Geometry Jurgensen Chapters 1-5, Geometry

a line that intersects two or more coplanar lines in different points

Transversal
(trans.)

two nonadjacent interior angles on opposite sides of a transversal

Alternate Interior Angles
(alt. int. ⦞)

two interior angles on the same side of a transversal

Same-side Interior Angles
(s-s. int. ⦞)

two angles in corresponding positions relative to the two lines

Corresponding Angles
(corr. ⦞)

Theorem 5-1

opposite sides of a parallelogram are congruent

Theorem 5-2

opposite angles of a parallelogram are congruent

Theorem 5-3

diagonals of a parallelogram bisect each other

parallelogram

A quadrilateral with both pairs of opposite sides parallel

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

Five ways to Prove that a quadrilateral is a parallelogram

1. Show that both pairs of opposite sides are parallel
2. Show that both pairs of opposite sides are congruent
3. Show that one pair of opposite sides are both congruent and parallel
4. Show that both pairs of opposite angles are congruent
5. Show that 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

Rectangle

a quadrilateral with four right angles; is always a parallelogram

Rhombus

a quadrilateral with four congruent sides; is always a parallelogram

Square

A quadrilateral with four right angles and four congruent sides. Is always a rectangle, a rhombus, and a parallelogram.

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

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

a quadrilateral with exactly one pair of parallel sides

Isosceles trapezoid

a 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

Polygon

A closed geometric figure in a plane formed by connecting line segements endpoint to endpoint with each segment intersecting exactly two others. No tow segments with a common endpoint are collinear.

Convex polygon

a polygon such that no line containing a side of the polygon contains a point in the interior of the polygon

Diagonal

a segment that joins two nonconsecutive vertices

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 measures of the exterior angles of any convex polygon, one angle at each vertex, is 360

Regular polygon

a polygon that is both equilateral and equiangular

Triangle

figure formed by 3 segments joining 3 noncollinear points

Vertex of triangle

each of the 3 points joining the sides of a triangle

Sides of triangle

the sides of a triangle are the segments that make up the triangle

Vertices

plural of vertex

Scalene triangle

a triangle with no congruent sides

Isosceles triangle

A triangle with at least two congruent sides

Equilateral triangle

a triangle with all sides congruent

Acute triangle

A triangle with three acute angles

Obtuse triangle

a triangle with one obtuse angle

Right triangle

a triangle with one right angle

Equiangular triangle

a triangle with all angles congruent

Auxiliary line

line (or ray or segment) added to a diagram to help in a proof; *shown as a dashed line

Theorem 3-11

the sum of the measures of the angles of a triangle is 180

Corollary

a statement that can be easily proved using a theorem

Corollary 1 of Theorem 3-11

if two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent

Corollary 2 of Theorem 3-11

each angle of an equiangular triangle has measure of 60

Corollary 3 of Theorem 3-11

in a triangle, there can be at most one right angle or obtuse angle

Corollary 4 of Theorem 3-11

the acute angles of a right triangle are complementary

Remote interior angles

the two nonadjacent interior angles corresponding to each exterior angle of a triangle

Exterior angle

an angle formed by one side of a triangle and the extension of another side

Theorem 3-12

the measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles

Postulate 11

if two lines are cut by a transversal and the 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 lines 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 lines perpendicular to the same line 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

Theorem 3-2

if two parallel lines are cut by a transversal, then the 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 one also

Parallel lines

coplanar lines that do not intersect

Parallel planes

planes that do not intersect

Theorem 3-1

If two parallel planes are cut by a third plane, then the lines of intersection are parallel

Alternate interior angles

two nonadjacent interior angles on opposite sides of the transversal; (to find alternate interior angles, look for the letter "Z" or backwards "Z")

Same-side interior angles

two interior angles on the same side of the transversal; (to find same-side interior angles, look for a square "C" or backwards square"C")

Corresponding angles

two angles in corresponding positions relative to the two lines; (To find corresponding <'s, look for the letter "F" or backwards "F, or upsidedown "F")

Median of a triangle

A segment from a vertex to the midpoint of the opposite side

Altitude of a triangle

the perpendicular segment from a vertx to the line containing the opposite side

Perpendicular bisector of a segment

a segment, ray, line, or plane that is perpendicular to a 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 4-8

If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle

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 right triangle, then the triangles are congruent

hypotenuse

the side of a right triangle opposite the right angle

Leg-Leg Method (LL)

If two legs of one right triangle are congruent to the two legs of another right triangle, then the triangles are congruent.

Hypotenuse-Acute Angle Method (HA)

If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and an acute angle of another right triangle, then the triangles are congruent.

Leg-Acute Angle Method (LA)

If a leg and an acute angle of one right triangle are congruent to the corresponding parts in another right triangle, then the triangles are congruent.

Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angles opposite those sides are congruent

Corollary 1 of Theorem 4-1

An equilateral triangle is also equiangular.

Corollary 2 of Theorem 4-1

an equilateral triangle has three 60° angles

Corollary 3 of Theorem 4-1

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 of Theorem Theorem 4-2

An equiangular triangle is also equilateral.

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 side of one triangle are congruent to two angles and the included side of the second, the triangles are congruent

Congruent triangles

Two triangles are congruent if and only if their vertices can be matched up so that all of their corresponding parts (sides and angles) are equal

Congruent polygons

Two polygons are congruent if and only if their vertices can be matched up so that their corresponding parts are congruent.

Non coplanar

Not in the same plane

Equidistant

equally distant from two points

set of all points

Space