Benchmark Review Flashcards
Point
– A point has no dimension . It is a location on a plane. It is represented by a dot.
Line
- A line has one dimension. Its is an infinite set of points represented by a line with two arrowheads that extends without end.
Plane
– A plane has two dimensions extending without end. It is often represented by a parallelogram.
Line segment
– A line segment consists of two endpoints and all the points between them.
Ray
– A ray has one endpoint and extends without end in one direction.
__
BC
(Geometric Notation)
Segment BC
–>
BC
(Geometric Notation)
Ray BC
BC
Geometric Notation
Line BC
BC
Geometric Notation
Length of BC
∠ABC
Geometric Notation
Angle ABC
m∠ABC
Geometric Notation
Measure of angle ABC
△ABC
Geometric Notation
Triangle ABC
॥
Geometric Notation
Is parallel to
⟂
Geometric Notation
Is perpendicular to
≅
Is congruent to
~
Geometric Notation
Is similar to
V
Logic Notation
Or
⋀
And
➝
Read “implies ” , if … then…
↔
Read “ if and only if ”
iff
Read “if and only if”
～
Not
∴
Therefore
Conditional Statement
A logical argument consisting of a set of premises.
Hypothesis ( p ) , and Conclusion ( q )
Conditional Statement
Symbolically:
If p , then q
p➝q
Converse
Formed by interchanging the hypothesis and conclusion of a conditional statement
Inverse
Formed by negating the hypothesis and conclusion of a conditional statement
Contrapositive
Formed by interchanging and negating the hypothesis and conclusion of a conditional statement
Conditional
If p , then q
p ➝ q
Converse
If q , then p
q ➝ p
Inverse
If not p , then not q
～p ➝ ～q
Contrapoative
If not q , then not p
～q ➝ ～p
Deductive Reasoning
Method using logic to draw conclusions based upon definitions, postulates, and theorems
Inductive Reasoning
Method of drawing conclusions from a limited set of observations
Proof
A justification logically valid and based on initial assumptions, definitions, postulates, and theorems
Law of Detachment
Deductive reasoning stating that if the hypothesis of a true conditional statement is true then the conclusion is also true
Law of Detachment
If p ➝ q is a true conditional statement and p is true, then q is true.
Law of Syllogism
Deductive reasoning that draw s a new conclusion from two conditional statements when the conclusion of one is the hypothesis of the other
Law of Syllogism
If p ➝ q and q ➝ r are true conditional statements, then p ➝ r is true.
Counterexample
Specific case for which a conjecture is false
Perpendicular Lines
Two lines that intersect to form a right angle
Parallel Lines
Lines that do not intersect and are coplanar
Skew Lines
Lines that do not intersect and are not coplanar
Transversal
A line that intersects at least two other lines
Corresponding Angles
Angles in matching positions when a transversal crosses at least two lines
Alternate Interior Angles
Angles inside the lines and on opposite sides of the transversal
Alternate Exterior Angles
Angles outside the two lines and on opposite sides of the transversal
Consecutive Interior Angles
Angles between the two lines and on the same side of the transversal
Midpoint
Divides a segment into two congruent segments
Midpoint Formula
Given Points
A ( x 1 , y 1 ) and B ( x 2 , y 2 )
midpoint M = (x1 -x2) (y1 - y2)
——— ———-
2 2
Slope Formula
Ratio of vertical change to horizontal change
slope = m = change in x = y2 - y1
—————– ————–
change in y x2 - x1
Slopes of Lines
Parallel lines have the same slope.
Slopes of Lines
Perpendicular lines have slopes whose product is - 1.
Slopes of Lines
Vertical lines have undefined slope.
Slopes of Lines
Horizontal lines have 0 slope.
Distance Formula
Given points A ( x 1 , y 1 ) and B ( x 2 , y 2 )
AB = √ (x2 - x1) 2 + (y2 - y1) 2
The distance formula is based on the Pythagorean Theorem.
Perpendicular Bisector
A segment, ray, line, or plane that is perpendicular to a segment at its midpoint
Constructions
Traditional constructions involving a compass and straightedge reinforce students’ understanding of geometric concepts.
Scalene
Classifying Triangles
No congruent sides
No congruent angles
Isosceles
Classifying Triangles
At least 2 congruent sides
2 or 3 congruent angles
Equilateral
Classifying Triangles
3 congruent angles
All equilateral triangles are isosceles
Acute
Classifying Triangles
3 acute angles
3 angles, each less than 90°
Right
Classifying Triangles
1 right angle
1 angle equals 90°
Obtuse
Classifying Triangles
1 obtuse angle
1 angle greater than 90°
Equiangular
Classifying Triangles
3 congruent angles
3 angles, each measures 60°
Triangle Sum Theorem
Measures of the interior angles of a triangle = 180°
m∠A + m∠B + m∠C = 180°
Exterior Angle Theorem
Exterior angle, m∠ 1, is equal to the sum of the measures of the two non adjacent interior angles.
Pythagorean Theorem
If △ABC is a right triangle, then a 2 + b 2 = c 2 .
