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Geometry I
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Terms in this set (156)
Definition of between
A is between G and I (written as G-A-I or I-A-G) iff GH+HI=GI
Definition of Collinear
A set of points are collinear iff they lie on the same line
Definition of Triangle
Non-collinear points ABC and the line segments AB BC and CA
Definition of Verticies
The vertices of triangle ABC are A, B and C
Definition of Midpoint
Point M is the midpoint of AB iff AM=MB and A-M-B
______ determine a line
Two points
Ruler Postulate
You can match up every real number with a point on the line
Definition of Line Segment
Segment AB is the points A and B together with points D so A-D-B
Definition of AB
[b-a]
Definition of Ray
Ray MP is the set of all points C such that M-C-P or M-P-C
Definition of Opposite Rays
Ray AB and Ray AC are opposite rays iff B-A-C
Definition of Angle
An Angle is two rays which share a common endpoint
Angle Measurement Postulate
You can match up every angle with a number between 0 and 180
Definition of Acute angle
An angle whose measure is less than 90 degrees
Definition of Right Angle
An angle which has a measure equal to 90 degrees
Definition of Obtuse Angle
An angle whose measure is greater than 90 degrees
Definition of congruent angles
Angle A and Angle B are congruent iff the measure of angle A is equal to the measure of angle B
Definition of Congruent Segments
Segment A is congruent to Segment B iff AB=CD
There are _____ minutes in a degree
60
There are _____ seconds in a minute
60
Definition of supplementary angle
Angle A and B are supplementary iff the measure of angle A is equal to the measure of angle B
Definition of Linear Pair
Angle ABC and Angle CBD form a linear pair iff Ray BA and Ray BD are opposite rays
Definition of Vertical Angles
Two angles form vertical angles if they are formed by intersecting lines and they are not a linear pair
If two angles form a linear pair then they are ______
Supplementary
Definition of Adjacent
Two angles are adjacent if they share a common side and a common vertex
Supplements of congruent angles are _____
congruent.
Vertical Angle Theorem
Vertical Angles are congruent
Supplements of the same angle are______
Congruent
For any two angles, AOB and BOC, if A-B-C, then ______
the measure of AOB + the measure of BOC is equal to the measure of AOC
Definition of Perpendicular
Two lines are perpendicular iff they form right angles
If two angles form a linear pair and are congruent, they are _____
Right angles
Definition of angle bisector
Ray OB is the angle bisector of Angle AOC iff the measure of AOB is equal to the measure of BOC
Definition of Right Triangle
A triangle with a right angle
CPCTC
Corresponding parts of congruent triangles are congruent
Definition of legs
The shorter sides of a right triangle
Definition of hypotenuse
The longest side of a right triangle
Definition of acute triangle
A triangle in which all of the angles are acute
Definition of obtuse triangle
A triangle that contains an obtuse angle
Definition of isosceles triangle
A triangle in which at least two sides are equal in length
Definition of a scalene triangle
A triangle in which no sides are congruent
Triangle ABC is congruent to triangle DEF iff
All of the angles and sides of ABC are congruent to their corresponding angle or side DEF
SSS
If all of the sides of ABC are congruent to all of the sides of DEF then ABC is congruent to DEF
SAS
If two sides of ABC and the angle in between them are congruent to their corresponding sides and angle in DEF then ABC is congruent to DEF
ASA
If two angles of ABC and the side between them are congruent to their corresponding angles and side in DEF then ABC is congruent to DEF
ITT
The base angles of an isoceles triangle are congruent
Converse of the ITT
If the base angles of a triangle are congruent then the triangle must be isoceles
Definition of equilateral
A triangle is equilateral iff all three sides are congruent
Definition of equiangular
A triangle is equiangular iff all three of its angles are congruent
Definition of Median
A median is a segment in a triangle whose endpoints are a vertex of the triangle and the midpoint of the opposite side
Definition of parallel lines
Two lines are parallel iff they don't intersect
The parallel postulate
If you have line l and a point p not on l then there is at most one line through P that is parallel to l
Definition of exterior angle
An exterior angle of triangle ABC is angle BCD where A-C-D
The Exterior Angle Theorem
The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle
AIP
If two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel
If you have line l and point P not on L then there is _______ line through P that is parallel to l
at least
CAP
If two lines cut by a transversal form congruent corresponding angles then the lines must be congruent
PAI
If two parallel lines are cut by a transversal, then the alternate interior angles must be congruent
The transitive property of congruent angles
If angle 2 is congruent to angle 1 and angle 1 is congruent to angle 3 then angle 2 is congruent to angle 3
The sum of the angles of any triangle add to _____
180
PCA
If two parallel lines are cut by a transversal, then the corresponding angles are congruent
Strong form of the Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles
No triangle can contain ______ right angles
Two
If P is not on line l there is at most ____ perpandicular from P to line l
one
Definition of Complementary
Two angles are complementary iff their measure add to 90 degrees
Definition of bisect
A point bisects a line segment if it cuts it into two congruent segments
The measures of each o the angles of an equilateral triangle is equal to ____
60 degrees
If one pair of alternate interior angles is congruent then ______
the other pair must also be congruent
AA-AAA
If two angles of one triangle are congruent, respectively, to two angles of a second triangle, then the third angles of each triangle must be congruent
AAS
If two angles in a triangle, and a side not between them, are congruent to their corresponding angles and side in another triangle, then the two triangles are congruent
If two lines are perpendicular to the same line, they are ____
parallel
If a line is perpendicular to one of two parallel lines then it is ______ to the other
perpendicular
Erect a perpendicular theorem
From every point on a line, there is a perpendicular to that line through that point
If two lines intersect and for a linear pair of congruent angles, then the lines are _____
perpendicular
Drop a perpendicular theorem
Given a point P not on l there is at least one perpendicular from P to l
Hypotenuse-Leg theorem
Two right triangles are congruent if their hypotenuses and one pair of corresponding sides are congruent
If two lines intersect and form one right angle, then all four angles are _______ and the are all congruent to each other
Right angles
Definition of perpendicular bisector
Line l is the perpendicular bisector of segment AB iff it is perpendicular to AB and it bisects AB
The perpendicular bisector theorem
A point is on the perpendicular bisector of a segment iff it is equidistant from the endpoints of the segment
The foot of the perpendicular from P to l is the point where ____
the perpendicular from P to l intersects l
Definition of distance from P to l
The distance from P to the foot of the perpendicular from P to l
Any point that is equidistant from the sides of an angle is ______
on the bisector of that angle
In an isosceles triangle the segment from the vertex which is perpendicular to the opposite side is also a _____
median
If two points, P and Q, are each equidistant from the endpoints of segment AB, then line PQ is the perpendicular bisector of segment AB then _______
It is the perpendicular bisector of AB
If angle A and angle B are right angles,
then angle A is congruent to angle B
In triangle ABC if segment CQ is a median and if ray CQ is also the angle bisector of angle ACB then ____
triangle ABC is isoceles
In an isosceles triangle the angle bisector is _______ to the opposite side
perpendicular
In any triangle, if the angle bisectors of two angles are congruent, then the triangle is _____
isosceles
The opposite sides of a parallelogram are ______
congruent
In parallelogram ABCD, triangle ABC is congruent to triangle _______
CDA
The opposite angles of a parallelogram are ____
congruent
If the opposite sides of a quadrilateral are congruent it is a _____
parallelogram
If one pair of opposite sides of a quadrilateral are both parallel and congruent, then it is a ______
parallelogram
If both pairs opposite angles of a quadrilateral are both congruent, then its is a ________
parallelogram
Definition of Midsegment
A segment joining the midpoints of two sides of a triangle
Midsegment theorem
In any triangle, a midsegment is parallel to the third side and is equal to half of the length of the third side
Converse of the Midsegment theorem
In any triangle, a line which passes through the midpoint of one side and is parallel to the third side must pass through the midpoint of the second side
If three parallel lines intercept congruent segments on one transversal then they will ______ on any transversal
intercept congruent segments
If three lines intercept congruent segments on two transversals and two of the lines are parallel, then __________
all three lines are parallel
The diagonals of a parallelogram ____ each other
bisect
If the diagonals of a quadrilateral bisect each other then the quadrilateral is a ______
parallelogram
Connect the midpoints of the sides of any quadrilateral and you get a _____
parallelogram
The base angles of an isosceles trapezoid are _____
congruent
the diagonals of an isosceles trapezoid are _____
congruent
In any quadrilateral, the segments joining the midpoints of the opposite side _____ each other
bisect
If the diagonals of a quadrilateral are perpendicular to each other and bisect each other, then it is a _____
rhombus
The area formula for triangles, trapezoids, rectangles, rhombuses and squares where m is the length of the mid segment and h is the height
A=mh
Every polygon has a number attached to it which we will call its area. These areas will be _____ numbers
positive
If two triangles are congruent, then their areas are _____
equal
The area of a polygon is the ____ of the areas of the (non overlapping) triangles inside of it.
sum
The area of a square is the length of a side ______
squared
the semiperimeter of a triangle whose sides are a, b, and c, is _____
1/2(a+b+c)
The area of a rectangle of width w and length l is given by the formula
A = wl
In any triangle, a^2+b^2=
c^2
The area of a triangle whose sides are a, b, and c is equal to ________ where s is the semiperimeter.
the square root of s(s-a)(s-b)(s-c)
In any triangle whose sides are a, b, and c, a+b_c, b+c_a
c+a_b
>
Converse of the Pythagorean theorem
In any triangle, if a^2 + b^2 = c^2, then it is a right triangle
Triangle ABC is similar to Triangle DEF iff
all the angles are congruent to their corresponding angles, and all corresponding sides are proportional
if two triangles have two sets of congruent corresponding angles, they are
similar
GCM Theorem
A line parallel to one side of a triangle intercepts proportional segments on the other two sides, assuming it hits them.
If the first triangle is similar to the second, and the second is congruent to the third, then the first must be _______ to the third
similar
SAS similiarity
If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar
SSS similarity
if the corresponding side lengths of two triangles are proportional, then the triangles are similar
An altitude of a triangle
The perpendicular segment from a vertex to the opposite side or to the line that centains the opposite side.
In similar triangles, the altitudes are _____ to the proportional sides
corresponding
Angle Bisector Theorem
In any triangle, an angle bisector divides the opposite side in the same ration as the other two sides
In similar triangles the areas are proportional to the _________ of corresponding sides
square of the ratio
Transitive property of similar triangles
If triangle a is similar to triangle b, and triangle b is similar to triangle c, then triangle a is similar to triangle c
Three parallel lines intercept _______ on any two transversals
proportionate
Converse of the Angle Bisector Theorem
If a segment from the vertex of a triangle to the opposite side divides the side in the same ration as the other two sides, the it is an angle bisector
In similar triangles, corresponding angle bisectors are _________ to corresponding sides
proportional
In similar triangles, corresponding medians are _______ to corresponding sides
proportional
In a right triangle the altitude to the hypotenuse divides the triangle into __________
three similar triangles
In a right triangle, the altitude to the hypotenuse is the ______ of the lengths of the two segments on the hypotenuse
mean proportional
A chord is
a straight line connecting two points on a circle
A diameter is
A straight line passing from side to side through the center of a circle or sphere.
A secant is
A line which contains a chord
A tangent is
a line which intersects a circle at only one point
Point P is in the interior of angle A if
it lies on a segment whose endpoints are on the sides of the angle
Two circles are concentric if
they have the same center
If a radius is perpendicular to a chord, it ____ it
bisects
A tangent is ________ to a radius drawn to the point of tangency
perpandicular
An angle is a central angle of circle o if
its vertex is at O
A minor arc is
all the points of a circle that are on the interior of a central angle
A major arc is
all of the points of a circle that are not on a minor arc
A semicircle is
the set of all points of a circle that are on ne side of a line that contains a diameter
The measure of minor arc AB in circle O is equal to
the measure of angle AOB
The measure of a semicircle is
180
the measure of a major ark is
360- the measure of its corresponding major ark
An angle BAC is an inscribed angle of a circle iff
A B and C are on the circle
Inscribed Angle Theorem
The measure of an inscribed angle is half of the measure of the intercepted arc
The intersecting chords theorem
If any two chords of a circle intersect, then ab=cd
if a radius bisects a non diameter chord, it is _______ to that chord
perpandicular
The perpendicular bisector of a chord ___________ the center of a circle
passes through
No circle contains three ______ points
collinear
Two distinct circles can intersect in at most ____ different points
two
Three noncollinear points can lie on at most ______ circle
one
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