182 terms

Geometry Honors Midterm

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Space
the set of all points
Collinear points
points all in one line
Coplanar points
points all in one plane
Intersection
the set of points that are in both figures
Postulates
Statements that are accepted without proof
Congruent
Two objects that have the same size and shape
Congruent segments
segments that have equal lengths
midpoint of a segment
the point that divides the segment into two congruent segments
bisector of a segment
a line, segment, ray, or plane that intersects the segment at its midpoint
angle
the figure formed by two rays that have the same endpoint
sides (angles)
The two rays of an angle
vertex
the common endpoint of an angle
Congruent angles
angles that have equal measures
Adjacent angles
two angles in a plane that have a common vertex and a common side but no common interior points.
bisector of an angle
is the ray that divides the angle into two congruent adjacent angles
theorems
Important statements that are proved
Segment Addition Postulate
If 𝐵 is between 𝐴 and 𝐶, then 𝐴𝐵 + 𝐵𝐶 = 𝐴𝐶
Angle Addition Postulate
If point 𝐵 lies in the interior of ∠𝐴𝑂𝐶, then 𝑚∠𝐴𝑂𝐵 + 𝑚∠𝐵𝑂𝐶 = 𝑚∠𝐴𝑂𝐶
line
contains at least two points
plane
contains at least three points not all in one line
space
contains at least four points not all in one plane
Through any two points
there is exactly one line
Through any three points
there is at least one plane
through any three noncollinear points
there is exactly one plane
If two points are in a plane
then the line that contains the points is in that plane
If two planes intersect
then their intersection is a line
If two lines intersect (point)
then they intersect in exactly one point
Through a line and a point not in the line
there is exactly one plane
If two lines intersect (plane)
then exactly one plane contains the lines
Three undefinable terms
point, line, plane
Conditional Statements
If p, then q (q if p) (p implies q)
Converse
If q, then p
counterexample
An if-then statement is false if an example can be found for which the hypothesis is true and the conclusion is false
biconditional
If a conditional and its converse are both true, they can be combined into a single statement by using the words "if and only if." A statement that contains the words "if and only if"
Deductive Reasoning
proving statements by reasoning from accepted postulates, definitions, theorems, and given information
Complementary angles
two angles whose measures have the sum 90°.
Supplementary angles
two angles whose measures have the sum 180°.
Vertical angles
two angles such that the sides of one angle are opposite rays to the sides of the other angle
Perpendicular lines
two lines that intersect to form right angles
APOE
If 𝑎 = 𝑏 and 𝑐 = 𝑑, then 𝑎 + 𝑐 = 𝑏 + 𝑑.
SPOE
If 𝑎 = 𝑏 and 𝑐 = 𝑑, then 𝑎 − 𝑐 = 𝑏 − 𝑑.
MPOE
If 𝑎 = 𝑏, then 𝑐𝑎 = 𝑐𝑏.
DPOE
If 𝑎 = 𝑏 and 𝑐 ≠ 0, then a/c = b/c
Subst
If 𝑎 = 𝑏, then either 𝑎 or 𝑏 may be substituted for the other in any equation (or inequality).
RPOE
𝑎 = 𝑎
SyPOE
If 𝑎 = 𝑏, then 𝑏 = 𝑎.
TPOE
If 𝑎 = 𝑏 and 𝑏 = 𝑐, then 𝑎 = 𝑐
Dist.
𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐
RPOC
𝐷𝐸 ≅ 𝐷𝐸
∠𝐷 ≅ ∠𝐷
SyPOC
If 𝐷𝐸 ≅ 𝐹𝐺, then 𝐹𝐺 ≅ 𝐷𝐸
If ∠𝐷 ≅ ∠𝐸, then ∠𝐸 ≅ ∠𝐷
TPOC
If 𝐷𝐸 ≅ 𝐹𝐺 and 𝐹𝐺 ≅ 𝐽𝐾, then 𝐷𝐸 ≅ 𝐽𝐾.
If ∠𝐷 ≅ ∠𝐸 and ∠𝐸 ≅ ∠𝐹, then ∠𝐷 ≅ ∠𝐹
Mdpt Thm
If 𝑀 is the midpoint of 𝐴𝐵, then 𝐴𝑀 = 1/2 AB and BM = 1/2 AB
Angle Bisector Thm
If ray BX is the bisector of ∠𝐴𝐵𝐶, then 𝑚∠𝐴𝐵𝑋 = 1 /2 𝑚∠𝐴𝐵𝐶 and 𝑚∠𝑋𝐵𝐶 = 1 /2
𝑚∠𝐴𝐵𝐶
Vertical Angle Thm
Vertical angles are congruent
If two lines are perpendicular,
then they form congruent adjacent angles
If two lines form congruent adjacent angles,
then the lines are perpendicular
If the exterior sides of two adjacent acute angles are perpendicular,
then the angles are complementary
If two angles are supplements of congruent angles (or of the same angle),
then the two angles are congruent.
If two angles are complements of congruent angles (or of the same angle),
then the two angles are congruent.
Skew lines
noncoplanar lines that are neither parallel nor intersecting
Parallel lines
coplanar lines that do not intersect
transversal
a line that intersects two or more coplanar lines in different points
Alternate interior angles
two nonadjacent interior angles on opposite sides of the transversal
Same-side interior angles
two interior angles on the same side of the transversal
Corresponding angles
two angles in corresponding positions relative to the two lines
If two parallel lines are cut by a transversal, (corresponding angles)
then corresponding angles are congruent
If two lines are cut by a transversal and corresponding angles are congruent,
then the lines are parallel
If two parallel planes are cut by a third plane,
then the lines of intersection are parallel.
If two parallel lines are cut by a transversal,
then alternate interior angles are congruent.
If two parallel lines are cut by a transversal,
then same-side interior angles are supplementary
If a transversal is perpendicular to one of two parallel lines,
then it is perpendicular to the other one also
If two lines are cut by a transversal and alternate interior angles are congruent,
then the lines are parallel
If two lines are cut by a transversal and same-side interior angles are supplementary,
then the lines are parallel
In a plane, two lines perpendicular to the same line
are parallel.
Through a point outside a line, (parallel)
there is exactly one line parallel to the given line
Through a point outside a line (Perpendicular)
there is exactly one line perpendicular to the given line.
Two lines parallel to a third line
are parallel to each other
triangle
the figure formed by three segments joining three noncollinear points
Vertex
Each of the three points of the triangle
sides (triangle)
The segments of the triangle
Scalene triangle
no congruent sides of triangle
Isosceles triangle
at least two congruent sides of triangle
Equilateral triangle
all sides congruent of triangle
acute triangle
three acute angles in triangle
obtuse triangle
one obtuse angle in triangle
right triangle
one right angle in triangle
Equiangular triangle
triangle where all angles are congruent
auxiliary line
a line (or ray or segment) added to a diagram to help in a proof
corollary
a statement that can be proved easily by applying the theorem
remote interior angles
The two interior angles of the triangle that are nonadjacent to the exterior angle
exterior angle
Formed when one side of a triangle is extended
convex polygon
a polygon such that no line containing a side of the polygon contains a point in the interior of the polygon
Quadrilateral
Polygon with 4 sides
Pentagon
Polygon with 5 sides
Hexagon
Polygon with 6 sides
Heptagon
Polygon with 7 sides
Octagon
Polygon with 8 sides
Nonagon
Polygon with 9 sides
Decagon
Polygon with 10 sides
Undecagon
Polygon with 11 sides
Dodecagon
Polygon with 12 sides
n-gon
Polygon with n sides
diagonal
A segment joining two nonconsecutive vertices
regular polygons
Polygons that are both equiangular and equilateral
(n-2)180
Sum of the interior angles of a convex 𝑛-gon
(n-2)180/n
Measure of one interior angle of a regular 𝑛-gon
360
Sum of the exterior angles of a convex 𝑛-gon
360/n
Measure of one exterior angle of a regular 𝑛-gon
n-3
Number of diagonals you can draw from one vertex of a convex 𝑛-gon
n(n-3)/2
Total number of diagonals you can draw in a convex 𝑛-gon
The sum of the measures of the angles of a triangle
180°
If two angles of one triangle are congruent to two angles of another triangle,
then the third angles are congruent.
Each angle of an equiangular triangle has measure
60°
In a triangle, (type of angles and how many)
there can be at most one right angle or obtuse angle
The acute angles of a right triangle
are complementary
The measure of an exterior angle of a triangle equals
the sum of the measures of the two remote interior angles
The sum of the measures of the angles of a convex polygon with 𝑛 sides
(𝑛 − 2)180°
The sum of the measures of the exterior angles of any convex polygon, one angle at each vertex
360°
All right angles are
congruent
Since congruent triangles have the same shape, their corresponding angles
are congruent
Since congruent triangles have the same shape, their corresponding sides
are congruent
Two triangles are congruent if and only if their vertices can be matched up so that
the corresponding parts (angles and sides) of the triangles are congruent
A line and a plane are perpendicular if and only if
they intersect and the line is perpendicular to all lines in the plane that pass through the point of intersection
isosceles triangle
triangle with at least two congruent sides
All equilateral triangles are
isosceles
Some isosceles triangles are
equilateral
hypotenuse
In a right triangle, the side opposite the right angle
legs (triangle)
other two sides of a triangle that isnt hypotenuse
SSS
If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent
SAS
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent
ASA
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
Isosceles Triangle Thm
If two sides of a triangle are congruent, then the angles opposite those sides are congruent
An equilateral triangle is also
equilangular
An equilateral triangle has
three 60° angles
The bisector of the vertex angle of an isosceles triangle is
perpendicular to the base at its midpoint
If two angles of a triangle are congruent,
then the sides opposite those angles are congruent.
AAS
If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent
HL
If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent
median (triangle)
a segment from a vertex to the midpoint of the opposite side. Every triangle has 3. Always lie inside the triangle
altitude
the perpendicular segment from vertex to the line that contains the opposite side. In an acute triangle, they are all inside the triangle. In a right triangle, two of them are parts of the triangle. They are the legs of the right triangle. The third one is inside the triangle. In an obtuse tringle, two of them are outside the triangle
perpendicular bisector
a line (or ray or segment) that is perpendicular to the segment at its midpoint. There are three of them that can be drawn in any triangle.
angle bisector
a segment, ray, or line which bisects an angle of the triangle
concurrent
When two or more lines interesct in one point they are said to be this
point of concurrency
the point in which concurrent lines intersect
centroid
the point where the medians of a triangle meet and always lies inside the triangle
If a point lies on the perpendicular bisector of a segment,
then the point is equidistant from the endpoints of the segment.
If a point is equidistant from the endpoints of a segment,
then point lies of the perpendicular sector of the segment
If a point lies on the bisector of an angle,
then the point is equidistant from the sides of the angle
If a point is equidistant from the sides of an angle,
then the point lies on the bisector of the angle
The centroid of a triangle is two-thirds of the distance from
each vertex to the midpoint of the opposite side
parallelogram
a quadrilateral with both pairs of opposite sides parallel
rectangle
quadrilateral with four right angles
rhombus
quadrilateral with four congruent sides
square
quadrilateral that is both a rectangle and a rhombus
trapezoid
quadrilateral with exactly one pair of parallel sides
bases
parallel sides of a trapezoid
legs (trapezoid)
sides of trapezoid that arent the bases
isosceles trapezoid
trapezoid with congruent legs
median (trapezoid)
the segment that joins the midpoints of the legs of a trapezoid
opposite sides of a parallelogram are
congruent
opposite angles of a parallelogram are
congruent
diagonals of a parallelogram
bisect each other
consecutive angles of a parallelogram are
supplementary
diagonal of a parallelogram forms
two congruent triangles
If both pairs of opposite sides of a quadrilateral are parallel
then the quadrilateral is a paralleogram
If both pairs of opposite sides of a quadrilateral are congruent
then the quadrilateral is a paralleogram
If one pair of opposite sides of a quadrilateral is both congruent and parallel
then the quadrilateral is a parallelogram
If both pairs of opposite angles of a quadrilateral are congruent
then the quadrilateral is a parallelogram
If the diagonals of a quadrilateral bisect each other
then the quadrilateral is a parallelogram
If two lines are parallel
then all points on one line are equidistant from the other line
If three parallel lines cut off congruent segments on one transversal
then they cut off congruent segments on every transversal
A line that contains the midpoint of one side of a triangle and is parallel to another side passes through
the midpoint of the third side
Midsegment
segment that joins the midpoints of two sides of a triangle
Midsegment is parallel to and half as long as
the third side
The diagonals of a rectangle are
congruent
The diagonals of a rhombus are
perpendicular
Each diagonal of a rhombus bisects
two angles of the rhombus
The midpoint of the hypotenuse of a right triangle
is equidistant from the three vertices
If an angle of a parallelogram is a right angle
then the parallelogram is a rectangle
If two consecutive sides of a parallelogram are congruent
then the parallelogram is a rhombus
Base angles of an isosceles trapezoid are
congruent
Median of trapezoid is
parallel to the bases and has a length equal to the average of the base lengths
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