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Title: Geometry Study Guide
Description: For students who have first year in Geometry
Description: For students who have first year in Geometry
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Sabrina Sarwar
Postulates:
Segment Addition Postulate:
If C is between AB then AC + CB = AB
Angle Addition Postulate:
If point B lies on the interior of
A line contains at least two points
A plane contains at least three points, not on the same line
Space contains at least four points, not on the same line
Through any two points there is exactly one line
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 on that plane
If two planes intersect, then their intersection is a line
If two lines intersect, only one plane contains both the lines
...
If two lines are cut by a transversal and the corresponding angles are congruent, then the lines
are parallel
SSS
If three sides of one triangle are congruent to three sides of another triangle, then the triangles
are congruent
...
ASA
If two angles and the included side of one triangle are congruent two angles and the included
side of another triangle, then the triangles are congruent
...
Arc Addition Postulate
The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs
...
5AB = AM and
...
5m
If two angles are supplements of congruent angles (or of the same angle) then the two angles
are congruent
...
a = a
Symmetric Property: If a = b, then b = a
Transitive Property: If a = b and b = c, then a = c
Addition Postulate: If equal quantities are added to equal quantities, the sums are equal
...
Multiplication Postulate: If equal quantities are multiplied by equal quantities, the products are
equal
...
Substitution Postulate: A quantity may be substituted for its equal in any expression
...
Sabrina Sarwar
Parallel Lines and Planes
If two parallel lines are cut by a third plane, then the lines of intersection are parallel
If two parallel lines are cut by a transversal
The alternate interior/exterior angles are congruent
The same side interior angles are supplementary
The corresponding angles are congruent
...
In a plane, two lines perpendicular to the same line are parallel
...
Two lines parallel to the third line are parallel to each other
The sum of the measures of the angles of a triangle is 180
If two angles of one triangle are congruent to two angles of another, then the third angles
are congruent
Each angle of an equiangular triangle measures 60
In a triangle, there can be most one right 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 exterior angles of any convex polygon, one angle at each
vertex is 360
...
If a point lies on the bisector of an angle, then the point is equidistant from the sides of an angle
...
Quadrilaterals
Sabrina Sarwar
Opposite sides of a parallelogram are congruent
Opposite angles of a parallelogram are congruent
Diagonals of a parallelogram bisect each other
If both pairs of opposite sides of a quadrilateral are congruent, then the quad
...
If one pair of opposite sides of a quad
...
Is a
parallelogram
...
are congruent, then the quad
...
If the diagonals of a quad
...
If two lines are parallel, then all points on one line are equidistant from the other line
If three parallel lines are 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 o f a triangle and is parallel to another side passes
through the midpoint of the third side
The segment that joins the midpoints of two sides of a triangle
Is parallel to the third side
Is 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
...
If three parallel lines intersect two transversals, then they divide the transversals
proportionally
Triangle Angle Bisector theorem
If a ray bisects an angle of a triangle, then it divides the opposite into segments proportional to
the other two sides
...
When the altitude is drawn to the hypotenuse if a right triangle, the length of the altitude
is the geometric mean between the segments of the hypotenuse
...
Pythagorean Theorem
a^2 + b^2 = c^2
If the square of one side of a triangle is equal to the sum of the squares of another two sides,
then the triangle is a right triangle
If the square of the longest side of a triangle is less than the sum of the squares of the other two
sides, then the triangle is an acute angle tri
...
454590:
Hypotenuse is radical 2 times as long as the leg
306090:
The hypotenuse is twice as long as the shorter leg, and the longer leg is radical 3 times as the
shorter leg
...
In the same circle or in congruent circles, two minor arcs are congruent if and only if their central
angles are congruent
...
Tangents to a circle is from a point are congruent
Congruent arcs have congruent chords
Congruent chords are equally distant from the center
If two inscribed angles intercept the same arc, then the angles are congruent
An angle inscribed in a semicircle is a right angle
If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary
Congruent arcs have congruent central angles
...
Sabrina Sarwar
The locus of points at a fixed distance, d, from a line, l, is a pair of parallel lines d distance from
l and on either side of l
...
The locus of points equidistant from two parallel lines, l1 and l2 , is a line parallel to both l1 and l2
and midway between them
...
Area
If the scale factor of two similar figures is a:b then
The ratio of the perimeters is a:b
Sabrina Sarwar
The ratio of the areas is a2 : b2
Square: s 2
Rectangle: bh
Parallelogram: bhl
Triangle: ½(bh)
Rhombus: A = ½ (d1 ∙ d2)
Trapezoid: ½ h (b1 + b2)
Regular Polygon: ½ ap
Circle: πr2
Any Sector: ½r2θ or θ/360πr2 or [(Arc Length)/2πr]πr2
Semicircle: ½ πr2
Quartercircle: ¼ πr2
Volume
Surface Area: Sum all of the areas of all of the surfaces (units squared) SA= LA+2B
Lateral Area: A lateral face is a face that is not the base (units squared)
If the scale factor of two similar solids is a:b then,
The ratio of corresponding perimeters is a:b
The ratio of the base areas, and of the lateral areas, and of the total areas a2 : b2
The ratio of the volumes is a3 : b3
Cube:
V = s 3
SA= 6s2
LA= 4s2
Sphere:
V = or 4/3πr3
SA = 4πr2 or πd2
Rectangular Prism:
V= lwh
SA= 2lh+2hw+2lw
LA= (2l+2w)h
Cone:
V = ⅓ πr2h
SA =πrl + πr2
LA = πrl
Cylinder:
V = πr2h
SA = 2πrh + 2πr2
LA = 2πrh (right cylinder)
Pyramid:
V = ⅓ Bh
SA = B + ½ Pl
LA = ½ Pl
Hemisphere:
V = (2/3)πr3
SA = 3πr2
Coordinate Geometry
Sabrina Sarwar
OR x2+ y2 = r2 origin circle
Reflection
Dilation
Rotation
Transformation
Transformations
Sabrina Sarwar
An isometry maps a shape to a congruent shape
...
An isometry maps a polygon to a polygon with the same area
...
distance (lengths of segments are the same)
angle measures (remain the same)
parallelism (parallel lines remain parallel)
collinearity (points stay on the same lines)
midpoint (midpoints remain the same in each figure)
orientation (lettering order NOT preserved
...
)
A translation is an isometry
...
same shape and size,
turned in different directions
...
A dilation maps an age to a congruent angle
A dilation DO,k maps any segment to a parallel segment, k times as long
A dilation DO,k maps any polygon to a similar polygon whose area is k2 times as large
The composite of two isometries is an isometry
...
The translation glides all points
through twice the distance from the first line of reflection to the second
...
The measure of the angle of rotation is twice the measure of the angle from the
first line of reflection to the second
...
Vocab
Sabrina Sarwar
Collinear Points: points that lie on the same line
...
Opposite Rays: 2 rays that lie on the same line, with a common endpoint and no other points in
common
...
Cross Section: Parallel to the base
Concurrent: Lines that contain the same point
Median: of a triangle is a segment joining any vertex to the midpoint of the opposite side
...
The medians of a triangle intersect in a
point that is two thirds of the distance from each vertex to the midpoint of the opposite side
Altitude: of a triangle is a segment from any vertex perpendicular to the line containing the
opposite side
...
Angle Bisectors: A line that bisects the angle
...
It
creates an inscribed circle
...
The point where they meet is
called the circumcenter
...
)
Negation: Opposite the statement
Conjunction (^): both statements must be true (and)
Disjunction (v): (or) either or both facts must be true
Conditional: IF
...
All other cases are TRUE
...
mathsisfun
...
html
Every other constructions:
http://www
...
com/constructions
Title: Geometry Study Guide
Description: For students who have first year in Geometry
Description: For students who have first year in Geometry