Geometry Quarterly #2

complementary

complementary

2 angles that add up to 90 degrees, don’t have to be adjacent

supplementary

2 angles that add up to 180 degrees, don’t have to be adjacent

Linear pair

adjacent supplementary angles

vertical angles

when 2 lines intersect, the non-adjacent angles are vertical angles

vertical angles theorem

all vertical angles are congruent

transversal

a line that intersects two coplanar lines at two different points

corresponding angles

lie on the same side of the transversal and same sides of the intersecting lines

same-side interior angles

lie on the same side of the transversal and between the intersected lines

alternate exterior angles

lie on opposite sides of the transversal and outside the intersected lines

alternate interior angles

are nonadjacent angles that lie on opposite sides of the transversal between the intersected lines

parallel lines

lie in the same plane and never intersect →- indicates parallel lines same direction. ll symbolizes parallel too.

same-side interior angles postulate

if two parallel lines are cut by a transversal, then the pairs of same-side interior angles are supplementary

alternate interior angles theorem

if two parallel lines are cut by a transversal, then the pairs of alternate interior angles have the same measure

corresponding angles theorem

if two parallel lines are cut by a transversal, then the pairs of corresponding angles have the same measure

alternate exterior angles are…

congruent

transitive property

if a=b and b=c then a=c

subtraction property

an equal value subtracted or removed from two equal items will result in a new equal amount

if a=b then a-c=b-c

substitution

if a=b then b can replace a in any case

converse of the same-side interior angles postulate

if two lines are cut by a transversal so that pair of same-side interior angles are supplementary, then the lines are parallel

converse of the alternate interior angles theorem

if two lines are cut by a transversal so that any pair of alternate interior angles are congruent, they are parallel

converse of the corresponding angle theorem

if two lines are cut by a transversal so that any pair of corresponding angles are congruent, then the lines are parallel

the parallel postulate

through a point “P” not on the “L” there is exactly one line parallel to “L”

Perpendicular bisector theorem

if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of a segment

Perpendicular line slope relationship

opposite reciprocal

Converse of the Perpendicular Bisector Theorem

If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment

distance formula

d=√(x₂-x₁)² + (y₂-y₁)²

point-slope formula

y - y₁ = m ( x - x₁ )

midpoint formula

(x₁ + x₂ / 2) , (y₁ + y₂ / 2)

slope formula

rise/ run y₂-y₁ / x₂-x₁

slope intercept form

y = mx + b

parallel line slope relationship

same slope

CPCFC

corresponding parts of congruent figures are congruent

CPCTC

corresponding parts of triangles are congruent (biconditional)

Bioconditional

p if and only if q - statement can be written in that form

Two triangles are congruent if and only if corresponding pairs of angles are congruent

Contrapositive

“if p then q” “if not q, then not p”

contrapositive of true statement is true

ex. if corresponding pairs of sides or corresponding pairs of angles are not congruent, then the triangles are not congruent

ASA

angle, included side, angle

ASA Triangle Congruence Theorem

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

reflexive property

a=a

midpoint

of a line segment is the point that divides the segment into two segments that have the same length

right angles

all right angles are congruent; 90 degrees

bisector

goes through an angle (angle bisector), segment (segment bisector), line to form two equal parts, angles, etc

congruent supplements theorem

if two angles are supplements of the same angle (or congruent angles) then the two angles are congruent

congruent complements theorem

if two angles are complements of the same angle (or congruent angles), then the two angles are congruent

regular polygon

all angle measures are the same, all side lengths are the same (equiangular and equilateral)

SAS

side, included angle, side

SAS Triangle Congruence Theorem

if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the the triangles are congruent

perpendicular bisector

perpendicular bisector of a line segment is a line perpendicular to the segment at the segments midpoint. Form right angles

SSS

side, side, side

SSS Triangle Congruence Theorem

if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent

two triangle congruence that don’t work

AAA & SSA

AAS

angle, angle, side

AAS Triangle Congruence Theorem

if two angles and a non-included side of one triangle are congruent to the corresponding angles and non-included side of another triangle, then the triangles are congruent.

HL

hypotenuse, leg

HL Triangle Congruence Theorem

if the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the triangles are congruent

pythagorean theorem

a² + b² = c² (a bottom, b side, c hypotenuse)

Triangle Sum Theorem

the sum of the angle measures of a triangle is 180 degrees

Polygon Angle Sum Theorem

the sum of the measures of the interior angles of a convex polygon with “n” sides is n-2(180)

concave

in, dip in shape “M”

convex

out, up ;“O”;”G"

exterior angle (polygon)

an angle formed by one side of a polygon and the extension of an adjacent side. Exterior angles form linear pairs with the interior angles

remote interior angle

an interior angle that is not adjacent to the exterior angle

exterior angle theorem

the measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles

isosceles triangle (know parts)

a triangle with at least two congruent sides

isosceles triangle theorem

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

Converse of the Isosceles Triangle Theorem

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

Equilateral Triangle

a triangle with 3 congruent sides

Equiangular Triangle

a triangle with 3 congruent angles

Equilateral triangle theorem

if a triangle is equilateral, then it is equiangular

Converse of the equilateral triangle congruence theorem

if a triangle is equiangular, then it is equilateral

Triangle Inequality Theorem

the sum of any two side lengths of a triangle is greater than the third-side length **tip - add two smallest sides**

Side-Angle Relationships in Triangles

if two sides of a triangle are not congruent, then the larger angle is opposite the longer side (if AC>BC, then <B > <A)

Angle-Side Relationship in Triangles

if two angles of a triangle are not congruent, then the longer side is opposite the larger angle. (if <B > <A then AC > BC)

A circle that contains all the vertices of a polygon is circumscribed about the polygon

circumcircle? circumcenter?

circle is called circumcircle

the center of the circle is called the circumcenter

Point of Concurrency

Three or more lines are concurrent if they intersect at the same point. The point of intersection is called the point of concurrency.

Circumcenter Theorem

the perpendicular bisectors of the sides of a triangle intersect at a point that is equidistant from the vertices of the triangle PA=PB=PC

Acute Triangles (circumcenter location)

circumcenter is inside the triangle

Right Triangle (circumcenter location)

circumcenter is on the triangle

Obtuse Triangle (circumcenter location)

circumcenter is outside the triangle

the point from a point to a line is the…

length of the perpendicular segment from the point to the line

Angle Bisector Theorem

if a point is on the bisector of an angle, then it is equidistant from the sides of the angle <APC≅<BPC so… AC = BC

Converse of the Angle Bisector Theorem

if a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle

AC = BC so… <APC≅<BPC

When is a circle inscribed in a polygon? What’s an inscribed circle called?

if each side of the polygon is tangent (on) to the circle. The inscribed circle is called the incircle

The center of a circle inscribed in a triangle is called?

The incenter of the triangle

Incenter Theorem

the angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle

Median

of a triangle is a segment whose endpoints are a vertex of a triangle and the midpoint

Centroid of a Triangle

the intersection (point of concurrency) of the three medians of a triangle

Where is the centroid regarding the triangle? What is it for a triangle?

inside the triangle; center of gravity of a triangle

Centroid Theorem

the centroid of a triangle is located 2/3 of the distance from each vertex to the midpoint of the opposite side

Altitude of a Triangle

perpendicular segment from a vertex to the line containing the opposite side; every triangle has 3 altitudes; not a bisector, doesn’t necessarily go to midpoint

Location of Altitude

can be inside, outside, or on the triangle

Orthocenter & Location?

the intersection (or point of concurrency) of the lines that contain the altitudes; can be inside, outside, or on the triangle

Acute Triangle (orthocenter triangle)

inside triangle

Right Triangle (orthocenter triangle)

on triangle

Obtuse Triangle (orthocenter triangle)

outside triangle

circumcenter

perpendicular bisectors

incenter

angle bisectors

centroid

medians

orthocenter

altitudes