Section 1-1 Points, Lines, and Planes

I CAN. Identify and model points, lines, and planes. Identify collinear and coplanar points and intersecting lines and planes in space.

Undefined Term- Words, usually readily understood, that are only explained using examples and descriptions. Point Line Plane

Point- A specific location. Drawn: as a dot Named by: a capital letter

Line- A straight path that continues forever in both directions. Drawn: with an arrowhead at each end Named by: the letters representing two points on the line or a lowercase script letter

Plane- A flat surface that extends indefinitely in all directions. Drawn: As a shaded, slanted 4-sided figure Named by: a capital script letter or by the letters naming three noncollinear points.

Collinear- Points that lie on the same line. Coplanar- Points that lie in the same plane.

Use the figure to name each of the following. a) A line containing point A b) A plane containing point C

Use the figure to name each of the following. a) A line containing point W b) A plane containing point X

Name the geometric terms modeled by each object. a) The long hand on a clock. b) A 10 X 12 patio c) The location where the corner of a driveway meets the road

Intersection A set of points common to two or more geometry figures. Two lines intersect at a point. Two planes intersect at a line.

Draw and label a figure for the relationship given. Point P represents the intersection of AB and CD

Draw and label a figure for the relationship given. TU lies in plane Q and contains point R.

Draw and label a figure for the relationship given. QR intersects plane T at point S.

Draw and label a figure for the relationship given. Lines GH and JK intersect at L for G(-1, -3), H(2, 3), J(-3, 2), and K(2, -3) on a coordinate plane. Point M is coplanar with these points, but not collinear with GH or JK

Draw and label a figure for the relationship given. Lines AB and CD intersect at E for A(-2, 4), B(0, -2), C(-3, 0), and D(3, 3) on a coordinate plane. Point F is coplanar with these points, but not collinear with AB or CD

Definitions (Defined Terms) terms that are explained using undefined terms and/or other defined terms. Space- A boundless three-dimensional set of all points.

A) How many planes appear in this figure? B) Name three points that are collinear. C) Are points G, A, B, and E coplanar? Explain. D) At what point do EF and intersect? AB

A) How many planes appear in this figure? B) Name three points that are collinear. C) Name the intersection of plane HDG with plane X. D) At what point do LM and EF intersect?

Section 1-2 Linear Measure

I CAN. Measure segments. Calculate with measures.

Line Segment (Segment)- A measurable part of a line that consists of two points, called endpoints, and all the points between them. The length or measure of a segment always includes a unit of measure, such as meter or inch. All measurements are approximations dependent upon the smallest unit of measure available on the measuring instrument.

*Metric measurements (centimeters, millimeters, etc.) are written as a decimal. *Standard measurements (inches, feet, etc.) are written as a fraction.

Find the length of CD using each ruler.

Find the length of AB using each ruler.

Find the length of CD using each ruler.

Between- For any two points, there is a point between them such that the two pieces add to equal to the whole piece. Point M is between points P and Q if and only if P, Q and M are collinear and PM + MQ = PQ

Find AC. Assume that the figure is not drawn to scale.

Find EG. Assume that the figure is not drawn to scale.

Find DE. Assume that the figure is not drawn to scale.

Find AB. Assume that the figure is not drawn to scale.

Find the length of LM if M lies between L and N, LN = 4 cm, and MN = 2.6 cm.

Find y and PQ if P is between Q and R, PQ = 2y, QR = 3y + 1, and PR = 21.

Find x and ST if T is between S and U, ST=7x, SU=45, and TU=5x 3.

Congruent- Having the same measure. Congruent Segments Segments that have the same measure. is read is congruent to. XY PQ Red slashes on the figure also indicates that segments are congruent.

In the graph, suppose a segment was drawn along the top of each bar. Which states have segments that are congruent? Explain.

Name the congruent segments in the sign shown.

Section 1-2 Extension Absolute Error and Relative Error

I CAN. Determine the absolute error of a measurement. Determine the relative error of a measurement.

Absolute Error one half the unit of measure for any measurement. 1 AE 2 smallest unit The smaller the unit of measure, the more precise the measurement.

Find the absolute error of each measurement. Then explain its meaning. A) 6.4 centimeters B) 4.2 meters

Find the absolute error of each measurement. Then explain its meaning. A) 2 ¼ inches B) 1 ½ inches

Significant Digits the digits that are used to express a measurement in precision. To determine whether digits are considered significant, use the following rules: Nonzero digits are always significant. In whole numbers, zeros are significant if they fall between nonzero digits. In decimal numbers greater than or equal to 1, every digit is significant. In decimal numbers less than 1, the first nonzero digit and every digit to the right are significant.

Determine the number of significant digits in each measurement. A) 430.008 meters B) 0.00750 centimeters C) 779,000 mi D) 50,008 ft E) 230.004500 m

Relative Error RE Absolute Error Expected Measure

A manufacturer measures each part for a piece of equipment to be 23 centimeters in length. Find the relative error of this measurement.

Find the relative error of each measurement. A) 3.2 mi B) 1 ft C) 26 ft

Section 1-3 Distance and Midpoints

I CAN. Find the distance between two points. Find the midpoint of a segment.

The distance between two points is the length of the segment with those points as its endpoints. Distance on a Number Line Find the absolute values of the difference between their coordinates. Coordinate Plane Distance Formula d 2 2 x x y y 2 1 2 1

Use the number line to find CD.

Use the number line to find the length of the indicated segment. A) BE B) AC C) CF

Irrational Number a number that cannot be expressed as a terminating or repeating decimal.

Find the distance between R(5, 1) and S(-3, -3).

Find the distance between C(-4, -6) and D(5, -1).

Midpoint- The point halfway between the endpoints of a segment. Number Line x 1 x 2 2 Coordinate Plane Midpoint Formula x x y y 1 2, 1 2 2 2

Ex. Find Coordinates of Midpoint Find the coordinate of the midpoint of PQ

Find the coordinates of M, the midpoint of for P(-1, 2) and Q(6, 1). PQ

Find the coordinates of M, the midpoint of for S(-6, 3) and T(1, 0). ST

To Find a Missing Endpoint 1. Find a pattern from the endpoint to the midpoint. 2. Continue the pattern to find the missing endpoint.

Find the coordinates of X if Y(-2, 2) is the midpoint of and Z has coordinates (2, 8). XZ

Find the coordinates of D if E(-6, 4) is the midpoint of and F has coordinates (-5, -3) DF

Segment Bisector- A segment, line, or plane that cuts a segment into two equal pieces.

What is the measure of BC if B is the midpoint of AC?

What is the measure of PR if Q is the midpoint of PR?

Section 1-4 Angle Measure

I CAN. Measure and classify angles. Identify and use congruent angles and the bisector of an angle.

Ray A part of a line that has one endpoint and extends indefinitely in one direction. To name a ray always start with the endpoint.

Opposite Rays- Two rays that form a straight line. Since both rays share a common endpoint, opposite rays are collinear.

Angle- The intersection of two noncollinear rays at a common endpoint. Sides- The rays of an angle. Vertex- The point on an angle where the two rays intersect.

An angle divides a plane into three distinct parts. Points A, D and E lie on the angle. Points C and B lie in the interior of the angle. Points F and G lie in the exterior of the angle.

Interior- A point that does not lie on the angle itself and lies inside the angle. Exterior- A point that does not lie on the angle itself and lies outside the angle.

To name an angle: Naming Angles Use the number inside the angle Use the three points that make up the angle, always putting the vertex in the middle.

A) Name all angles that have W as a vertex. B) Name the sides of 1. C) Write another name for WYZ.

Use the map of a high school shown. A) Name all angles that have B as a vertex. B) Name the sides of 3. C) Write another name for GHL. D) Name a point in the interior of DBK.

Degree- A unit measure used in measuring angles and arcs. The degree results from dividing the distance around a circle into 360 parts.

To measure an angle, you can use a protractor. Angle PQR is a 65 degree angle. We say the degree measure or m PQR = 65

Right Angle- An angle with a degree measure of 90. Acute Angle- An angle with a degree measure less than 90.

Obtuse Angle- An angle with degree measure greater than 90 and less than 180.

Use a protractor to measure the angle to the nearest degree. Classify each angle as right, acute, or obtuse. A) PMQ B) PMR C) QMS

Congruent Angles Angles that have the same measure. Arcs on the figure also indicate which angles are congruent.

Angle Bisector- A ray that divides an angle into two congruent angles.

ABC DBF. If m ABC = 6x +2 and m DBF = 8x 14, find the actual measurements of ABC and DBF.

In the figure, and are opposite rays, and bisects JKL. If JKN 8x 13 and find m JKN. KJ KM KN m m NKL 6x 11

Section 1-5 Angle Relationships

I CAN. Identify and use special pairs of angles. Identify perpendicular lines.

Adjacent angles- Two angles beside each other, that share a side and a vertex. Examples Nonexample

Vertical angles- Two angles across from each other formed by intersecting lines.

Linear pair- Two angles that make a straight line.

Name an angle pair that satisfies each condition. A) Two obtuse vertical angles B) Two acute adjacent angles C) Name a linear pair

Complementary angles- Two angles with measures that have a sum of 90.

Supplementary angles- Two angles with measure that have a sum of 180.

Find the measures of two complementary angles if the difference in the measures of the two angles is 12.

Find the measures of two supplementary angles if the difference in the measures of the two angles is 18.

Perpendicular- Two lines that intersect at a 90 degree angle. Symbol: is read is perpendicular to. Perpendicular lines intersect to form four right angles. Perpendicular lines intersect to form congruent adjacent angles. Segments and rays can be perpendicular to lines or other line segments and rays. The right angle symbol in the figure indicates that the lines are perpendicular.

Find x and y so that BE and AD are perpendicular.

Find x and y so that PR and SQ are perpendicular.

Determine whether each statement can be assumed from the figure. A) LPM and MPO are adjacent. B) OPQ and LPM are complementary. C) LPO and QPO are a linear pair.

Determine whether each statement can be assumed from the figure. A) KHJ and GHM are complementary. B) GHK and JHK are a linear pair. HL HM C) and are perpendicular.

Section 1-6 Two-Dimensional Figures

I CAN. Identify and name polygons. Find perimeter, circumference, and area of two-dimensional figures.

Polygon- A closed figure formed by a finite number of coplanar segments called sides. The sides that have a common endpoint are noncollinear, and Each side intersects exactly two other sides, but only at their endpoints, called the vertices. Vertex the point where two sides intersect.

Concave- A polygon for which there is a line containing a side of the polygon that also contains a point in the interior of the polygon. On a concave polygon, part of the figure caves in.

Convex- A polygon for which there is no line that contains both a side of the polygon and a point in the interior of the polygon.

n-gon- A polygon with n sides. Number of Sides Polygon 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 12 Dodecagon n n-gon

Regular polygon- A convex polygon in which all of the sides are congruent and all of the angles are congruent. Octagon PQRSTUVW is a regular octagon.

Irregular polygon- A polygon in which all the sides are not congruent, or all the angles are not congruent, or a polygon that is concave.

Name each polygon by its name of sides. Then classify it as convex or concave and regular or irregular.

Perimeter- The sum of the lengths of the sides of a polygon. Circumference- The distance around a circle. Area- The number of square units needed to cover a surface.

Perimeter, Circumference, and Area

Find the perimeter or circumference and area of the figure.

Yolanda has 26 centimeters of cording to frame a photograph in her scrapbook. Which of these shapes would use most or all of the cording and enclose the largest area? A) right triangle with each leg about 7 cm long B) Circle with a radius of about 4 cm C) Rectangle with a length of 8 cm and a width of 4.5 cm D) Square with side length of 6 cm

Find the perimeter and area of triangle PQR if P(-1, 3), Q(-3, -1), and R(4, -1). 2 Use the Distance Formula d x x y 2 2 1 2 y1

Section 1-7 Three-Dimensional Figures

I CAN. Identify and name three-dimensional figures. Find surface area and volume.

Polyhedron Closed three-dimensional figures made up of flat polygonal regions. Face A flat surface of a polygon. Edges The line segments where the faces intersect. Vertices The point where three or more edges intersect.

Solids that are Polyhedrons Prism A polyhedron with two parallel congruent faces called bases connected by parallelogram faces. Pyramid A polyhedron that has a polygonal base and three or more triangular faces that meet at a common vertex.

Polyhedrons or polyhedra are named by the shape of their bases.

Solids that are not Polyhedrons Cylinder A solid with congruent parallel circular bases connected by a curved surface. Cone A solid with a circular base connected by a curved surface to a singular vertex. Sphere A set of points in space that are the same distance from a given point.

Determine whether the solid is a polyhedron. Then identify the solid. If it is a polyhedron, name the bases, faces, edges, and vertices.

Regular polyhedron A polyhedron in which all of the faces are regular congruent polygons. Platonic solids The five regular polyhedra: tetrahedron, hexahedron, octahedron, dodecahedron, or icosahedron.

Surface Area A two-dimensional measurement of the surface of a solid. Volume The measure of the amount of space enclosed by a solid figure.

Surface Area and Volume

Find the surface area and volume of the square pyramid.

Find the surface area and volume of the cylinder.

The diameter of the pool Mr. Sato purchased is 8 feet. The height of the pool is 20 inches. Find each measure to the nearest tenth. A) Surface Area of Pool B) Volume of Pool