Geometry and Measurement

Combine and Dissect Figures

Sometimes it is necessary to look at the different parts of a figure to determine properties such as area and volume. Rearranging a complicated figure can make the math simpler to do. When this is done it is called composing (putting together) or decomposing (pulling apart) figures.

Direction and Structure

Spatial concepts are important in geometry and they begin in early elementary. Ideas such as above, below, together, apart, inside, and other direction words have significant meaning with shapes. As students grow in their geometric knowledge, they will begin to put multiple shapes together to create complex structures. However, they need a strong foundation of spatial concepts in order to advance to a higher level.

Area and Perimeter of Polygons

The perimeter of a figure is the distance around the outside of that figure. The area is the amount of surface inside of a figure.

Angles in Triangles

The sum of the interior angles of a triangle is always 180°. In other words, all three angles will always add up to 180°.

Proportional Reasoning and Similar Shapes

If shapes are similar, proportions can be set up and used to solve for unknown sides or angles. - When a pair of figures is declared to be similar (with the symbol ∼), the corresponding parts of the two figures are named in a consistent order. - The corresponding angles within similar figures are congruent angles (with the symbol \cong≅), which means they are equal in measure. - The corresponding side lengths are proportional sides, which means they can be used to create ratios.

Pyramids - Slant Heights

In three-dimensional figures that have slanted faces, such as cones and pyramids, there is also a third measurement called slant height (s). Slant height is the distance from the apex (top) to an edge of the base. - The height, h, is the distance from the apex (the top vertex) to the base of the pyramid. - The slant height, s, is the distance from the apex to the middle of an edge of the base.

What is an axiom

a self-evident truth that requires no additional proof axiom (or postulate) is a truth that is accepted as being self-evident. In mathematics, an axiom is a beginning point from which all other statements are derived.

theorems

must be proven before they can be accepted or used.

Spatial Reasoning

spatial reasoning can be very helpful in solving problems. Spatial reasoning is the ability to think about how things appear in real life. This means drawing a diagram of the situation to make it more evident what the question is asking. - This problem is very difficult without a drawing. Drawing a picture makes this question much easier to understand.

Attributes of Polygons

- A circle is the set of all points equidistant (the same distance) from a given point (the center). Circles have no sides or interior angles. A polygon is a multi-sided, closed figure made up of vertices (corners) and sides (segments connecting consecutive corners).

Relationships Between Angles - Multiple Lines

- Corresponding Angles a pair of angles that lie in matching corners of each intersection of the transversal When two parallel lines are crossed by a transversal, corresponding angles are equal. - Alternate Interior Angles a pair of angles formed on opposite sides of the transversal and between the two lines When two parallel lines are crossed by a transversal, alternate interior angles are equal. - Vertical Angles a pair of opposite angles made by two intersecting lines When two lines intersect, vertical angles are equal. - Alternate Exterior Angles a pair of angles formed on opposite sides of the transversal and on the outer sides of the two lines When two parallel lines are crossed by a transversal, alternate exterior angles are equal. - Supplementary Angles a pair of angles formed on the same side of the transversal and between the two lines Any pair of angles that form a straight angle is also supplementary.

How do you set up a proportion from a similar-shapes problem

- Draw the outline of a proportion (two equal fractions) inside of a rectangle. Label the left side of the rectangle with the two different shapes. On the top of the rectangle, label the first fraction "Side a" and the second fraction "Side b." - In the first fraction, place the value of "Side a" from each of the shapes, making sure the labels to the left correspond appropriately. - In the second fraction, write the known amount of "Side b." Place an xx for the unknown amount. - Cross multiply and divide to solve for xx.

Symmetry

- If a shape is symmetric, you can draw a line to create a mirror image. Otherwise, it is asymmetric.

Examples of theorems:

- If a transversal intersects two parallel lines, then alternate interior angles are congruent. - In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other. - The sum of the measures of the angles of a triangle is 180°.

Irregular Shapes

- Sometimes polygons have non-standard shapes. In this case, a polygon can be divided into multiple simpler shapes. For example, the polygon below can be broken up into a rectangle and a triangle.

Area v Perimeter

- The perimeter of an object is the distance around the outside of that figure. It is a one-dimensional measurement so it will always be labeled in units such as: inches (in.) feet (ft.) miles (mi.) centimeters (cm) - The area is the amount of surface inside of a figure. It is a two-dimensional measurement (length × width) so it will be labeled in square units, such as: square inches (in.2) square miles (mi.2) square centimeters (cm2) square kilometers (km2)

Circles Measurements

- The radius (r) is the distance from the center of a circle to any point on the circle. - The diameter (dd) is the distance from one point on a circle, through its center, to another point on the other side of the circle. The diameter is always twice as long as the radius (d=2r). - The circumference of a circle is the distance around the circle. It can be found by the formula: c = c=2πr=πd - The area of the circle is the area within it. It can be found by the formula: A = {A=πr2 (where r = radius)

properties of equality

- The transitive property states if a=b and b=c, then a=c. - The reflexive property states that a=a. - The symmetric property states if a=b, then b=a Using multiple axioms, theorems, and properties of equality a proof can create a new, true statement for a given scenario.

Examples of axioms/postulates:

- Through any two points there is exactly one line. - If two lines intersect, they intersect in a point. - Through any three non-collinear points there exists exactly one plane. - If a transversal intersects two parallel lines, then corresponding angles are congruent.

Parallel and Perpendicular Lines

A pair of linear equations could be parallel, perpendicular, or neither. It depends on their slopes. It's easy to tell the relationship if they are in slope-intercept form (y=mx+by=mx+b) since mm provides the slope. - Parallel lines are lines that never intersect because they are changing at the same rate. Therefore, lines that are parallel have the same slope. The graph below shows parallel lines with the equations y=2x+3 and y=2x+1. - Perpendicular lines are lines that intersect at a right (90º) angle. Lines that are perpendicular have slopes that are opposite reciprocals, meaning their signs are opposite and their fractions are flipped (ex: 3/4 and -4/3) *Parallel lines have the same slopes and different y-intercepts. Perpendicular lines have opposite and reciprocal slopes and same or different y-intercepts*

Types of Quadrilaterals

A quadrilateral is any polygon with four vertices and four sides. - Trapezoid: quadrilateral with one pair of parallel sides and one pair of non-parallel sides - Parallelogram: quadrilateral with two pairs of opposite sides that are parallel to each other. Both pairs of opposite angles are equal - Rectangle: parallelogram with four right (90°) angles. Both pairs of opposite sides are parallel. - Rhombus: parallelogram with all sides of congruent length. - Square: parallelogram with four right (90°) angles and four equal sides. Remember: ALL squares are rectangles, but not all rectangles are squares. ALL rectangles are parallelograms, but not all parallelograms are rectangles.

Tessellations

A tessellation is a pattern of shapes that fit perfectly together.

Rigid Transformations

A transformation is a change to a geometric object's placement. Just like algebraic functions use rules to take in an input (x) and create an output (y), geometric transformations use rules to take in pre-images (the original object) and create images (the resulting object).

types of angles

Acute Angle An acute angle has a measure between 0° and 90°, such as \angle ABE∠ABE. Right Angle A right angle has a measure of exactly 90°, such as \angle D∠D. Obtuse Angle An obtuse angle has a measure between 90° and 180°, such as \angle ABC∠ABC. Straight Angle A straight angle measures exactly 180°, such as \angle EBC∠EBC. Reflex Angle A reflex angle has a measure between 180° and 360°, such as \angle B∠B. Reflex angles are rarely discussed or studied, so when in doubt, the assumption should be that \angle B∠B is the angle that is less than 180°, unless otherwise specified.

Solving with Shapes - Rectangles and Squares

All quadrilaterals have four sides and their interior angles sum to 360

All three rigid transformations

All three rigid transformations maintain congruent angles and side lengths. Translations slide the object Reflections flip the object Rotations turn the object around the center of rotation

What is an angle

An angle is formed by the intersection of two rays, lines, line segments, or by two rays at a common endpoint. The point of intersection is the angle's vertex. The symbol for an angle is \angle∠

Pyramids and Prisms

Aside from spheres, most solids have either one base and an apex (top-most point) or two bases. Pyramids and prisms are both three-dimensional figures that are sometimes confused with each other. - A pyramid has a base, triangular faces, and an apex (where the faces meet). - A prism has quadrilateral faces with two congruent bases (and no apex). - The two bases of a prism are parallel to each other, but not necessarily aligned directly above/below each other. - A solid figure is named by its base: triangular prism, square pyramid, pentagonal pyramid, hexagonal prism, etc. *Both pyramids and prisms are named for their bases. Pyramids have one base and one apex. Prisms have two bases. A prism is a shape in three dimensions that has two of the same polygons for the base; prisms are named by the shape of their polygonal bases. A pyramid is a shape in three dimensions that has one polygon for a base and triangular faces that meet at a point (the apex); pyramids are named by the shape of their polygonal base.

What are Equilateral and Equiangular polygon

Equilateral polygon: a polygon with congruent (equal length) sides equilateral pentagon Equiangular polygon a polygon with congruent (equal measure) angles equiangular pentagon Remember: All regular polygons are equilateral, equiangular, and convex.

The Coordinate Plane

Lines can be graphed on the coordinate plane. The coordinate plane has an x axis and a y axis. The x-axis lies horizontally, while the y-axis stands up vertically. The intersection of the x-axis and y-axis is called the origin. Each of the four areas are a quadrant. *Remember that where x and y are positive is quadrant I. From there, the names number counterclockwise.*

Surface Area - From Nets

Nets can also be used to compute the surface area of a three-dimensional figure. This is particularly helpful for pyramids and prisms. When calculating surface area of 3D figures, sketch the net and find the area of each face.

Ordered Pairs

On the coordinate plane, points are represented by ordered pairs. An ordered pair is always in the form (x, y). The origin is the point located at (0,0). - On the x-axis, which goes horizontally, find the x coordinate. For positive values of x, go right; for negative, go left. - On the y-axis, which goes vertically, find the y coordinate. For positive values of y, go up; for negative, go down. - Place a point at the intersection of the x and y value.

Relationships Between Planes

Parallel Planes Intersecting Planes Two parallel planes never intersect each other. They are consistently equidistant from each other. Two intersecting planes intersect at one line, just as two intersecting lines intersect at one point.

Relationships Between Lines

Perpendicular lines are lines that intersect each other and form right angles. Parallel lines are lines that are coplanar and never intersect each other. They are consistently equidistant from each other. Intersecting Lines Intersecting lines are two unique lines that touch at one point. Skew lines are lines that are NOT coplanar and do not intersect.

Components of a Plane

Point A point is a location in space without any dimension. It has no length or width. A point is represented by a small dot and identified by a single letter. Line A line is a straight path of infinite length but no thickness. It is identified by any two points found on the line. Plane A plane is a flat surface of infinite length and width but no depth. It is identified by any three noncollinear points found within the plane or by a capital script letter. Line Segment A line segment is a straight path with two endpoints. It has a defined length, but no defined width. It is identified by the two endpoints of the segment. Ray A ray is a straight path that has one defined endpoint and extends infinitely in the other direction. It is identified by the endpoint and any other point on the ray. Collinear Points Collinear points are any two or more points on the same line. Coplanar Points Coplanar points are any points found within the same plane. Any three points will determine a unique plane.

Perimeters and Areas of Common Shapes

Rectangle P=2(w+l) A=l×w Square P=4s A = s^2 Triangle P = sum of all three sides A=1/2​bh Parallelogram P=2(a+b) A=bh Trapezoid P = sum of all sides A = ( b1+b2/2) hA=(2b1​+b2​​)h or A= 1/2( b1+b2)h w = width; l = length; s = side; b = base; h = height

Regular tessellations

Regular tessellations are made of only the same regular polygon. They can only be made using a regular triangle, square, or hexagon. - Tessellations can be thought of as a way to tile an infinitely large floor. - They are made of the same regular polygon repeating over and over again. - Each vertex must look identical.

Calculating Area and Perimeter of Parallelograms

Remember that opposite sides of a parallelogram are congruent. In this figure, the base is 5 centimeters while the height is 7 centimeters. Since the needed dimensions are known, the formulas can be used directly. P=2(b+h)=2(5+7)=24 A=bh=5(7)=35 cm

Semiregular tessellations

Semiregular tessellations are made of different regular polygons arranged in a pattern.

Converting Between Area and Perimeter

Since perimeter and area formulas use the same measurements, you can find one if you know the other. Since all four sides of a square are the same, you can easily go between perimeter and area. Example 1: Given a square with an area of 49\text{ u}^249 u2 find the perimeter. To find the perimeter, first, find the length of one side of the square. Since all four sides of a square are the same length, the length of a side can be found by taking the square root of the area. - To find the area, first, find the length of one side of the square. Since all four sides of a square are the same length, the length of a side can be found by dividing the perimeter by four. - While a square is a type of rectangle, it is important to remember that all rectangles do not have 4 equal sides. Therefore, you can't find both side lengths from just the area of a rectangle.

Volume

The volume of three-dimensional figures is the measure of the total three-dimensional space contained in an object. These objects include spheres, rectangular and triangular prisms, rectangular and triangular pyramids, cylinders, and cones.

Calculating Area of Irregular Shapes

To find the area of an irregular shape, break the shape into smaller common shapes.

Relationships Between Angles

Vertical Angles Vertical angles are the pair of opposite angles created when two lines (or line segments) intersect. In the diagram, two pairs of vertical angles exist: \angle FEI = \angle GEH∠FEI=∠GEH, and \angle FEG = \angle IEH∠FEG=∠IEH. - The two angles in a pair of vertical angles are congruent to each other. Complementary Angles Complementary angles are two angles that have a sum of 90° - Think of the 'C' in complementary as the front part of the number 9 in 90. Supplementary Angles Supplementary angles are two angles that have a sum of 180° - Think of the 'S' in supplementary as part of the 8 in 180. *When looking at two angles, vertical angles are equal, complementary angles add up to 90°, and supplementary angles add up to 180°*