# geometry chapters 1-11

## 133 terms

X1+X2/2, Y1+Y2/2

### distance formula

(Square root) v| (x2-x1)^2+ (y1-y2)^2

AB+BC=AC
<ab+<bc=<ac

### regular polygon

all sides and angles are congruent

### counterexample

The example that proves the statement wrong

### converse of a conditional statement

if the conclusion, then the hypothesis

### inverse of a conditional statement

if the negative hypothesis, then the negative conclusion

### contrapositive of the conditional statement

if the negative conclusion, then the negative hypothesis

### biconditional statement

"if and only if" format, only true if conditional and converse are true

### substitution property

a=b,b=a in any situation if stated earlier

a(b+c)= ab+ac

a=a

a=b, b=a

### transitive property of equality

a=b,b=c, then a=c

### congruence of segments theorem

AB=CD, CD=AB, same as with conngruence of angles theorem but with angles

### right angles congruence theorem

right angles are always congruent

### congruent supplements theorem

if 2 angles are supplementary to the same angle, they are congruent. If <a+<b=180', and <b+<c=180', <a=<c

### congruents complements theorem

if 2 angles are complementary to the same angle, they are congruent.

### linear pair postulate

if 2 angles form a linear pair, they are supplementary

### vertical angles congruence theorem

vertical angles are congruent

### triangle sum theorem

the sum of the interior angles of a triangle is 180'

### exterior angles theorem

the measure of an exterior angle of a triangle is equal to the sum of the measures of the two adjacent interior angles

### third angles theorem

if 2 angles in one triangle are congruent to 2 angles of another triangle, then the triangles are congruent (SSS)

### SAS congruence postulate

if 2 sides and the angle between them are congruent to the same part in another triangle, then they are congruent

### SSS congruence postulate

if 3 sides of a triangle is congruent to another triangle, then the two are congruent

### ASA congruence postulate

2 angles and the side in between them are congruent in two triangles, then the triangles are congruent

### AAS congruence postulate

2 angles and a side not in between them are congruent in two triangles, then the triangles are congruent

### HL congruence theorem

in a right triangle, if the hypotenuse and a leg are congruent in two triangles, then the triangles are congruent

### CPCTC

corresponding parts of congruent triangles are congruent
all corresponding parts and sides make a triangle congruent and vise versa

### base angles theorem

if two sides of a triangle are congruent, then the angles opposite them are congruent(opposite for the converse)

### AA similiarity postulate

angle-angle 2 angles are congruent in two triangles, then the triangles are similiar

### SSS similiarity postulate

side-side-side if all the sides are in the same ratio to eachother in two triangles, the triangles are similar to eachother

### SAS similarity postulate

side-angle-side if two sides are in the same ratio to eachother in two triangles, and the angle between the sides is congruent in two triangles, then the triangles are similar

### triangle proportionality theorem

if a line parallel to one side of a triangle intersectsthe other two sides, then it divides the two sides proportionally.
Converse- if a line divides two sides of a triangle proportionally, then it's parallel to the third side

### parallel lines proportionality theorem

if three parallel lines intersect at 2 transversals, then they divide the transversals proportionally

### angle bisectors theorem

if a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of two other sides

### concurrency of angle bisectors of a triangle (incenter)

concurrency is equidistant from the sides of the triangle

### median of a triangle

goes from a vertex to the midpoint of the opposite side

### concurrency of medians of a triangle (centroid)

concurrency is 2/3 the distance from each vertex to the midpoint of the opposite side

### altitude of a triangle

goes from a vertex to the opposite side and the opposite side and height line form a right angle

### concurrency of the altitudes of a triangle (orthocenter)

concurrency is on intersection of legs for a right triangle
outside the triangle for obtuse triangle
inside the triangle for acute triangle

### triangle inequality theorem

the sum of the lengths of any two sides of a triangle is greater than the length of the of the third side

### hinge theorem

2 sides are congruent in two triangles, but included angle is larger in one of the triangles, than that triangle's third side is bigger. Converse of this is opposite.

### midsegment theorem

the segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that side.

### perpendicular bisector theorem

in a plane, if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoint of the segments. converse; the opposite of this

### concurrency of perpendicular bisectors of a triangle theorem (circumcenter)

equidistant from the vertices of the triangle

### angle bisector theorem

if a point is on the bisector of an angle, then it is equidistant from the two sides of an angle. converse; the opposite of this.

### classifying acute triangles theorem

with pythagorean theorem: if hypotenuse^2<leg^2+leg^2, then it's acute

### classifying obtuse triangles theorem

with pythagorean theorem: if hypotenuse^2>leg^2+leg^2, then the triangle is obtuse

### geometric mean (altitude) theorem

in a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altituse is the geometric mean of the lengths of the two segments

### geometric mean (leg) theorem

in a right triangle, altitute from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each leg of the right triangle is the geometric means of the lengthsof the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.

### 45'-45'-90' triangle

the hypotenuse= v|2 times a leg

### 30'-60'-90' triangle

the hypotenuse=2 times the shortest leg
the longer leg= v|3 times the shortest leg

### transformation

the changing of a shape's position, size, or direction.

### isometry

a transformation where the original and new image are congruent

### translation

moving from one place to another on a coordinate plane
(x,y)-(x+a,y+b), or T(a,b)

### reflection

reflecting of a pre-image over a line of reflection to create an image. Every point is reflected. Written as R(line).
r(x-axis)=(a,b)-(a,-b)
r(y-axis)=(a,b)-(-a,b)

### rotation

a figure is turned about on a fixed point, called the center of rotation. Figure makes a circle around this point, in a COUNTER-CLOCKWISE motion. As a rotation around the orgin: R(90')=(a,b)-(-b,a) R(180')=(a,b)-(-a,-b)
R(270')=(a,b)-(b,-a)

### dilation

results in image similar to figure, increase or decrease. Scale factor=k if |k|<1 REDUCTION
if |k|>1 EN:ARGEMENT

### composition of transformations

2 or more transformations are combined to form a single transformation. EX. R(90') * r(x-axis) r(axis) is done first

### glide reflection

any combo of translation and reflection

### polygon interior angles theorem

the sum of the interior angles of a polygon is equal to the nuber of sides minus two times 180 degrees. n-gon,
(n-2)x 180'

### polygon exterior angles theorem

the sum of the measures of the exterior angles of a convex polygon is 360'

### ways to prove a quadrilateral is a parallelogram

* opposite sides are parallel
*opposite sides are congruent
*opposite angles are congruent
* opposite sides are both parallel and congruent
*diagonals bisect eachother

### rhombus

*parallelogram, square, rectangle
*diagonals are perpendicular bisectors
*opposite sides are parallel
*all sides are equal in length
*opposite angles are congruent

### rectangle

*diagonals are congruent
*opposite sides are parallel and congruent
*all vertices are right angles
*can be a square or a rhombus

### square

*diagonals are congruent perpendicular bisectors of eachother
*opposite sides are parallel
*all sides are equal
*all vertices are right angles

### trapezoid (non isosceles)

*base sides are parallel
*consecutive angles are supplementary
*interior angles add up to 360'

### isosceles trapezoid

*base sides and angles are congruent
*the diagonals are congruent

### kite

*the diagonals are perpendicular
*one pair of opposite angles is congruent
*two pairs of congruent sides

### tangent

one point of tangency that intersects with the circle at one point. Tangents from a common external point of a circle are congruent. Always perpendicular to a radius

### secant

two points of intersetion with a circle. if there is an angle formed by them and they intersect a circle, there will be a correllation.

### Formula for the equation of a circle

(h,k) is the center of the circle. (x,y) is a location on the outside of the circle.

### locus

the set of all points in a plane that satisfy a given condition or set of given conditions. e.g; c is a point. the locus of all points 3 units from c is a circle with a radius of 3 units.

### Prism

Surface area: The length x the width x the height
volume: The area of the base x the height (V=b*h)

### cylinder

surface area: (2piradius^2) + (2piradius*height) this is basically 2 x the area of a circle multiplied by height x the circumference, which is a rectangle wrapped around the two circles.
Volume: piradius^2height, Like the area of the circle multiplied by the height of the cylinder.

### pyramid

surface area: base+1/2baselength*slant height (B+1/2 Pl) B=base P=base length l=slant height
volume: 1/3 (base area*height) (1/3 Bh )

### cone

(area of a circle) + (lateral area of the cone)
1/3 (base) * (height)

### sphere

surface area: 4piradius^2 (think like a baseball, basically made of four circles sewn together to cover)

### definitioin of congruecy

segment: when two or more segments have the same length.
angle: when two or more angles have the same angle measure.
general definition: when things have the same measure

### If-Then Form of a Conditional Statement

The statement says, basically, "if this, then this"/ if cause, then effect.

### Perpendicular Lines

perpendicular lines are lines that intersect eachother in such a way that the angles at the intersection are 90 degrees.

### Two-Column Proof

like a "T" chart, where the postulate/theorem/reason is on one side, and what it means on the other

### Parallel Lines

they are lines with the same slope that never intersect

### transversal

it is is a line that intersects two other lines

### Corresponding Angles (Include Corresponding Angles Postulate and Corresponding Angles Converse)

they are angles that are on the same side of the transversal, but is outside the parallel lines, and the other is not, and the angles are in the same relative position
postulate: if the two angles are corresponding angles, then they are congruent
converse: if the two angles are congruent corresponding angles, then the lines they touch, aside from the transversal, are parallel

### Alternate Interior Angles (Include Alternate Interior Angles Theorem and Alternate Interior Angles Converse)

They are angles on opposite sides of the transversal, but are within the parallel lines
postulate: if the two angles are alternate interior angles, then they are congruent
converse: if the two angles are congruent alternate interior angles, then then the lines they touch, aside from the transversal, are parallel

### Alternate Exterior Angles (Include Alternate Exterior Angles Theorem and Alternate Exterior Angles Converse)

they are angles on either side of the transversal, and are outside of the parallel lines
postulate: if the two angles are alternate exterior angles, then they are congruent
converse: if the two angles are congruent alternate exterior angles, then the lines they touch, aside from the transversal, are parallel

### Consecutive Interior Angles (Include Consecutive Interior Angles Theorem and Consecutive Interior Angles Converse)

they are angles on the same side of the transversal, and within the parallel lines
postulate: if two angles are consecutive interior angles, then they are supplementary
converse: if two angles are supplementary consecutive interior angles, then the lines they touch, except for the transversal, are parallel

### Slopes of Parallel Lines Postulate

lines are only parallel if they have the same slope

### Slopes of Perpendicular Lines Postulate

the product of the slope of two lines must equal -1 if they are perpendicular

### Perpendicular Transversal Theorem

If a transversal is perpendicular to one of the parallel lines, then it is perpendicular to the other

### Lines Perpendicular to a Transversal Theorem

if two lines are perpendicular to the same line, then they are parallel to each other

### Standard Equation of a Line

Ax+By=C
A,B, and C are integers
A is greater than 0

### Slope-Intercept Form of a Line

y=mx+b
m=slope, x=x-intercept, b=y-intercept

### Point-Slope Form of a Line

y#1-y#2=m(x#1-x#2)

used to solve for a parabola, one of the terms is raised to the second power.
ax^2+bx+c=0

### Midsegment

the midpoint on a segment. It is equidistant from either end of a segment

### Perpendicular Bisector

A line that is perpendicular to a segment at its midpoint.

### Angle Bisector

a ray that goes from the vertex, and divides the angle into two adjacent congruent angles

### Pythagorean Theorem

the sum of one leg to the second power and the other leg to the second power is equal to the hypotenuse to the second power in a right triangle
L1^2+L2^2=H^2

### Converse of the Pythagorean Theorem

the hypotenuse to the second power is equal to the sun of one leg to the second power and the other leg to the second power in a right triangle
H^2=L1^2+L2^2

SOHCAHTOA,

### Sine Ratio

SOHCAHTOA,
sine=opposite/hypotenuse

SOHCAHTOA,

### Line Symmetry

a line that divides an image in half, so that each half is the mirror image of the other

### Rotational Symmetry

a figure that is able to be rotated any degree around the center and still be in the same position as the original image

### Orientation

the position and placement of an image

### Parallelogram

a quadrilateral whose opposite sides are parallel and congruent

### linear pair postulate

if two angles form a line, then they are supplementary

### vertical angles congruence theorem

vertical angles are congruent, and they are angles that are opposite eachother in a system of crossing lines

### Tangent (include any theorems and properties)

Definition: it is a line that touches the edge of the circle, the point of tangency is where the line touches the circle.
-In a plane, a line tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle.
-Tangent segments from a common external point are congruent

### Central Angle (include its corresponding arc measure)

it is an angle from the central point of a circle that reaches to the edge of a circle.
-Its measure is the same as the minor arc, which is between the angle.
-the measure of the major arc (the part of the circle that isn't the minor arc) is 360 degrees-the minor arc

### Chord (include properties of chords from section 10.3)

it is a segment with endpoints on a circle
-In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent
-if one chord is a perpendicular bisector of another chord, then the first chord is a diameter
-if the diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc
-in the same circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center

### Inscribed Angle (include how to find its measurement)

the measure of an inscribed angle is one half the measure of its intercepted arc

### Inscribed Polygon (include what you know about inscribed right triangles and quadrilaterals)

an inscribed polygon is a shape with all vertices on the circle
-the diameter is the hypotenuse of all inscribed right triangles, and the opposite angle is the right angle.
-A quadrilateral can only be inscribed is and only if its opposite angles are supplementary

### Angle Formed by a Tangent and Chord

the measure of each angle formed is one half of its intercepted arc

### Angles Inside the Circle Theorem

if two chords intersect inside a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle

### Angles Outside the Circle Theorem

if a tangent and a secant, two tangents, or two secants intersect outside the circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs

### Segments of Chords Theorem

if two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord

### Segments of Secants Theorem

If two secant segments share the same endpoint outside a circle, then the product of the lengths of one secant segment and one external segment equals the product of the lengths of the other secant segment and its external segment.

### Segments of Secants and Tangents Theorem

If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segments equals the square of the length of the tangent segment.

### Standard Equation of a Circle

(x-h)²+(y-k)²=r²
a circle with radius (r) and center (h,k)

base x height

### area of a triangle

one half base x height

### area of a rhombus and a kite

one half the product of the lengths of the diagonals

### area of similar polygons

if polygons are similar, and their ratio is a:b, then the ratio of a^2:b^2 would also work

### area and circumference of a circle

circumference: pi x diameter (radius x 2)

### area of a sector

the area of the sector to the area of the circle is equal to the measure of the intercepted arc to 360 degrees

### area of a regular polygon

apothem= center of the polygon to a side of the polygon
area= one half the apothem x the perimeter

### Similar Solids Theorem

If 2 similar solids have a scale factor of a : b,
then corresponding areas have a ratio of a^2 : b^2,
and corresponding volumes have a ratio of a^3 : b^3.