Geometry

Line

A line is a one dimensional geometric abstraction, infinitely long, with no width. A straight line is the shortest distance between two points. There is exactly one straight line that passes through any two points. A line segment is a section of a straight line finite blank, with two endpoints.

What is an angle bisector

A line or line segment bisexts an angle if it splits the angle into two smaller, equal angles. line segment BD bisects ABC. ABD has The same measure as DBC. they are both half the size of ABC

Diagonal of a polygon

A line segment connecting any to nonadjacent vertices

Radius

A line segment that connects the center of the circle with any point on the circle. The radius of a circle is 1/2 the length of the diameter

Diameter

A line segment that connects two points on a circle and passes through the center of the circle

Chords and Tangents

A line segment that joins two points on the circle. The longest chord of a circle is its diameter. AT is a chord of P A line that touches only one point on the circumference of a circle. A line drawn tangent to a circle is perpendicular to the radius at the point of tangency. Line L is tangent to P at point T

Transversal's and their intersection with parallel lines

A line that intersects two parallel lines is called a transversal. when this happens, each of the parallel lines intersect the transversal line at the same angle. when two or more parallel lines are cut by a transversal, all the acute angles are equal to each other and all the obtuse angles are equal to each other. You can also say that any acute angle is supplementary to any obtuse angle

Parallelogram

A parallelogram is a quadrilateral with two pairs of parallel sides. Opposite sides are equal in length; opposite angles are equal in measure; and angles that are not opposite are supplementary to each other

Vertex of a polygon

A point where two sides intersect. Polygons are named by assigning each vertex a letter and listing them in order as in pentagon ABCDE

Polygon

A polygon is a closed figure whose sides are straight line segments. classes of polygons are named according to their number of sides. A triangle has three sides, equadrilateral has four sides, a pentagon has five sides, and hexagon has six sides.

Regular polygon

A polygon with sides of equal length and interior angles equal measure

Triangle

A polygon with three straight sides and three interior angles

Quadrilaterals

A quadrilateral is a four sided polygon. Regardless of its shape, the four interior angles will sum to 360° there are five types of quadrilaterals. trapezoids, parallelograms, rectangles, rhombuses, and squares These are ranked in order of descendants. A parallelogram is a trapezoid, a square is all of the above.. though a rhombus is on its own as a parallelogram

A rectangle

A rectangle is a parallelogram with four right angles. Opposite sides are equal and diagonals are equal

Rhombus

A rhombus is a parallelogram with four equal sides. Opposite angles are equal to each other, but they do not have to be right angles

Square

A square is a rectangle with equal sides

Trapezoid

A trapezoid is a quadrilateral with at least one pair of parallel sides

Equilateral triangle

A triangle has three sides are all equal in length and whose three interior angles each measure 60°

Isosceles triangle

A triangle with two equal sides which are opposite to equal angles.

Vertex

A vertex is the point at which two lines or line segments intersect to form an angle. Angles are measured in degrees.

Quantitative comparison questions with triangles

All quantitative comparison questions require you to judge if there is enough information to make a comparison. In geometry, this judgment is often a matter of knowing the correct definition of formula. for triangles: - if you know two angles, you know the third - to find the area, you need the base and the height - in a right triangle, if you have two sides, you can find the third. And if you have two sides, you can find the area - in an isosceles right triangle and 30° 60° 90° triangles, if you know one side, you can find everything

Central angle

An angle formed by two radii.

The arc of a circle

An arc is a section of the circumference. Any arc can be thought of as the portion of a circle cut off by a particular central angle. in Circle Q, arc ABC is the portion of the circle that is cut off by the central angle AQC The measurement of an arc is equal to that of the central angle that cuts it off. So ABC and AQC would have the same degree measurement an arc that is exactly half the circumference of its circle is called a semi circle The length of an arc is the same fraction of a circle circumference as its degree measure is of 360° len(arc) = (n/360) * 2*Pi*radius or circumference

Sum of angles around a point and along a straight line

The sum of the measures of angles around a point is 360° The sum of the measures of angles on one side of a straight line is 180°. Two angles are supplementary to each other if their measures sum to 180°

legs (triangle)

The two equal sides of an isosceles triangle or the two shorter sides of a right triangle, often the one forming the right angle. The unequal side of an isosceles triangle is sometimes called the base

True or false: the sum of the lengths of any two sides of a triangle is greater than the length of the third side

True

True or false the perimeter of a square is equal to the sum of the lengths of the four sides

True, because all four sides are the same length, we just multiply one side times four

True or false: the right angle is always the largest angle in a right triangle

True, therefore the hypotenuse, which lies opposite the right angle, is always the longest side

Perpendicularity

Two lines are perpendicular if they intersect at a 90° angle (right angle). to find the shortest distance from a point to a line, draw a line from the point to the line such that it is perpendicular. You can then measure the length of this new line.

Pythagorean triples

certain ratios of integers always satisfy the Pythagoras and theorem, we call these the Pythagorean triples three, four, and five are very common as is six, eight, and 10

Area of a parallelogram

designate one side as the base. Then, draw a line segment from one of the vertices opposite the base down to the base. That line will be the height. base * height

True or false: if the lengths of two sides of a triangle are equal, the greater angle lies opposite the longer side and vice versa

false, if the lengths of two sides of a triangle are UNEQUAL, then the greater angle lies opposite the longer side and vice versa

Circumference

if the distance around a polygon is called its perimeter, then the distance around a circle is called at circumference The ratio of the circumference of any circle to its diameter is a constant called Pi, which is equal to 3.14 so to find circumference multiply the diameter of the circle and pi C = Pi(D)

Adjacent and vertical angles

imagine two lines intersecting with each other to form four angles The adjacent angles are angles that are next to each other, or we might say are supplementary because they lie along a straight line. The vertical angles are not adjacent to each other, they are considered opposite each other. These opposite angles are equal in measure because each of them is supplementary to the same adjacent angles

Area of a rectangle

length * width

Area of a square

length of side^2

Pi

pi is the ratio of the circumference of any circle to its diameter. Pi is approximated as 3.14 since pi equals the ratio of the circumference to the diameter, we can say that Pi = C/D

The sector of a circle

sector is a portion of a circles area that is bounded by two radii and arc. sector AXB is in the picture to find the area of a sector, find the degree measure of the sector central angle and figure out what fraction that degree measure is of 360° then multiply the area of the whole circle by that fraction A = (n/360) * Area of the circle

Special right triangles

special right triangles refer to an isosceles right triangle that has a ratio of 45-45-90 or 30-60-90 the 45-45-90 have a ratio of x:x:x * Square root of two, with the hypotenuse equaling X times the square root of two the 30-60-90 have a ratio of x:x * square root of 3: 2x

angle A of triangle ABC is a right angle. Is side BC longer or shorter than AB?

this question seems very abstract, until you draw a diagram of a right triangle, labeling the vertex with a 90° angle as point A The side opposite a 90° angle is called the hypotenuse and it is always the longest side of a right triangle

Complementary, acute, obtuse, and supplementary angles

two angles are complementary to each other if their measures sum to 90° an acute angle is any angle that measures less than 90° an obtuse angle is any angle that measures between 90° and 180° two angles are supplementary if their measures sum to 180°

Which is larger? A: Area of right triangle ABC, where AB = 5 and BC = 4 B: 6

we do not have enough information to answer this. We know the two sides of the right triangle, but we don't know if they are legs or the hypotenuse and therefore can't find the area.

Perimeters of quadrilaterals

you can find the perimeter of any polygon, by simply adding the length of its sides. However, the properties of rectangles and squares lead to a very simple formula that you can use to speed up calculations. P = 2(L+W) The perimeter is two times length plus width

How to find the sum of the interior angles of a polygon

Simply divide the polygon into triangles. Draw diagonals from any vertex to all the nonadjacent vertices then multiply the number of triangles by 180°. This works because the sum of the interior angles of any triangle is always 180° in the example we see three triangles, so 3×180° = 540°

How to find the perimeter of a triangle

Sum of the lengths of the sides. if A, B, and C are integers and A and B both have a length of 4, what is the largest possible perimeter of the triangle perimeter is 4+4+ C and we know that the largest possible third side of a triangle must be smaller than the sum of the other two sides. So side C must be smaller than eight. because all sides are integers inside C should be as large as possible to create the largest possible perimeter, C = 7 4+4+7=15

The Pythagorean theorem

The Pythagorean theorem which holds for all right triangles and no others, states that the square of the hypotenuse is equal to the sum of the squares of the legs a^2 + b^2 = c^2 The Pythagorean theorem is very useful whenever you're given the lengths of any two sides of a right triangle, because as long as you know weather the remaining side is a leg or hypotenuse, you can find its length with the theorem

The altitude of a triangle

The altitude or height of a triangle is the perpendicular distance from a vertex to the side opposite the vertex. The altitude may fall inside or outside the triangle or it may coincide with one of the sides

Area of a circle

The area of a circle is Pi(r)^2

Area of a triangle

The area of a triangle is 1/2(b)(h) base * height * 0.5 The area of a right triangle is very easy to find. Think of one leg is the base and the other as the height. The area is 1/2 the product of those two legs

Perimeter

The distance around a polygon; the sum of the lengths of its sides

Circle

The set of all points in a plane at the same distance from a certain point. AB is the diameter OA, OB, and OC are the radii A circle's total measurement is 360°. Half of a circle is 180°

What do small slash marks mean on the sides of polygons or on the angles?

The slash marks provide information, angles with the same number of slash marks through their arcs have the same measure and likewise for the sides of a polygon.

Interior and exterior angles of a triangle

The sum of the interior angles of any triangle is 180°. Therefore, in any triangle, all of the angles must add up to 180° An exterior angle of a triangle is when the vertex of a triangle meet another line and creates another angle with it. that angle is equal to the sum of the remote interior angles. this means that an exterior angle of a triangle is whatever the two other interior angles are added together. The three exterior angles of any triangle add up to 360°

Triangle Inquality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

What is the value of X?

X and the angle adjacent and to the left of the 70° angle on L2 are corresponding angles. Therefore the angle marked X must be supplementary to the 70° angle. 70° + X = 180°, so X must equal 110°

what is the length of the hypotenuse of a right triangle with legs of lengths 9 and 10

a^2 + b^2 = c^2 9^2 + 10^2 = 181 The hypotenuse equals the square root of 181

Areas of quadrilaterals

all area formulas involve multiplication, and the results are always stated in square units.

Angle

an angle is the result of a vertex (where two point meet). We measure angles in degrees. angles may be named according to their vertices. Sometimes, an angle is named according to three points, rather than the vertex