Geometry 2025 Summer

Segment

A line that ends on two endpoints

Ray

A line with one endpoint

Colinear

Points that all sit on the same line segment

Angle

An angle is formed by two rays with the same endpoint

Vertex

Endpoint of an angle

Adjacent angles

Two angles with a common side and a common vertex

Vertical angles

Two angles whose sides are opposite rays

Complementary angles

Two angles whose measures have a sum of 90°

Supplementary angles

Two angles whose measures have a sum of 180°

Linear pair

Two adjacent angles that share a side and make a line

Angle bisector

A ray that divides an angle into two congruent angles

Midpoint in one dimension (formula)

(a+b)/2

Midpoint in two dimensions (formula)

[(x1+x2)/2 , (y1+y2)/2]

Distance formula

d=/(x2-x1)²+(y2-y1)²

Area of a circle (formula)

A=pi×r²

Circumference of a circle (formula)

C=2(pi)r

Inductive reasoning

Reasoning based on patterns

Conjecture

An unproven statement that is based on observations

Counterexample

An example that shows a conjecture is false

Conditional statement

An if-then statement

p →q

A way to write a conditional statement

p (conditional statement)

Hypothesis

q (conditional statement)

Conclusion

Negation

The negative of the statement

~p or ~q

~p or ~q represents “not p or q" or the negation of p or q

Equivalent statements

When two statements are true and false at the same time

Conditional

p → q

Converse

q → p

Inverse

~p → ~q

Contrapositive

~q → ~p

A conditional statement is equivalent to…

It’s equivalent to its contrapostive

The inverse and the …. of any conditional are equivalent

Converse

Biconditional

A single true statement that combines a true conditional and its true converse

q <---> p

Sometimes how a biconditional is written

If and only if

A phrase used when you join two parts of each conditional to create a biconditional

A statement is reversible if…

It’s converse is true

A biconditional is a…

Definition

A definition is “good" if…

It can be written as a biconditional and if it is reversible

Deductive reasoning

The process of reasoning logically from one or more general statements to reach a conclusion

Law of detachment

If the hypothesis (p) of a true conditional is true, then the conclusion (q) is true. If p → q is true and p is true, then q is true.

Law of syllogism

A principal that allows you to draw a conclusion from two conditional statements. If p → q is true and q → r is true, then p → r is true.

What can you conclude? : Ray RS divides angle ARB so that angle ARS us congruent to SRB

Ray RS is an angle bisector. Use of the law of detachment

What can you conclude? : If you play football, then you will get sweaty. If you get sweaty, then you should take a shower.

If you play football, then you should take a shower.

Addition property of equality

a+c = b+c

Subtraction property of equality

a-c = b-c

Multiplication property of equality

a×c = b×c

Division property of equality

a÷c = b÷c

Reflexive property of equality

a=a

Symmetric property of equality

If a=b, then b=a

Transitive property of equality

If a=b and b=c, then a=c

Substitution property of equality

If a=b, then b can replace a in any expression

Distributive property

a(b+c) = ab+ac

Reflexive property of congruence

Everything is congruent to itself

Symmetric property of congruence

You can flip the congruence sign

Transitive property of congruence

If line AB is congruent to line CD and line CD is congruent to line EF, then line AB is congruent to line EF

Segment additon postulate

AB+BC = AC