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Geometry Chapter 4 Cumlative
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Terms in this set (83)
congruent polygons
polygons in which all sides and all angles are congruent
corollary
A statement that follows immediately from a theorem
hypotenuse
The longest side of a right triangle, opposite the right angle
base angles of an isosceles triangle
The 2 non-vertex angles of an isosceles triangle, formed by the legs and base
base of an isosceles triangle
the side opposite the vertex angle, the non-congruent side
legs of an isosceles triangle
the 2 congruent sides of an isosceles triangle
legs of a right triangle
the 2 shortest legs of a right triangle, adjacent to the right angle
vertex angle of an isosceles triangle
the angle formed by the 2 congruent legs of an isosceles triangle
3rd Angles Theorem
If 2 angles of 1 triangle are congruent to 2 angles of another triangle, then the 3rd angles are also congruent
SSS Postulate
If the 3 sides of 1 triangle are congruent to the 3 sides of another triangle, then the triangles are congruent (SSS)
SAS Postulate
If 2 sides and the included angle of 1 triangle are congruent to 2 sides and the included angle of another triangle, then the triangles are congruent (SAS)
AAS Theorem
If 2 angles and a non-included side of 1 triangle are congruent to 2 angles and the corresponding non-included side of another triangle, then the triangles are congruent. (AAS)
ASA Postulate
If 2 angles and the included side of 1 triangle are congruent to 2 angles and the included side of another triangle, then the triangles are congruent (ASA )
Isosceles triangle theorem
If 2 sides of a triangle are congruent, then the angles opposite those sides are congruent.
Converse of isosceles triangle theorem
If 2 angles of a triangle are congruent, then the sides opposite those angles are congruent.
Hypotenuse Leg Theorem
If the hypotenuse and a leg of 1 right triangle are congruent to the hypotenuse and a leg of another triangle, then the triangles are congruent.
CPCTC
Corresponding Parts of Congruent Triangles are
bisector of isosceles vertex is Perpendicular bisector of base
If a line bisects the vertex angle of an isosceles triangle, then the line is also the perpendicular bisector the base
If a triangle is equilateral... (Corollary 1)
If a triangle is equilateral, then it is also equiangular
Corollary 2 (An equilateral triangle has...)
An equilateral triangle has 3 60 degrees angles
Corollary 3 (The bisector of the vertex of an isosceles triangle...)
The bisector of the vertex of an isosceles triangle is perpendicular to the base at its midpoint.
An equiangular triangle is...
An equiangular triangle is also equilateral.
Median
A segment from a vertex to the midpoint of the opposite sides
Altitude
The perpendicular segment from a vertex to the line that contains the opposite side
Where do some altitudes fall in a right triangle?
2 of the 3 altitudes fall on the same line as the legs.
Where do altitudes fall in an obtuse triangle?
2 of the 3 altitudes fall outside the triangle
Perpendicular bisector
In a given plane, there is exactly one line perpendicular to a segment at its midpoint.
Theorem 4-5 (If a point lies on the perpendicular bisector...)
If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of a segment
Theorem 4-6 (If a point t from the endpoints of a segment...)
If a point is equidistant from the endpoints of a segment, then the point lies in on the perpendicular bisector of a segment
Right angles theorem
All right angles are congruent
Theorem 4-7 (If a point lies on the bisector of an angle...)
If a point lies on the bisector of an angle, then the point is equidistant from the sides of angle.
Theorem 4-8 (If a point is equidistant from the sides of angle....)
If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle.
Equilateral Triangle
A triangle with three congruent sides
Equiangular Triangle
A triangle with 3 congruent angles
Centriod
Point of concurrency formed by the medians of a triangle
Orthocenter
Point of concurrency formed by the altitudes of a triangle
The orthocenter of a right triangle
Falls on the vertex of the right angle
Circumcenter
point of concurrency of the perpendicular bisectors of the three sides of a triangle
The distance from a point to a line
The length of the perpendicular segment from the point to the line
incenter
Point of concurrency for the angle bisectors of a triangle
CA Angles
Perpendicular lines form congruent adjacent angles
shared lines...
shared lines are congruent
line Perpendicular to a plane
a line that intersects the plane in a point and is perpendicular to every line in the plane the passes through the line of intersection
A line and a plane are perpendicular
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
Measure of one interior angle of a polygon equation
(n-2)180/n
Corollary 4 (Acute angles of a right triangle)
The acute angles of a right triangle are complementary.
Sum of interior angles equations
(n-2)180
Sum of exterior angles of a polygon
360
Measure of one exterior angle equation
360/n
Number of diagonals equation
n(n-3)/2
Posutlate ten (corr, parralel)
If two parallel lines are cut by a transversal, then corresponding angles are congruent.
third plane theorem
If two parallel planes are cut by a third plane, then the lines of intersection are parallel
S-S int theorem
If two parallel lines are cut by a transversal, then same-side interior angles are supplementary
2 lines perpendicular theorem
In a plane two lines perpendicular to the same line are parallel
Theorems 3.8 (outside ll)
Through a point outside a line there is exactly one line parallel to the given line
Theorem 3.9 (outside perpendicular)
Through a point outside a line, there is exactly one line perpendicular to the given line
Theorem 3.12 (Measure of exterior angle of a triangle)
The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles.
Remote interior angle
An interior angle that is not adjacent to the exterior angle
Exterior angle of a triangle
equals sum of opposite interior angles
Theorem 3.13 (angles of convex polygon)
The sum of the measures of the angles of a convex polygon with n sides is (n-2)180
Scalene triangle
Zero congruent sides
Isosceles triagnle
2 or more equal sides
Equilateral
All sides are equal
What polygon has 3 sides?
Triangle
A polygon with 4 sides
Quadrilateral
A polygon with 5 sides
Pentagon
A polygon with 6 sides
Hexagon
Polygon with 7 sides
Heptagon
A polygon with 8 sides
Octogon
polygon with 9 sides
nonagon
A polygon with 10 sides
Dodecagon
A polygon with 12 sides
dodecagon
Regular polygon is...
Equilateral and equiangular
Ruler Postulate
The points on a line can be put into a one-to-one correspondence with the real numbers.
Protractor Postulate
the measurement of an angle between two rays can be designated as a unique number, and this number would be between 0 and 180 degrees
Theorem 2.8 (complement of congruent angles)
If two angles are complements of congruent angles, then the two angles are congruent
Addition Postulate
If B is between A and C, then AB + BC is = AC
Angle Addition Postulate
the measure of the larger angle is the sum of the measures of the two smaller ones.
Postulate 6 (through any two points...)
Through any two points there is exactly one line
A line contains at least...
A line contains at least two points; a plane contains at least three points not all in one line; space contains at least four points not all in one plane.
Midpoint theorem
If M is the midpoint of AB, then AM=1/2AB and MB=1/2AB
Angle bisector Theorem
If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle
Skew lines
If lines are skew, then they are not coplanar and they do not intersect
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