Geometry

Polygon

-A closed figure formed with line segments which are called the sides -Poly means "many" and gon means "sides"; thus, polygon means "a many-sided figure"

Line segment

-A piece of a line -Has two endpoints and is named by these two endpoints -The symbol — written on top of two letters is used to denote that line segment

Altitude/height

-A segment that goes from a vertex of the triangle and makes a 90° angle with the opposite side, known as the base -Sometimes, the opposite side needs to be extended in order to accomplish this In △ABC, segment AD is the altitude from vertex A. In this case, the base, segment BC, needs to be extended in order for the segment AD to be able to make a 90° angle with the opposite side. In △QRS, segment RT is the altitude from vertex R. Segment RT makes a 90° angle with the base, segment QS.

Transversal

-A third line that cuts two parallel lines, forming two sets of four angles -Angles in the same relative positions have the same measures -For any two angles you select, if they are not equal to one another, they will be supplementary to one another In the diagram: angle 1 = angle 5 angle 2 = angle 6 angle 3 = angle 7 angle 4 = angle 8 But since vertical angles are equal, angle 1 = angle 3 angle 2 = angle 4 angle 5 = angle 7 angle 6 = angle 8 From this, we can see angle 1 = angle 3 = angle 5 = angle 7 angle 2 = angle 4 = angle 6 = angle 8

Triangle

-A three-sided polygon -It has three angles, or angular rotations, in its interior -The sum of the angles (or angular rotations) is always 180° -The symbol for triangle is △ -Named by naming its vertices or corners The diagram is △ABC.

Pythagorean triples

-Any three sides of a right triangle -There are many Pythagorean triples with sides that are natural numbers -Common Pythagorean triples: 3-4-5, 5-12-13, 7-24-25, 8-15-17 -Any multiple of one of these will also form a Pythagorean triple (for example, if each side of a 3-4-5 triangle were doubled, it would form a 6-8-10 Pythagorean triple)

Various ways an angle can be named

-By the letter of the vertex; therefore, the angle above could be named ∠A -By the number (or small letter) in its interior; therefore, the angle above could be named ∠1 -By the letters of the three points that formed it; therefore, the angle above could be named ∠BAC, or ∠CAB. The center letter is always the letter of the vertex.

Angle

-Formed by two rays that start from the same point -This point is called the vertex; the rays are called the sides o the angle -Measured in degrees -The degrees indicate the size of the angle, from one side to the other

Right angle

-Has a measure of 90° -The symbol in the interior of an angle designates the fact that a right angle is formed In the diagram, ∠ABC is a right angle.

Equiangular triangle

-Has all of its angles of equal measure -Thus, each angle has a measure of 60° -An equiangular triangle is also equilateral

Scalene triangle

-Has all three of its sides of different lengths -The angles in a scalene triangle will all have different measures -The largest angle will be opposite the longest side -The smallest angle will be opposite the shortest side

Equilateral triangle

-Has all three of its sides of equal length, which in turn make each of the three angles equal in measure -Therefore, each angle in an equilateral triangle has a measure of 60° since the sum of the angles in any triangle is 180°

Ray

-Has only one endpoint and continues forever in one direction -Could be thought of as a half-line -Named by the letter of its endpoint and any other point on the ray -The symbol —> written on top of the two letters is used to denote that ray

Isosceles triangle

-Has two of its sides equal in length -The angles opposite the equal sides in an isosceles triangle have equal measure

Vertical angles

-If two straight lines intersect, they do so at a point -Four angles are formed -Those angles opposite each other are called vertical angles -Those angles sharing a common side and a common vertex are adjacent/supplementary angles -Vertical angles are always equal

Pythagorean theorem

-In any right triangle, there is a relationship between the lengths of the three sides -This relationship is referred to as the Pythagorean theorem -If the sides of a right triangle are labeled a, b, and c with c representing the longest of the three sides (the one opposite the right angle), then a² + b² = c²

Point

-The most fundamental idea in geometry -Represented by a dot and named by a capital letter

Line

-The shortest path connecting two points -Continues forever in opposite directions -Consists of an infinite number of points -Named by any two points on the line -The symbol <—> written on top of two letters is used to denote that line

The 45-45-90 right triangle

-This right triangle is an isosceles right triangle -If each of the sides that form the right angle has a measure of 1, then using the Pythagorean theorem, you find that the hypotenuse has the value √2 -If an isosceles right triangle had each of the equal sides with a measure of 5, then the hypotenuse would have a measure of 5√2

Perpendicular lines

-Two lines that meet to form right angles (90°) -The symbol ⟂ is used to denote perpendicular lines In the diagram, l ⟂ m.

Intersecting lines

-Two or more lines that cross each other at a point -That point would be on each of those lines

Parallel lines

-Two or more lines that remain the same distance apart at all times -Parallel lines never meet -The symbol || is used to denote parallel lines In the diagram, l || m.

Angle bisector

A ray from the vertex of an angle that divides the angle into two equal pieces In the diagram, ray AB is the angle bisector of ∠CAD. Therefore, ∠1 = ∠2.

Angle bisector (triangle)

A segment that goes from one vertex of a triangle and divides the angle at that vertex into two smaller but equal angles In △GHI, segment HJ is an angle bisector from vertex H.

Median

A segment that goes from one vertex of a triangle to the midpoint of the opposite side In △ABC, segment BD is a median from vertex B. Point D is the midpoint of segment AC, and so AD = CD.

Obtuse angle

Any angle whose measure is larger than 90° but less than 180° In the diagram, ∠4 is an obtuse angle.

Acute angle

Any angle whose measure is less than 90° In the diagram, ∠b is acute.

Adjacent angles

Any angles that share a common side and a common vertex In the diagram, ∠1 and ∠2 are adjacent angles.

Straight angle

Has a measure of 180°; also known as a line. In the diagram, ∠BAC is a straight angle.

Acute triangle

Has each of its angles with measures less than 90°

Right triangle

Has one of its angles with a measure equal to 90°

Obtuse triangle

Has one of its angles with a measure greater than 90°

Triangle rules: exterior angle

If one side of a triangle is extended, the exterior angle formed by that extension is equal to the sum of the other two interior angles

Solid geometry

The study of shapes and figures in three dimensions

Plane geometry

The study of shapes and figures in two dimensions (the plane)

Triangle rules: sides

The sum of the lengths of any two sides of a triangle must be larger than the length of the third side In the diagram of △ABC: AB + BC > AC AB + AC > BC AC + BC > AB

Supplementary angles

Two angles whose sum is 180° In the diagram, since ∠ABC is a straight angle, ∠3 + ∠4 = 180°. Therefore, ∠3 and ∠4 are supplementary angles. If ∠3 = 122°, its supplement, ∠4, would be: 180° - 122° = 58°.

Complementary angles

Two angles whose sum is 90° In the diagram, since ∠ABC is a right angle, ∠1 + ∠2 = 90°. Therefore, ∠1 and ∠2 are complementary angles. If ∠1 = 55°, its complement, ∠2, would be: 90° - 55° = 35°.