangle are complementary |
| |
Theorem 3-12 |
the measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles |
| |
Polygon |
many angles |
| |
Convex Polygon |
polygon such that no line containing a side of the polygon contains a point in the interior of the polygon |
| |
Theorem 3-13 |
the sum of the measures of the angles of a convex polygon with n sides is (n-2)180 |
| |
Theorem 3-14 |
the sum of the the measures of the exterior angles of any convex polygon, one angle at each vertex, is 180 |
| |
Inductive Reasoning |
reasoning that is widely used in science and everyday life |
| |
Deductive Reasoning |
conclusion based on accepted statements (definitions, postulates, previous theorems, corollaries, and given information) conclusion MUST be true if hypotheses are true |
| |
Inductive Resoning |
Conclusion based on several past observations. conclusion is PROBABLY true, but not necessarily true |
| |
Congruent |
having the same size and shape |
| |
SSS postulate |
if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent |
| |
SAS postulate |
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 |
| |
ASA postulate |
if two angles and the included sides of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent |
| |
A way to prove two segments or two angles are congruent |
identify two triangles in which the two segments or angles are corresponding parts |
| |
A way to prove two segments or two angles are congruent |
prove that the triangles are congruent |
| |
A way to prove two segments or two angles are congruent |
state that the two parts are congruent, using the reason CPCTC |
| |
Isosceles Triangle Theorem |
if two sides of a triangle are congruent, then the angles opposite those sides are congruent |
| |
Corollary 4-1 |
an equilateral triangle is also equiangular |
| |
Corollary 4-2 |
an equilateral triangle has three 60° angles |
| |
Corollary 4-3 |
the bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint |
| |
Theorem 4-2 |
if two angles of a triangle are congruent, then the sides opposite those angles are congruent |
| |
Corollary 4-4 |
an equiangular triangle is also equilateral |
| |
AAS theorem |
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 |
| |
HL theorem |
if the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another triangle, then the triangle are congruent |
| |
Ways to prove triangles congruent |
SSS, SAS, ASA, AAS |
| |
Ways to prove right triangles congruent |
HL |
| |
Median |
segment from a vertex to the midpoint of the opposite side in a triangle |
| |
Altitude |
perpendicular segment from a vertex to the line that contains the opposite side in a triangle |
| |
Perpendicular bisector |
line (or ray or segment) that is perpendicular to the segment at its midpoint |
| |
Theorem 4-5 |
if a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment |
| |
Theorem 4-6 |
if a point is equidistant from the endpoints of a segment, then the point is on the perpendicular bisector of the segment |
| |
Theorem 4-7 |
if a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle |
| |
Theorem 4-1 |
opposite sides of a parallelogram are congruent |
| |
Parallelogram |
quadrilateral with both pairs of opposite sides parallel |
| |
Theorem 5-2 |
opposite sides of a parallelogram are congruent |
| |
Theorem 5-3 |
diagonals of a parallelogram bisect each other |
| |
Theorem 5-4 |
if both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram |
| |
Theorem 5-5 |
if one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram |
| |
Theorem 5-6 |
if both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram |
| |
Theorem 5-7 |
if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram |
| |
Ways to prove that a quadrilateral is a parallelogram |
show that BOTH pairs of opposite sides are parallel |
| |
Ways to prove that a quadrilateral is a parallelogram |
show that BOTH pairs of opposite sides are congruent |
| |
Ways to prove that a quadrilateral is a parallelogram |
show that ONE pair of opposite sides are both congruent and parallel |
| |
Ways to prove that a quadrilateral is a parallelogram |
show that both pairs of opposite angles are congruent |
| |
Ways to prove that a quadrilateral is a parallelogram |
show that the diagonals bisect each other |
| |
Theorem 5-8 |
if two lines are parallel, then all points on one line are equidistant from the other line |
| |
Theorem 5-9 |
if three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal |
| |
Theorem 5-10 |
a line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side |
| |
Theorem 5-11 |
the segment that joins the midpoints of two sides of a triangle is parallel to the third side and is half as long as the third side |
| |
Theorem 5-12 |
the diagonals of a rectangle are congruent |
| |
Theorem 5-13 |
the diagonals of a rhombus are perpendicular |
| |
Theorem 5-14 |
each diagonal of a rhombus bisects two angles of the rhombus |
| |
Rectangle |
quadrilateral with four right angles |
| |
Rhombus |
quadrilateral with four congruent sides |
| |
Square |
quadrilateral with four right angles and four congruent sides |
| |
Theorem 5-15 |
the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices |
| |
Theorem 5-16 |
if an angle of a parallelogram is a right angle, then the parallelogram is a rectangle |
| |
Theorem 5-17 |
if two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus |
| |
Trapezoid |
quadrilateral with exactly one pair of parallel sides |
| |
Isosceles trapezoid |
trapezoid with congruent legs |
| |
Theorem 5-18 |
base angles of an isosceles trapezoid are congruent |
| |
Theorem 5-19 |
the median of a trapezoid is parallel to the bases and has a length equal to the average of the base lengths |
