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.