list triangle congruence theorems/postulates
SAS, SSS, ASA, AAS, HL
biconditional?
conditional statement with "if and only if"; only if both conditional and converse are true
law of syllogism?
if p -> q and q -> r are true statements, then p -> r is a true statement
law of detachment?
if a conditional is true and its hypothesis is true, then the conclusion is true
if a transversal intersects two parallel lines, then alternate interior angles are
congruent
if a transversal intersects two parallel lines, then same side interior angles are
supplementary
if a transversal intersects two parallel lines, then same side exterior angles are
supplementary
if a transversal intersects two parallel lines, then corresponding angles are
congruent
if a transversal intersects two parallel lines, then alternate exterior angles are
congruent
if two lines and a transversal form corresponding angles that are congruent, then the two lines are
parallel
if two lines and a transversal form alternate interior angles that are congruent, then the two lines are
parallel
f two lines and a transversal form same side interior angles that are supplementary, then the two lines are
parallel
if two liens and a transversal form alternate exterior angles that are congrent, ten the two lines are
parallel
if two lines and a transversal form same side exterior angles that are supplementary, then the two lines are
parallel
if two lines and a transversal form same side exterior angles that are ______, then the two lines are parallel
supplementary
if two lines are parallel to the same line, then
they are parallel to each other
if two lines are perpendicular to the same line, then
they are parallel to each other
the sum of the measures of the angles of a triangle is
180
list classifications of a triangle by angle
equiangular, acute, right, obtuse
list classifications of a tirangle by side
equilateral, isosceles, scalene
what is an isosceles triangle?
at least two sides congruent
what is a scalene triangle?
no sides congruent
exterior angle of a polygon?
angle formed by a side and an extension of the adjacent side
the measure of an exterior angle of a polygon is the sum of what two angles?
the two remote interior angles corresponding to the exterior angle
convex polygon?
polygon with no diagonal with points otuside the polygon
concave polygon?
has at least one diagonal with points outside the polygon
sum of measures of angles of ngon?
(n-2)180
the sum of the measures of the exterior angles of a polygon, one at each vertex, is
360
slope intercept form?
y = mx + b
point slope form?
y-y1 = m(x-x1)
standard form of a linear equation?
ax + by - c
slopes of parallel lines are
equal
perpendicular lines have a slope product of
-1
if two angles of one triangle are congruent to two angles of another triangle , then
the third angles are congruent
describe SAS postulate
if two sides and included angle are congruent....triangle is congruent
describe ASA postulate
if two angles and included side are congruent....congruent triangles
describe AAS theorem
f two angles and a NONINCLUDED side .....congruent
isosceles triangle theorem?
if two sides of a triangle are congruent, then the angles opposite those sides are congruent
triangle midsegment theorem?
if a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side, and is half its length
if a segment joins the midpoints of two sides of a triangle, then the segment is _______ to the third side, and is ______ its length
parallel, half
if a pt is on the perpendicular bisector of a segment, then
it is equidistant from the endpoints of the segment
if a point is equidistant from the endpoitns of a segment,
it is on the perpendicular bisector of a segment
If a pt is on the bisector of the angle, then
that point is equidistant from the sides of the angle.
if a point is equidistant from the sides of the angle, then
the point is on the bisector of the angle.
distance from point to line?
perpendicular distance from point to line
concurrent lines?
three or more lines that intersect in the same point
state the converse of this statement: if it rains, then they will cancel school
if they cancel school, it will rain
state inverse of this statement: if it rains, then they will cancel school
if it does not rain, they will not cancel school
state the contrapositive of this statement: if it rains, then they will cancel school
if they do not cancel school, it is not raining
the statement and the ______ have the same truth value
contrapositive
the converse and the ______ have the same truth value
inverse
what are concurrent lines?
3 or more lines intersecting in one point
what is the point at which concurrent lines intersect?
point of concurrency
the perpendicular bisectors of the sides of a triangle intersect at a point of concurrency equidistant from the ______
the perpendicular bisectors of the sides of a triangle intersect at a point of concurrency equidistant from the vertices
the ______- of the _____ of a triangle intersect at a point of concurrency equidistant from the vertices
perpendicular bisectors, sides
the _______ of a triangle are concurrent at a point equidistant from the ______
angle bisectors, sides
circumcenter of a triangle?
point of concurrency of the perpendicular bisectors of a triangle
what it is called when a triangle is inside a circle?
circle is circumscribed about the triangle
Find the center of the circle that you can circumscribe aobut the triangle with vertices (0,0), (-8,0), and (0,6)
-4,3
find the center of the circle that you can circumscribe about a triangle with vertices (0,0), (-4,0), (0,-6)
-2,-3
point of concurrency of the angle bisectors of a triangle?
incenter of the triangle
what is it called when a circle is inside a triangle?
circle inscribed in a triangle
what is the median of a triangle?
segment whose endpoints are a vertex and the midpoint of the opposite side
the medians of a triangle are concurrent at a point ____ the distance from each _____ to the ____ of the opposite side
2/3; vertex; midpoint
what is the centroid?
the point of concurrency of the medians in a triangle
what is the difference between a median and a perpendicular bisector?
a median does not necessarily bisect the segment at a right angle, also it has an endpoint at a vertex
go to the following link: http://image.absoluteastronomy.com/images/encyclopediaimages/t/tr/triangle.centroid.median.png. O is the centroid of ABC. AE = 12, find OE.
4
go to the following link: http://image.absoluteastronomy.com/images/encyclopediaimages/t/tr/triangle.centroid.median.png. O is the centroid of ABC. OE = 6, find DE.
(1/3DE = 6; DE = 18) 18
what is an altitude of a triangle?
the perpendicular segment from a vertex to the line containing the opposite side (see link: http://www.geogebra.org/en/wiki/images/e/ea/TriangleAltitudePicture.jpg)
go to the link: http://www.icoachmath.com/Sitemap/images/Altitude%20of%20a%20Triangle2.jpg which of the images is an altitude?
figure 3
in an acute triangle, where is the altitude?
inside the triangle
in a right triangle, where is the altitude?
a side
in an obtuse triangle, where is the altitude?
outside the triangle
t/f: altitudes can lie outside the triangle
T
what is the orthocenter of a triangle?
point at which altitudes intersect
go to textbook page (lib) 260, #15. what is AB?
altitude
t/f: if a segment is perpendicular to another segment in a triangle, then it must be the bisector.
F
what is the inverse of the following statement? If a triangle is isosceles, it has two equal sides.
if a triangle is not isosceles, it does not have two equal sides.
T/F a conditional statement and its inverse always have the same truth value.
F
T/F: the inverse of a biconditonal is always true.
T
What is the negation of the following statement: If an angle is a straight angle, its measure is 180.
an angle is not a straight angle.
if a condition's symbolic form is p -> q, what is the symbolic form for the negation?
~p
what is an indirect proof?
proof where statement and its negation are only possibilities; and negation is proved wrong
if a = b + c and c > 0, then?
a > b
The measure of an exterior angle of a triangle is ______ than the measure of each of its remote interior angles
greater
if two sides of a triangle are not congruent, then the larger angles lies opposite
the longer side
consider this triangle: http://2000clicks.com/mathhelp/TriangleABC.gif none of the sides are congruent. which is the largest angle?
B
list the angles in this triangle (http://2000clicks.com/mathhelp/TriangleABC.gif) in order from smallest to largest
B, A, C
if two angles of a triangle are not congruent, then teh longer side lies oppose the
larger angle
in general, what can be said about the relationshiup between the size of an angle and the length of the opposite side?
the bigger the angle, the bigger the side
What is the triangle inequality theorem?
the sum of the lengths of any two sides of a triangle is greater than the length of the third side
the sum of the lengths of any two sides of a triangle is _____ ythan the length of third side
greater
can a triangle have side with the given lengths? 5, 12, 10
yes
can a triangle have sides with the given lengths? 2,4,7
no
write (a) the inverse and (b) the contrapositive of the statement.if a polygon has eight sides, then it is not an octagon.
(a) if a polygon does not have eight sides, then it is not an octagon. (b) if a polygon is not an an octagon, then it does not have eight sides.
what is the converse of a conditional?
switch of hypothesis and conclusion
describe the relatioship between a midsegment and the third side of the triangle?
parllel, and half itse length
identify the two statements that contradict each other: I. pqr is a right triangle II. pqr is an obtuse triangle III. pqr is scalene
I and II
parallelogram
quad with both pairs of opposite sides parallel
rhombus
parallelogram with four congruent sides
rectangle
parallelogram with four right angles
square
parallelogram with four congruent sides and four right angles
kite
quad with two pairs of adjacent sides congruent and no opposite sides congruent
trapezoid
quad with exactly one pair of parallel sides
quadrilateral
polygon with four sides
isosceles trapezoid
trapezoid whose nonparallel opposite sides are congruent
t/f: a square is a rhombus
T
how many pairs of adjacent sides congruent does a kite have?
2
how many pairs of opposite sides congruent does a kite have?
0
t/f: a trapezoid can have two pairs of parallel sides
f
what is the differences between an isosceles trapezoid and a regular trapezoid?
the nonparallel opposite sides of the issosceles trapezoid are congruent
how many pairs of parallel sides does a kite have?
0
a square can be considered the intersection of what two shapes?
rectangle, rhombus
Two sides of a rhombus are 3b + 2 and 4b - 2. Find b.
4
opposite sides of a paralleogram are
congruent
what are consecutive angles?
angles of a polygon that share a side (i.e. same side interior angles in a parallelogram)
in this parallelogram (http://kathrynportfolio.us/jpgs/parallelogram2.jpg), consider that angle A = 112. What is angle D? B? C?
68, 112, 68
opposite angles of a parallelogram are
congruent
In this paralel.ogram (http://kathrynportfolio.us/jpgs/parallelogram2.jpg) consider that DC is 3x - 15 and AB is 2x + 3. What is x?
18
The diagonals of a parallelogram have what relationship to each other?
bisect each other
if three or more parallel lines cut off congruent segments on one trasversal, then
they cut off congruent segments on every transversal
name four methods of proving that a quadrilateral is a parallelogram
1) diagonals bisect each other2) one pair of opposite sides is both congruent and parallel3) both pairs of opposite sides are congruent4) if both pairs of opposite angles are congruent
relationship between diagonal and angles of rhombus?
each diagonal bisects two angles
the diagonals of a rhombus are ____ to each other
perpendicular
the diagonals of a rectangle are
congruent
if one diagonal of a parllelogram bisects two angles of the parallelogram, then it is a
rhombus
if the diagonals of a paralloalgram are perpendicular, then
it is a rhombus
if the diagonals of a parallelogram are congruent, then the parallelogra m is a
rectangle
list ways to determine if pgram is rhombus:
1) each diagonal bisects two angles 2) diagonals are perpendicular
list ways to determine if pgram is rectangle:
1) diagonals are congruent
t/f: diagonals of all parallelograms are congruent
FALSE
t/f: diagonals of parallelograms are perpendicular
F
the base angles of an isosceles trapezoid are
congruent
the diagonals of an issosceles trapezoid are
congruent
list four characteriztics of isosceles trapzeoids
1) legs congruent 2) bases parallel 3) base angles congruent 4) diagonals congruent
diagonals of a kite are
perpendicular
list characteristics of a kite
1) twopairs of adjacent sides congruent 2) no opposite sides congruent 3) perpendicular diagonals
a/b = c/d is equivalent to
1) ad = bc 2) b/a = d/c 3) a/c = b/d 4) (a+b)/b = (c+d)/d
list four other expressions that x/y = 5/6 is equivalent to
1) 6x = 5y 2) y/x = 6/5 3) x/5 = y/6 4) (x+y)/y = 11/6
golden rectangle definition
rectangle that can be divided into a square and a rectangle that is similar to the original rectangle
definition of similar?
1) corresponding angles are congruent 2) corresponding sides are proportional
t/f: figures fcan be congruent if angles are proportional
F
approximate golden ratiom
0.470844907
list similarity postulates/theorems for triangles
AA~, SAS~, SSS~,
what is the AA~ postulate?
if two angles of one triangle are congruent to two angles of another triangle, then they are similar
what is hthe SAS~ theorem?
if an angle across two triangles is congruent, and the surrounding two sides are proportional, the two triangles are similar
what is SSS~ theorem?
if corresponding sides of two triangles are proportional , then the triangles are similar
the altitude to the hypotenuse of a right triangle divides the triangle into two tirnagles. describe these.
they are similar to the original triangle and to each other
what is the geometry mean of 6 and 4?
2sqrt(6)
the length of the altitude to the hyp of a right triangle is the ______ of the lengths of the segments of the hyp
geometric mean
the altitude to the hyp of a right triangle separates the hyp so that the length of each leg of the triangle is the geometric of what two lengths?
the length of adjacent hyp segment and the length of the hyp itself
go to textbook page 441, example 2. solve for x and y.
y = 2sqrt(5); x = 6
go to textbook page 441, example 2 CU. solve for x and y.
y = 4sqrt(3); x = 8
go to tb p 441, example 3. how far is Marla from the info center? (what is AD)?
AD = 180 m
go to tb p 441, example 3 CU. answer the q.
240 m
p. 442, #15.
9
p. 442,#17.
10
p. 442, # 19.
12
What are the two requirements for the Side Splitter Theorem (i.e. dividing sides of traingle proportinally)?
1) line is parallel to one side of a triangle 2) intersects other two sides
If three paralle lines intersect two transversals, then the segments intercepted on the transversals are
PROPORTIONAL
p. 447, CU 1; show work
(x+1.5)/x = 2; x + 1.5 = 2x; 1.5 = x
if a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are _________ to the other two sides of the triangle.
proportional
p. 448, CU 3. solve for y, show work.
3.6/y = 5/8. 5y = 28.8. y = 5.76
if the similarity ratio of two similar figures is a/b, then
1) the ratio of their perimeters if a/b and 2) the ratio of their areas is a^2/b^2
p. 456, example 3. solve.
6.25 * 13 = ~81 bushels
p. 456, CU 3. solve.
$6.95
p. 456 #1.
1/2, 1/4.
p. 457, #9.
216/108 = x/192; $384.
what is the tangent ratio?
opposite/adjacent
cosine ratio?
adjacent/hypotenuse
sine ratio?
opposite/hypotenuse
what is the relationship between the sine and cosine of an angle?
sin x = cos (90-x)
what is a vector?
any quantity with magnitude and direction
what is the sum of vtwo vectors with coordinates (6, 5) + (7, 2)
(13, 7)
what is pythag equation of obtuse triangle?
c^2 > a^2 + b^2
what is pythag equatin of acute triangle?
c^2 < a^2 + b^2
what are leg lengths of of 45 45 90 triangle with hyp = sqrt(2)
1, 1
what are leg lengths of 30 60 90 triangle with hyp = 1
1/2, sqrt(3)/2
area of a trapezoid?
1/2h(b1+b2)
area of rhombus or kite?
1/2d1d2
area of rhombus/?
1/2 d1 * d2
area of kite?
1/2 d1 * d2
what is apothem?
perpendicular distance from center to a side
what is the area of a regular polygon (using apothem?)
1/2ap
what are adjacent arcs?
arcs of same circle that have exactly one point in common
what is the circumference of a circle?
c = 2pir
what is the area of a circle?
pir^2
what is the formula for arc length?
measure of arc/360 * circumference
Find arc length of an arc with degree 100 and a circle with radius 15.
8.3pi
what is the area of a sector of the circle?
measure of arc/360 * area of circle
what is the area of a sector of circle with degree 60, and with a circle with radius 5.
25/6pi
geometric probability?
where points represent outcomes
elena's bus runs every 25 min. if she arrives at her bus stop at arandom time, what is the probability that she will have to wait at least 10 minutes for the bus?
05-Mar
find the area of a regular pentagon with 8 cm sides.
110.11
what is the area of a tirangle given SAS?
one half product of lengths of two sides and sine of included angles
find the mangitude and direction of the vector on p. 494 #15.
21.8 degrees north of east, mangitude 53.85
is the resultant the sum of the initial points of the terminal points?
temrinal points