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Geometry Mastery: A Review Guide
Many students struggle in Geometry for two reasons:
1) Students are not familiar or comfortable with this style of learning. Proofs in Geometry are rooted in logical reasoning, and it takes hard work, practice, and time
for many students to get the hang of it.
2) Students do not study for Geometry enough. Not only must students learn to use logical reasoning to solve proofs in Geometry, but they must be able to recall
many theorems and postulates to complete their proof. In order to recall the theorems, they need to recognize which to use based on the information provided and
the figure, and they must have the information stored in memory to actually retrieve it.
This guide lists the theorems you will need to master in order to succeed in your Geometry class. This does not list every theorem proven in Geometry, but it should
cover the content you will see in your Geometry class. Follow the below tips to ensure you are well- prepared on your Geometry tests.
1.
Highlight your theorems. Read each definition, theorem, postulate, or property in this guide, then find it in your notes. Highlight the theorem and its
corresponding example or definition
2.
Create pictures to help you recall your theorems. Draw a picture next to each theorem that confuses you in this guide. The picture should represent
literally what the theorem states. Ask your teacher or a tutor for help if you have trouble determining what picture to draw.
Here is an example of a picture you should draw next to the theorem and its definition:
3.
Vertical Angles Theorem
If two angles are vertical, then they are congruent
A
C
A≅C
4.
Create flashcards. If you’re still having trouble remembering these concepts, create a flashcard for each theorem with its definition and example picture.
You will use this to memorize each theorem. On one side of the flashcard you should write the theorem’s name. On the other side you should write the
definition on the left and its picture on the right:
Theorem
Vertical Angles Theorem
Definition
Picture
If two angles are vertical,
then they are congruent
C
A≅C
5.
Refer to this guide when you are working on practice proofs, and use your flashcards each night to help you memorize the theorems. As you learn more
theorems in class, you will memorize more theorems, so this process is continuous.
6.
Finally, practice! Many students realize that the homework that his or her teacher assigns is not enough to completely master the concepts. You may
need to request extra work from your teacher or find additional problems elsewhere to give you plenty of practice.
7.
If you are serious about you class and you are determined to make a good grade, then preparation, practice, and memorization is essential. Using this
guide or creating one for yourself is the first step to organizing your theorems and effectively learning them.
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Basic Geometry
Postulates
Ruler Postulate
Each point on a number line can be paired to a unique real number
Segment addition postulate
Where B is between points A and C on a line: AB+BC= AC
Angle addition postulate
Where point P is inside ABC, ABP + PBC = ABC
Linear pair postulate
If two angles forma linear pair (on a line), they are supplementary
Theorems
Segment Congruence Theorem
Two segments are congruent if and only if they have the same length
Common Segments Theorem
For collinear points A, B, C, and D: If AB ≅ CD, then AC ≅ BD.
Angle Congruence Theorem
Two angles are congruent if and only if they have the same measure
Right Angle Congruence Theorem
If two angles are right angles, then they are congruent
Vertical Angles Theorems
If two angles are vertical, then they are congruent
Congruent Complements Theorem
If two angles are complementary to the same angles, then they are congruent
Congruent Supplements Theorem
If two angles are supplementary to the same angle, then they are congruent
Linear Pair Theorem
If two angles form a linear pair, then they are supplementary.
Basic Properties
Addition property of equality
If a=b, then a+c = b+c
Subtraction property of equality
If a=b, then a-c= b-c
Multiplication property of equality
If a=b, then ac=bc
Division property of equality
If a=b, then 𝑎⁄𝑐 = 𝑏⁄𝑐
Symmetric property
If a=b, then b=c
Transitive property
If a=b and b= c, then a=c
Substitution property
If a= b and b= 25, then a=25
Distributive property
If c(a+b)= ca + cb
Definitions
Midpoint
When P bisects segment AB, AP is congruent to PB
Angle Bisector
When segment BP bisects angle ABC, ABP is congruent to PBC
Congruent Segments
Line segments that have the same length
Congruent Angle
Angles that have the same measure
Perpendicular Lines
Two lines that form congruent adjacent angles that measure 90 degrees
Complementary Angles
Two adjacent angles whose sum measure 90 degrees
Supplementary Angles
Two adjacent angles whose sum measure 180 degrees
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Lines and angles
Postulates
Parallel postulate
If there is a line and point somewhere not on that line, then there exists exactly one line through the point that is
parallel to the other line
Perpendicular postulate
If there is a line and one point somewhere not on that line, then there exists exactly one line through the point
that is perpendicular to the other line.
Slope of parallel lines
Two lines are parallel if and only if they have the same slope
Slope of perpendicular lines
Two lines are perpendicular if and only if the product of their slopes is equal to -1
Corresponding angles postulate
If two parallel lines are cut by a transversal, then the corresponding angles are congruent
Corresponding angles converse
If two coplanar lines cut by a transversal result in congruent corresponding angles, then the two lines are parallel
Theorems
Same Side Interior Angles Theorem
If two parallel lines are cut by a transversal, then the two pairs of same side interior angles formed are
supplementary.
Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal, then the two pairs of alternate interior angles formed are congruent.
Alternate Exterior Angles Theorem
If two parallel lines are cut by a transversal, then the two pairs of alternate exterior angles formed are congruent.
Same side Interior Angles Converse
If two coplanar lines cut by a transversal result in a pair of same side interior angles that are supplementary, then
the two lines are parallel
Alternate Interior Angles Converse
If two coplanar lines cut by a transversal result in congruent alternate interior angles, then the two lines are
parallel
Alternate Exterior Angles Converse
If two coplanar lines cut by a transversal result in congruent alternate exterior angles, then the two lines are
parallel
Perpendicular Transversal Theorem
If a transversal line is perpendicular to one of two parallel lines, then it is also perpendicular to the other parallel
line.
Definitions
Coplanar
Points or lines existing in the same plane
Parallel Lines
Two coplanar lines that do not intersect
Transversal line
A line that intersects two or more coplanar lines
Skew Lines
Two lines that are not coplanar and do not intersect
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Triangles
Postulates
Side- Side- Side (SSS) Congruence Postulate
If all three sides of one triangle are congruent to the three sides of another triangle, then those triangles are
congruent.
Side- Angle- Side (SAS) Congruence
Postulate
If two sides and the included angle of one triangle are congruent to two sides and the included angle of
another triangle, then those triangles are congruent.
Angle- Side- Angle (ASA) Congruence
Postulate
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.
Angle- Angle (AA) Similarity Postulate
If two angles of one triangle are congruent to two angles of another triangle then those two triangles are
similar.
Theorems
Triangle Sum Theorem
The sum of the angles of a triangle always equals 180
Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the two non-adjacent angles.
Third Angle Theorem
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are
congruent.
Angle- Angle- Side Congruence Theorem
If two angles and a non-included side of one triangle are congruent to the corresponding two angles and a
non- included side of another triangle, then the two triangles are congruent.
Hypotenuse- Leg Theorem
If the hypotenuse and one leg of a triangle are congruent to the hypotenuse and leg of another triangle then
the triangles are congruent
Isosceles Triangle Theorem
If two sides of a triangle are congruent, then their opposite angles are also congruent
Perpendicular Bisector Theorem
If a perpendicular line is drawn so that a point is on the bisector of a segment, then the point is equidistant
from the endpoints of the segment
Angle Bisector Theorem
If a line is drawn so that a point is on the bisector of an angle, then the point is equidistant from the sides of
the angle.
Circumcenter Theorem
The circumcenter of a triangle is equidistant from that triangle’s vertices.
Centroid Theorem
The centroid of a triangle occurs where the three lines created from each vertex to the midpoint of the
opposite side intersect
Incenter Theorem
The incenter of a triangle is equidistant from a triangles sides. It is created at the intersection of the three
angle bisectors
Hinge Theorem
If two sides of one triangle are congruent to two sides of another triangle, but their included angles are not
congruent, then the third side across from the longer included angle is the larger side.
Triangle Midsegment Theorem
The midsegment of a triangle is half the length of the side it is parallel to.
Triangle Inequality Theorem
The sum of any two sides of a triangle is greater than the triangle’s third side.
Pythagorean Theorem
If the triangle is a right triangle, then the sum of the squares of the lengths of the legs is equal to the square
of the length of the hypotenuse.
Pythagorean Inequalities Theorem
If two sides of a triangle are congruent to two sides of another triangle; then when the included angle of the
first triangle is greater than the included angle of the second triangle, the third side of the first triangle must
be longer than the third side of the second triangle.
45-45-90 Triangle Theorem
In a triangle with angle measures 45°, 45°, and 90°, the legs are congruent and the length of the hypotenuse
is the length of the legs multiplied by √2.
30-60-90 Triangle Theorem
In a triangle with angle measure 30, 60, and 90, the longer leg is the length of the shorter legs times √3 and
the length of the hypotenuse is 2 times the length of the shorter leg.
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Perpendicular Bisector Converse
The point equidistant from the endpoints of a segment lies on the perpendicular bisector of that segment.
Isosceles Triangle Theorem Converse
If two angles of a triangle are congruent, then their opposite sides are also congruent.
Angle Bisector Converse
If a point located on the interior of an angle is equidistant from its sides, then it lies on the bisector of the
angle.
Hinge Theorem Converse
If two sides of one triangle are congruent to two sides of another triangle, but their third sides are not
congruent, then the third angle across from the longer side is the larger angle.
Pythagorean Theorem Converse
If the sum of the squares of the length of the legs is equal to the square of the length of the hypotenuse in a
triangle, then the triangle is a right triangle.
Side- Side- Side (SSS) Similarity Theorem
If three sides of a triangle are proportional to three sides of another triangle, then the triangles are similar.
Side- Angle- Side (SAS) Similarity Theorem
If two sides and their included angle of a triangle are proportional to two sides and their included angle of
another triangle, then the triangles are similar.
Triangle Proportionality Theorem
If a parallel line to one side of a triangle intersects the other two sides, then the sides are divided
proportionally.
Triangle Proportionality Theorem Converse
If a line intersects two sides of a triangle and those sides are divided proportionally, then the line is parallel
to the third side.
Triangle Angle Bisector Theorem
An angle bisector of an angle of a triangle divides the opposite side in two segments that are proportional to
the other two sides of the triangle
The Law of Sines Theorem
For any triangle ΔABC with sides a, b, and c,
The Law of Cosines Theorem
For any triangle ΔABC with sides a, b, and c:
a2 = b2 + c2 - 2bcosA; b2 = a2 + c2 – 2bcosB; c2 = a2 + b2 – 2abcosA
sin 𝐴
𝑎
=
sin 𝐵
𝑏
=
sin 𝐶
𝑐
The altitude of the hypotenuse of a right triangle forms two triangles that are similar to each other and are
similar to the original triangle
If two sides of a triangle are not congruent to each other, then the largest angle is always opposite the
longer side.
If two angles of a triangle are not congruent to each other, then the longer side is always opposite the
largest angle.
Triangle Properties
Reflexive
Symmetric
A triangle is congruent to itself.
If ΔABC
≅ ΔDEF, then ΔDEF ≅ ΔABC
If ΔABC
≅ ΔDEF and ΔDEF ≅ ΔGHK, then ΔABC ≅ ΔGHK
Transitive
Definitions
Included angle
The angle between two lines
Included side
The side between two angles
Triangle
A figure with three sides and three angles
Equilateral Triangle
A triangle with three equal sides and each angle measures 60°
Isosceles Triangle
A triangle with two equal sides and two equal angles
Scalene triangle
A triangle with no equal sides and no equal angles
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Quadrilaterals
Postulates
Area Addition Postulate
The area of a region is the sum of the areas of each non- overlapping part.
Area Congruence Postulate
If two polygons are congruent, then they have the same area.
Area of a Square Postulate
The area of a square is the square of the length of a side.
Theorems
Interior Angles of a Quadrilateral
The sum of the measures of the interior angles of a quadrilateral is 306°.
Polygon Angle Sum Theorem
The sum of the measures of the interior angles of a polygon with n sides is
(n-2) 180°.
Polygon Exterior Angle Sum
The sum of the measures of the exterior angles of a polygon is 360°.
Trapezoid Midsegment Theorem
The midsegment of a trapezoid is parallel to each of the trapezoids’ bases, and its length is one half the sum
of the lengths of the bases.
Area of a Parallelogram
The area of a parallelogram is the product of a base and its corresponding height.
Area of a Rectangle
The area of a rectangle is the product of the base and its height.
Area of a Rhombus
The area of a rhombus is on half the product of the lengths of its diagonals.
Area of a Kite
The area of a kite is one half the product of the length of its diagonals.
Area of a Trapezoid
The area of a trapezoid is one half the product of the height and the sum of the base.
Area of a Triangle
The area of a triangle is one half the product of the base and its corresponding height.
Proportional Perimeters and Areas
Theorems
If the ratio of two similar figures is , then the ratio of their perimeters is and the ratio of their areas is
𝑎
𝑎
𝑎2
𝑏
𝑏
𝑏2
.
If a quadrilateral is a parallelogram, then its opposite sides are congruent.
If a quadrilateral is a parallelogram, then its opposite angles are congruent.
If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a
parallelogram.
If a quadrilateral is a parallelogram, then its diagonals bisect each other.
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
If both pairs of the opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
If both pairs of the opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
If a pair of opposite sides are parallel and congruent to each other, then the quadrilateral is a parallelogram.
If a quadrilateral is a rectangle, then it is a parallelogram.
If a parallelogram is a rectangle, then its diagonals are congruent.
If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle.
If a quadrilateral is a rhombus, then it is a parallelogram.
If a parallelogram is a rhombus then its diagonals are perpendicular to each other.
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If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.
If a parallelogram is a rhombus, then each of its diagonals bisects a pair of opposite angles.
If a diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.
If a pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.
If a quadrilateral is a kite, then its diagonals are perpendicular.
If a quadrilateral is a kite, then exactly one pair of its opposite angles are congruent.
If a quadrilateral is an isosceles trapezoid, then each pair of its base angles are congruent.
If a pair of base angles or a trapezoid are congruent, the trapezoid is isosceles.
A trapezoid is isosceles if and only if its diagonals are congruent.
Definitions
Polygon
A plane figure with at least three straight sides and angle
Parallelogram
A quadrilateral with pairs of parallel opposite sides.
Rectangle
A parallelogram with four congruent angles.
Rhombus
A parallelogram with four congruent sides.
Square
A parallelogram with four congruent angles and four congruent sides.
Kite
A quadrilateral with two pairs of consecutive congruent sides, with the opposite sides not congruent.
Trapezoid
A quadrilateral with exactly one pair of parallel sides.
Equilateral Polygon
Any polygon with all sides of the same length.
Isosceles Polygon
A polygon with two sides of the same length.
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Circles
Postulates
Arc Addition Postulate
The measure of an arc that is formed by two adjacent arcs is equal to the sum of those two arcs.
Theorems
Inscribed Angle Theorem
The measure of an inscribed angle is half the measure of its central angle (or half the measure of its
intercepted arc)
Chord- Chord Power Theorem
If two chords intersect in the interior of a circle, then the product of the segments of one chord is equal to the
product of the segment of the other chord.
Secant- Secant Power Theorem
If two secants intersect in the exterior of a circle, then the product of the measures of one secant segment
and its external segment is equal to the product of the measures of the other secant segment and its external
segment.
Secant- Tangent Power Theorem
If a secant segment and a tangent segment intersect on the exterior of a circle, then the product of the
measures of the secant segment and its external segment is equal to the square of the measure of the
tangent segment.
Equation of a Circle
The equation of a circle with center (a,b) and radius r is
(x-a)2 + (y-b)2 = r2
If a line is tangent to a circle, then it is perpendicular to the radius of the circle drawn to the tangent point.
If a line is perpendicular to the radius of a circle at a point on the circle, then the line is tangent to the circle.
If two segments are tangent to a circle from the same point external to the circle, then the segments are
congruent.
Congruent central angles also have congruent cords; Congruent cords also have congruent arcs; Congruent
arcs also have congruent central angles.
If the radius of a circle is perpendicular to a chord, then it bisects both the chord and its arc.
The perpendicular bisector of a chord is a radius or diameter of the circle.
If inscribed angles of a circle intercept the same arc then the angles are congruent.
An angle inscribed in a semicircle is always a right angle
If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
If a tangent and a secant or chord intersect on a tangent point of a circle and the sides of the angles intercept
arcs on the circle, then the measure of the angle formed is half the measure of the intercepted arc.
If two secants or two chords intersect in the interior of a circle and the sides of the angles intercept arcs on
the circle, then the measure of each of the angles formed is half the sum of the measures of the intercepted
arc.
If two secants intersect in the exterior of a circle, and the sides of the angle intercept arcs on the circle, then
the measure of the angle formed is equal to half the difference of the measures of its intercepted arcs.
Definitions
Chord
A segment whose endpoints both lie on a circle
Secant
A segment that intersects two points on a circle
Tangent
A line that intersects the circle (or any curve) on a point where the slope of the line is equal to the slop of the
circle
Arc
A closed segment on a circle (or any curve)
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