AOPS Geometry RB
Euclid's famous five axioms
1. Any two points can be connected by a straight line segment. 2. Any line segment can be extended forever in both directions, forming a line. 3. Given any line segment, we can draw a circle with the segment as a radius and one of the segment's endpoints as center. 4. All right angles are congruent. (We'll talk about right angles and what we mean by 'congruent' shortly!) 5. Given any straight line and a point not on the line, there is exactly one straight line that passes through the point and never meets the first line.
Congruent Theorem
SSS . SAS works. ASA, and AAS, too. But SSA, not so much
semicircle
half of a circle
supplemetary angles
two angles that add up to 180 degrees
base angles
two equal angles are often called the base angles of the triangle
parallel
two lines do not meet, we say that they are parallel
How to use Protractor
-Place the protractor on the angle so that the vertex of the angle is exactly where the center of the circle would be if the protractor were a whole circle. Your protractor should clearly show this center point; it's near the middle of the straight side. -Turn the protractor so that one side of the angle is along the 'zero line'; i.e., the line through the center point along the straight edge of the protractor. -Find where the other side of the angle meets the curved side of the protractor. The number there tells you the measure of the angle.
diameter
A chord that passes through the center of a circle is a diameter.
protractor
A tool used to measure angles
AAS Congruence Theorem
the Angle-Angle-Side Congruence Theorem, or AAS for short. AAS states: If two angles and a side of one triangle equal the corresponding angles and side in another triangle as shown below, then the triangles are congruent. When using AAS, the equal sides must be adjacent to corresponding equal angles. For example, the triangles shown below are not congruent because the equal sides are not adjacent to corresponding equal angles!
sides
the rays of an angle
tangent line
A line that touches a circle at a single point is a tangent line.
conjecture
A mathematical statement that is not an axiom but hasn't been proved false or true is called a conjecture.
vertical angles
A pair of opposite congruent angles formed by intersecting lines. *Vertical angles are equal.*
ray
A part of a line, with one endpoint, that continues without end in one direction
rectangle
A rectangle has four sides and four right angles, as shown. Furthermore, opposite sides of a rectangle equal each other in length.
degree
A unit of measurement for angles
legs
The two sides of a right triangle that form the right angle
secant line
A line that hits a circle at two points is a secant line.
chord
A segment that connects two points on a circle is a chord.
Point
Point has 0 dimensions. We usually label them with capital letters.
Fibonacci
he first two terms of the sequence are both 1, and each subsequent term is the sum of the previous two terms. Calculate the ratio between each term and the term before it, such as $\color[rgb]{0.11,0.21,0.37}34/21 \approx 1.619$.
a whole circle
is 360 degree
Sum of angles in a triangle
is always 180*
Line
no endpoints, goes on forever
Collinear
points that lie on the same line
arc
the portion of a circle that connects two points on a circle is called an arc of that circle.
remote interior angles
the angles of a triangle that are not adjacent to a given exterior angle
isosceles triangle
A triangle in which two sides are equal is called an isosceles triangle. The equal sides are sometimes called the legs of the triangle, and the other side the base.
obtuse angle
An angle between 90 and 180 degrees
reflex angles
Angles that are greater than $180^\circ$ are called reflex angles
scalee triangle
If no two sides of a triangle are equal, the triangle is scalene.
Midpoint
One special point on a segment is the segment's midpoint, which is the point halfway between the endpoints. Because the midpoint is the same distance from both endpoints, we say it is equidistant from the endpoints.
SsA Congruence Theorem
SSA (Side-Side-Angle) is not a valid congruence theorem. If two sides of one triangle are equal to two sides of another, and the two triangles have equal corresponding angles that are not the angles between the equal corresponding sides, then the two triangles are not necessarily congruent!
locus
The fancy name we have for a group of points that satisfy certain conditions is a locus.
major arc
The longer arc that connects the two points is a major arc of the circle.
perimeter
The perimeter of a closed figure is how far you travel if you walk along its boundary all the way around it once.
altitude
The perpendicular segment from the vertex opposite the base to the base (extended if necessary) is the altitude.
vertices
The points are called vertices of the triangle
complementary angles
Two angles whose sum is 90 degrees
congruent
Two figures are congruent if they are exactly the same — in other words, we can slide, spin, and/or flip one figure so that it is exactly on top of the other figure.
SAS Congruence Theorem
We call the principle illustrated in Problem 3.6 the Side-Angle-Side Congruence Theorem, or SAS for short. SAS states: If two sides of one triangle and the angle between them are equal to the corresponding sides and angle of another triangle, then the two triangles are congruent.
minor arc
We call the shorter of the two arcs that connect two points on a circle a minor arc of the circle.
angle
When two rays share an origin, they form an angle. two intersecting lines also make angles.
exterior angle
When we extend a side past a vertex of a triangle, we form an exterior angle of the triangle Any exterior angle of a triangle is equal to the sum of its remote interior angles.
Transversal Line
a line that cuts across parallel lines a transversal
square
a rectangle with all sides the same length
Straightedge
a ruler with no markings, used for line segments.
right triangle
a triangle with one right angle
all exterior angles of a triangle
all exterior angles of a triangle sum to 360
reflex angle
an angle that is greater than 180 degrees because the angle is bent beyond the straight line
straight angle
an angle that measures 180 degrees
acute angle
angle smaller than 90 degrees
adjacent angles
angles that share a side
All three angles of an equilateral triangle
are 60.
triangle
connect three points with line segments, we form a triangle
ASA Congruence Theorem
the Angle-Side-Angle Congruence Theorem, or ASA for short. ASA states that If two angles of one triangle and the side between them are equal to the corresponding angles and side of another triangle, then the two triangles are congruent.
sides
the segments are called sides.
Any exterior angle of a triangle is equal to
the sum of its remote interior angles.
If all three angles of a triangle are equal,
then so are all three sides.
If a radius of a circle bisects a chord of the circle that is not a diameter,
then the radius must be perpendicular to the chord.
concurrent
three or more lines all pass through the same point, we say the lines are concurrent.
plane
If the page extended forever in every direction, we'd call it a plane.
line segment
If there were a straight path from one point to another, that path would be called a line segment. The two points at the ends of a segment are cleverly called the endpoints of the segment. We use these endpoints to label the segment.
Valid congruent theorems
SSS, SAS, ASA, AAS, SAA are theroems that are congruent. SSA, AAA, ASS are not.
proof of contradiction
Sometimes using a proof by contradiction is much easier than proving a statement directly. To prove a statement by contradiction, we start by assuming the statement is false. Then we show that this assumption leads us to an impossible statement, which tells us that the assumption itself is false. Having proved the statement cannot be false, we have shown it must be true.
vertex
The < angle symbol tells us we're referring to an angle. The common origin is called the vertex of the angle
area
The area of a closed figure is the number of 1 X 1 squares (or pieces of squares) needed to exactly cover the figure. We sometimes use brackets to denote area, so that $\color[rgb]{0.11,0.21,0.37}[ABC]$ means the area of $\color[rgb]{0.11,0.21,0.37}\triangle ABC$.
angle chasing
Using information about angles to find information about other angles is often called angle-chasing.
axiom
We call such a statement that must be regarded as fact without proof an axiom. also known as postulates
theorems
We call such proven mathematical statements theorems.
Golden Ratio Spiral
\frac{1+\!\sqrt{5}}{2}\approx 1.618034
right angle
an angle that measures 90 degrees
A triangle in which all three sides are equal is
an equilateral triangle.
