Unformatted text preview: Geometry Summary Sheet
Welcome. This is the third iteration of a summary sheet started by Brian Kwong. Special thanks
to Shadman Khandaker, Daniel Roz, as well as my old peers who served as brilliant editors
from the start. Finally, I personally hope that y’all will find this summary sheet useful, and as
always, cheers. The express version can be found here. Also, we are now accepting new
editors! The application form can be found here. Table of Contents:
3. Congruence & Equality
5. Parallel Lines & Quadrilaterals
6. Similarity & Proportions
8. Right Triangles
12. The Coordinate Plane
13. Applied Algebra
14. Locus, Constructions & Concurrence Theorems
16. Solid Geometry
17. Planar Geometry Logic:
Statements can be written in two forms, either as a declarative, or a conditional.
1. A declarative statement directly states a fact as true, such as “Two straight ∡ are ≅”.
2. A conditional statement has two parts, an antecedent and a consequent. The
antecedent justifies the statement stated true in the consequent. An example is “If two
angles are straight angles, then they are congruent”. Also known as “If… then” form.
The negation of a statement is its opposite. In notation, it is represented as ~P (not P).
The law of the contrapositive states that given a statement, P → Q, its contrapositive, ~Q → ~P
is always true as well.
The chain rule states that if P → Q, and Q → R, then P → R, acting transitive-like in its
This is a truth table, showing what each statement’s truth value is, depending on what the
truth values of statements P and Q are.
p q p Λ q ( If p AND q is
true, p ^ q is true) p → q (if p, then q)
Conditional p ←→ q (p if and only
if q) Biconditional T T T T T T F F F F F T F T F F F F T T Types of Logic Statements:
- Original: p → q - Converse: q → p - Inverse: ~p → ~q - Contrapositive: ~q → ~p Back to Top Fundamentals:
The Structure of Geometry:
A deductive structure is where new conclusions are justified through proven conclusions and
1. Undefined Terms:
2. Postulate: An unproved assumption (Not always reversible)
3. Definitions: States the meaning of a term / idea. (Always reversible)
4. Theorems / Conclusions: A statement that can be proven. (Not always reversible)
Alright, it’s great that we can prove statements, but how do we go about proving them in the
Well, proofs are lists of steps leading up to the statement one is trying to prove (theorem).
There are two main ways of proving a statement, either directly or indirectly.
1. A direct proof is a proof where one tries to prove a statement using the given
information, assumptions (postulates), and proven theorems to reach a conclusion.
2. An indirect proof is a proof where one tries to prove a statement by disproving all of the
other possibilities, therefore, making the desired conclusion true.
Proofs can also be written in two ways, either the two-column proof, or in paragraph form.
1. The two-column proof is a type of proof in which one column, labeled statements, is a
specific list of steps relating to only the diagram at hand. The other column, reasons, is a
more generalized list that justifies the conclusions made in the statement column. These are definitions, postulates, and theorems that can be applied in numerous proofs.
The reasons given usually have two parts, the antecedent and consequent. The
consequent corresponds to the statement, while the antecedent acts as a justification to
concluding that statement.
2. The paragraph proof is a type of proof where only the statements are given. These are
also written in If… Then form, or antecedent then consequent form. These are more
commonly used in more advance courses.
Theorems for Indirect Proofs:
1) Law of the excluded middle: Either p OR ~p is true. No other possibility exists.
2) Law of Contradiction: Both q AND ~q cannot be true at the same time.
3) Postulate of Elimination: If all other possibilities are false, then the only remaining one
must be true.
1. A midpoint is a point that divides a segment into two congruent segments.
2. A segment bisector is a segment, ray, or line, that intersects another segment at its
3. An angle is the union of two rays with a common endpoint and no other points in
4. An angle bisector is a segment, ray, or line, that divides an angle into two congruent
5. Auxiliary Lines are lines, rays, or segments that do not appear in a diagram, but their
existence is guaranteed by a postulate.
a. Two points determine exactly one line.
b. Every ∡ has a bisector.
6. If two lines intersect to form right ∡s and ≅ adjacent ∡s, then they are ⏊.
7. If two ∡s are suppl. & ≅, then they are right ∡s. Theorems:
1) Theorems of Equidistance:
- If 2 points are equidistant from the endpoints of a segment, then the line joining them is
the ⏊ bisector of that segment. - If a point lies on the ⏊ 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 it lies on the ⏊ bisector
of that segment. 2) Theorems of Inequality:
- If 2 ∡s of a △ are =/ , then the sides opposite are =/ AND the side opposite the larger ∡
is the larger side. - If 2 sides of a △ are =/ , then the ∡s opposite are =/ AND the ∡ opposite the larger side
is the larger ∡. - The measure of an exterior ∡ of a △ is greater than either remote interior ∡. 3) Theorems of Uniqueness:
- At a point on a given line, one and only ⏊ can exist on the given line. - At a point not on a given line, one and only ⏊ can exist on the given line. - A line segment has one and only 1 midpoint. Back to Top Congruence & Equality:
Theorems to Know:
- If 2 ∡s are (right/straight) ∡s, then they are ≅. - If 2 ∡s are (suppl./comp.) to the same ∡, then they are ≅. - If 2 ∡s are (suppl./comp.) to ≅ ∡s, then they are ≅. - If a (segment/∡s) is (added/subtracted) (to/from) ≅ (segments/∡s), then their
(sums/differences) are ≅. - If ≅ (segments/∡s) are (added/subtracted) (to/from) ≅ (segments/∡s), then their
(sums/differences) are ≅. - If (segments/∡s) are ≅, then their like (multiples/divisions) are ≅. - If (segments/∡s) are ≅ to the same (segment/∡s), then they are ≅ to each other. - If (segments/∡s) are ≅ to ≅ (segments/∡s), then they are ≅ to each other. - If 2 ∡s are vertical, then they are ≅. - If 2 sides of a △ are ≅, then the ∡s opposite those sides are ≅. - If 2 ∡s of a △ are ≅, then the sides opposite those ∡s are ≅. Postulates of Equality:
- Reflexive: Anything is equal to itself. Ex) a = a - Symmetric: If a = b, then b = a. - Transitive: If a = b & b = c, then a = c. - Substitution: If a = b, then b can substitute for a & a can substitute for b. Postulates of Operations:
- Addition: If a = b & c = d, then a + c = b + d. - Subtraction: If a = b & c = d, then a - c = b - d. - Multiplication: If a = b & c = d, then a * c = b * d.
a. If equals are multiplied by equals, then the products are equal. - Division: If a = b & c = d & c =/ 0 and d =/ 0, then a / c = b / d.
a. If equals are divided by equals, then the quotients are equal. Back to Top Triangles:
Proving Triangles Congruent:
1) SSS Theorem of ≅ △s: Needs all 3 sides of a △ ≅ to the corresponding sides of a 2nd △.
2) SAS Postulate of ≅ △s: Needs 2 sides and the included ∡ of a △ ≅ to the corresponding
parts of a second △.
3) ASA Postulate of ≅ △s: Needs 2 ∡s and the included side of a △ ≅ to the corresponding
parts of a second △.
4) AAS Theorem of ≅ △s: Needs 2 ∡s and a non-included side of a △ ≅ to the
corresponding parts of a second △.
- If a △ is equilateral, then it is equiangular. - If a △ is equiangular, then it is equilateral. - If a line bisects 1 side of a △ and is parallel to a 2nd side, then it bisects the 3rd side. Triangle Definitions:
- If a line segment is drawn from any vertex to the midpoint of the opposite side, then it
is a median of a △. - If a line segment is drawn from any vertex to the opposite side, & is ⏊ to that side, then
it is an altitude of a △. Measures and Angles:
- The sum of the measures of three ∡s in a triangle is 180. - If 2 ∡s of a △ are ≅ to 2 ∡s of another △, then the 3rd ∡s are ≅. - The acute ∡s of a rt. △ are complementary. Back to Top Parallel Lines & Quadrilaterals:
Proving Lines Parallel:
1) If 2 lines are cut by a transversal such that the alternate interior ∡s are ≅, then the lines
2) If 2 lines are cut by a transversal such that the alternate exterior ∡s are ≅, then the lines
3) If 2 lines are cut by a transversal such that the corresponding ∡s are ≅, then the lines are
4) If 2 lines are ⏊ to the same line, then they are parallel to each other.
5) If 2 lines are parallel to the same line, then those 2 lines are parallel.
Angles from Parallel Lines:
1) If 2 parallel lines are cut by a transversal, then the alternate interior ∡s are ≅.
2) If 2 parallel lines are cut by a transversal, then the alternate exterior ∡s are ≅.
3) If 2 parallel lines are cut by a transversal, then the corresponding ∡s are ≅.
1) Polygon: The union of a set of points P1,P2,P3,...,Pn-1,Pn with the line segments
that if any segments intersect, they only intersect at
- Quadrilateral: A polygon with four sides. - Parallelogram: A quadrilateral in which both pairs of opposite sides are parallel.
Properties are: opposite sides are ≅ and parallel, opposite angles are ≅, consecutive
angles are supp, and diagonal bisect each other. - Rectangle: A parallelogram in which at least one angle is a right ∡. - Rhombus: A parallelogram in which at least one pair of consecutive sides are ≅. - Square: A parallelogram that is both a rectangle and a rhombus. - Trapezoid: A quadrilateral that has at least one pair of opposite sides parallel. - Isosceles Trapezoid: A trapezoid in which the non-parallel sides are ≅. 3) Supplementary Definitions:
- Regular Polygon: A polygon that is both equilateral and equiangular. - Convex Polygon: A polygon in which each interior angle has a measure less than 180. - Exterior Angle of a Polygon: An ∡ that is adjacent and supplementary to an interior ∡ of
the polygon. Parallelogram:
1) Proving a Quadrilateral a Parallelogram:
- If two pairs of opposite sides of a quadrilateral are ≅, then it’s a parallelogram. - If two pairs of opposite sides of a quadrilateral are parallel, then it’s a parallelogram.
[Def] - If one pair of opposite sides of a quadrilateral are both ≅ and parallel, then it’s a
parallelogram. - If the diagonals of a quadrilateral bisect each other, then it’s a parallelogram. 2) Parts Proven given a Parallelogram:
- Opposite sides of a parallelogram are ≅. - Opposite sides of a parallelogram are parallel. [Def] - Opposite ∡s of a parallelogram are ≅. - Diagonals of a parallelogram bisect each other. - Consecutive ∡s of a parallelogram are supplementary. Rectangle:
1) Proving a Quadrilateral a Rectangle:
- If a parallelogram has at least one right ∡, then it is a rectangle. [Def] - If the diagonals of a parallelogram are ≅, then it is a rectangle. Rhombus:
1) Proving a Quadrilateral a Rhombus:
- If either diagonal of a parallelogram bisect 2 ∡s of a Parallelogram, then it is a Rhombus. - If a pair of consecutive sides of a parallelogram are ≅, then it is a Rhombus. [Def] Isosceles Trapezoid:
1) Parts Proven given an Isosceles Trapezoid:
- Diagonals of an Isosceles Trapezoid are ≅. - Upper/Lower Base ∡s of an Isosceles Trapezoid are ≅. - Non-Parallel sides of an Isosceles Trapezoid are ≅. [Def] Back to Top Similarity & Proportions:
- Proportion: An equation that states 2 or more ratios. - Ratio: The quotient of 2 #s. - Rate: The ratio of 2 measures. - Similar Polygons: Polygons such that...
1. The ratio of corresponding sides are equal.
2. Corresponding ∡s are ≅. - Golden mean: The shorter part is to the longer part as the longer part is to the whole. - M is the mean proportional between a and b or m is the geometric mean of a and b, if
m^2 = (square root of) a X b. - M is the arithmetic mean of a and b if and only if m = (a + b)/2. Proportion Theorems:
1) If 2 △s are similar, then the ratio of their corresponding sides are equal.
2) If a line is parallel to 1 side of a △ & intersects the other 2 sides, then it divides those 2
3) If 3 or more parallel lines are intersected by 2 transversals, then the parallel lines divide
the transversals proportionally.
4) If a ray bisects an ∡ of a △, then it divides the opposite into segments that are
proportional to the adjacent sides.
5) The ratio of the areas of 2 ~ △s are equal to the square of the ratio of any pair of
6) The ratio of the (perimeter/altitude) of 2 ~ △s are equal to the ratio of any pair of
7) In a proportion, the product of the means is equal to the product of the extremes.
8) If the product of 2 #s is equal to the product of 2 other #s, then either pair may be the
extremes while the other pair is the means. Proving Triangles Similar:
- AAA Postulate of ~ △s: Needs all 3 ∡s of a △ ≅ to the corresponding ∡s of a second △. - AA Theorem of ~ △s: Needs 2 ∡s of a △ ≅ to the corresponding ∡s of a second △. - SSS Postulate of ~ △s: Needs all the ratios of corresponding sides to be equal to each
other. - SAS Postulate of ~ △s: Needs the ratios of 2 corresponding sides to be equal to each
other, as well as the included ∡s of the 2 pairs of corresponding sides ≅. Ratios based on ~ △s:
- If 2 △s are ~, then the ratio of their areas is equal to the square of any ratio of any pair
of corresponding sides. - If 2 △s are ~, then the ratio of any pair of corresponding altitudes are equal to the ratio
of any pair of corresponding sides. Back to Top Trigonometry
Note: SohCahToa works only if the triangle is right. They can only be applied to either acute ∡.
For example, it is not possible to have Sin90, Cos90, or Tan90.
All you need to know is SohCahToa. The Sine of one of the acute ∡ is equal to the Cosine
of the other acute ∡. Same applies for the reverse. The Tangent of one of the acute angles is
equal to the reciprocal of the Tangent of the other acute angle.
The three trigonometric functions are:
1) Sine = Length of Side Opposite ∡ / Length of Hypotenuse
2) Cosine = Length of Side Adjacent ∡ / Length of Hypotenuse
3) Tangent = Length of Side Opposite ∡ / Length of Side Adjacent ∡
Note: To find sides, use Sine, Cosine, and Tangent. To find angles, use Sin-1 , Cos-1 , and Tan-1
Note: Angle of elevation goes upwards, such as looking up to a point while on the ground, while
the angle of depres...
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