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Geometry Semester 2 Terms jdevota
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Terms in this set (95)
properties of parallelogram
- parallel sides are parallel by definition
- opposite sides are congruent
- opposite angles are congruent
- diagonals bisect each other
- any pair of consecutive angles are supplementary
properties of rectangles
- all properties of parallelogram apply
- all angles are right angles
- diagonals are congruent
properties of kites
- two disjoint pairs of consecutive sides are congruent
- diagonals are perpendicular
- one diagonal is the perpendicular bisector of the other
- one diagonal bisects a pair of opposite angles
- one pair opposite angles are congruent
properties of rhombuses
- all properties of parallelogram apply
- two consecutive sides are congruent
- all sides congruent
- diagonals bisect angles
- diagonals are perpendicular bisectors of each other
- diagonals divide rhombus into four congruent right triangles
properties of squares
- all properties of rectangle apply
- all properties of rhombus apply
- diagonals form four isosceles right triangles
properties of isosceles trapezoids
- legs are congruent
- bases are parallel
- lower base angles are congruent
- upper base angles are congruent
- diagonals are congruent
- any lower base angle is supplementary to any upper base angle
parallelogram
a quadrilateral with both pairs of opposite sides parallel
rhombus
a parallelogram with four congruent sides
rectangle
a parallelogram with four right angles
square
a parallelogram with four congruent sides and four right angles
kite
a quadrilateral with two pairs of adjacent, next to, sides congruent and no opposite sides congruent
trapezoid
a quadrilateral with exactly one pair of parallel sides and at the right of a isosceles trapezoid at the right a trapezoid whose nonparallel opposite sides are congruent
consecutive angles
angles of a polygon that share a side
quadrilateral is a parallelogram if...
1. if both pairs of opposite sides of a quadrilateral are parallel
2. if both pairs of opposite sides of a quadrilateral are congruent
3. if one pair of opposite sides of a quadrilateral are both parallel and congruent
4. if the diagonals of a quadrilateral bisect each other
5. if both pairs of opposite angles of a quadrilateral
quadrilateral is a rectangle if...
1. if a parallelogram contains at least one right angle
2. if the diagonals of a parallelogram are congruent
3. if all four angles of a quadrilateral are right angles
quadrilateral is a kite if...
1. if two disjoint pairs of consecutive sides of a quadrilateral are congruent
2. if one of the diagonals of a quadrilateral is the perpendicular bisector of the other diagonal
quadrilateral is a rhombus if...
1. if a parallelogram contains a pair of consecutive sides that are congruent
2.if either diagonals of a parallelogram bisects two angles of the parallelogram
3.if the diagonals of a quadrilateral are perpendicular to the bisectors of each other
quadrilateral is a square if...
1. if a quadrilateral is both a rectangle and a rhombus
trapezoid is an isosceles trapezoid if...
1. if the nonparallel sides of a trapezoid are congruent
2. if the lower or upper base angles of a trapezoid are congruent
3.if the diagonals of a trapezoid are congruent
area equation for rhombus
A=1/2(d₁)(d₂)
area of rectangle equation
A=bh
area of parallelogram equation
A=bh
base of parallelogram
any of its sides
altitude
a segment perpendicular to the line containing that base drawn from the side opposite the base
height
the length of an altitude
a diagonal divides any parallelogram into how many congruent triangles
2
the area of each congruent triangle is how much of the parallelogram
1/2
area of a triangle equation
A=1/2bh
what is the Pythagorean theorem
- in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse
- a²+b²=c²
Pythagorean triple
a set of nonzero whole numbers a, b, and x that satisfy the equation a²+b²=c²
True or False: if you multiply each number in a Pythagorean triple by the same whole number, the three numbers that result also form a Pythagorean triple
True
converse of the Pythagorean theorem
if the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle
proving a triangle is obtuse
c²>a²+b²
proving a triangle is acute
c²<a²+b²
name for an isosceles right triangle
45⁰-45⁰-90⁰ triangle
45⁰-45⁰-90⁰ triangle
45⁰-45⁰-90⁰ triangle equation
hypotenuse= √2(leg)
What is a 30⁰-60⁰-90⁰ triangle
when the length of the hypotenuse is twice the length of the shorter leg and the length of the longer leg is √3 times the length of the shorter leg
30⁰-60⁰-90⁰ triangle
equations pertaining to 30⁰-60⁰-90⁰ triangle
hypotenuse= 2(shorter leg)
longer leg= √3(shorter leg)
area of kite equation
A=1/2(d₁)(d₂)
area of trapezoid equation
A=1/2h(d₁+d₂)
area of regular polygon
A=1/2ap
apothem is
perpendicular and bisects the base and angle
area of a circle
A=πr²
sector of a circle
a region bounded by an arc of the circle and the two radii to the arcs endpoints
area of a sector
(mAB/360)(πr²)
circle
a set of all points equidistant from a given point called the center
radius
a segment that has one endpoint at the center and the other endpoint on the circle
congruent circles have
congruent radii
diameter
a segment that contains the center of a circle and has both endpoints on the circle
central angle
an angle whose vertex is the center of the circle; to find the corresponding percent of 360
semicircle
half a circle
minor arc
smaller than a semicircle
major arc
greater than a semicircle
arc addition postulate
mABC=mAB+mBC
circumference
distance around a circle
equation for circumference
C=dπ of C=2rπ
pi
ratio of the circumference of a circle to its diameter
equation for arc length
length of AB=(mAB/360)(2rπ)
congruent arcs
have same measure and are in the same circle or in congruent circles
similar figures
two figures that have the same shape but are not necessarily the same size
2 things that determine if polygons are similar
1)corresponding angles are congruent
2)corresponding sides are proportional
similarity ratio
the ratio of lengths of the corresponding sides
angle similarity postulate(AA)
if two angles of one triangle are congruent tot two angles of another triangle then the triangles are similar
side angle side theorem(SAS)
if an angle of one triangle is congruent to an angle of a second triangle and the sides including the two angles are proportional then the triangles are similar
side side side theorem(SSS)
if the corresponding sides of two triangles are proportional then the triangles are similar
indirect measuring
using similar triangles and measurements to find distances that are difficult to measure directly
ratio
comparison of two quantities
proportion
a statement that two ratios are equals
cross product property
states the product of the extremes is equal to the product of the means; ex- a:b=c:d & a/b=cd implies that ad=bc
scale drawing
the scale compares each length in the drawing to the actual length; lengths used in scale can be different units
Triangle-Angle-Bisector Theorem
if a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle
if the similarity ratio of two similar figures is a/b, then
1) the ratio of their perimeters ia a/b and 2) the ratio of their areas is a²/b²
corollary to side-splitter theorem (triangle proportionality)
if three lines intersect two transversals, then two segments intercepted on the transversals are proportional
side-splitter theorem (triangle proportionality)
if a line is parallel to one side of a triangle and interacts the other two sides, it divides those sides proportionally
triangle-angle-bisector theorem
perimeter ratio of similar figures
a/b
area ratio of similar figures
a²/b²
angle of elevation and depression
- elevation - up
- depression - down
- need a horizontal line
(x,y) -> (-y,x) - transformation
90° rotation
(x,y) -> (-x,-y) - transformation
180° rotation
(x,y) -> (y,-x) - transformation
270° rotation
(x,y) -> (x,-y)
reflection across x-axis
(x,y) -> (-x,y)
reflection across y-axis
(x,y) -> (y,x)
reflection across y=x
(x,y) -> (-y,-x)
reflection across y=-x
Circle
The set of all points in a plane at a given distance from a given point is a circle
Radius
The given distance is the radius of the circle. A radius is also any segment joining the center of the circle to a point of the circle
- all radii of a circle are congruent
Chord
A segment whose endpoints lie on a circle
Diameter
A chord that contains the center of a circle
- a length equal to twice a radius
Secant
A line that contains a chord of a circle
Tangent
A line or part of a line in the plane of a circle that intersects the circle in exactly one point
- the point of tangency is the point of intersection
Minor arcs
<180º
- named by two letters
Major arcs
>180º
- named by three letters (to show where it travels to)
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