Chapter 1 Tools of Geometry

1-1: Points, Lines and Planes The students will… identify and model points, lines, and planes. identify intersecting lines and planes.

Some terms in geometry are considered undefined terms because they are only explained using examples and descriptions. A point is a location. It does not have size or shape. A point A

A line is a set of points that extends forever in two opposite directions. Has no thickness or width There is exactly one line through any two points. line m line AB line BA AB BA A B m

A plane is a set of points that extends forever in all directions. There is exactly one plane through any three points not on the same line. plane K plane ABC plane ACB plane BAC plane BCA plane CBA plane CAB B K C A

Collinear points are points that lie on the same line. Noncollinear points do not lie on the same line. Coplanar points are points that lie in the same plane. Noncoplanar points do not lie on the same plane.

Example: Use the figure to name each of the following: A line containing point C A plane containing point A F E B A s C D P r

The intersection of two or more geometric figures is the set of points they have in common. X b

Example: Draw and label a diagram for each of the following: Points J(-4,2), K(3,2), and L are collinear Line p lies in plane N and contains point L Line s intersects plane A at point P

Example: Use the figure below: How many planes appear in the figure? Name three points that are collinear. Name the intersection of plane HDG with plane X. At what point do LM and EF intersect? Are points E,D,F, and G coplanar? At what point or in what line do planes JDH, JDE, and EDF intersect? Pg. 8-12 #14-36 even, 53, 59

Review How many points determine a line? How many points determine a plane? Refer to the points below: Are these points collinear? Why or why not? Are these points coplanar? Why or why not?

What is the intersection of two nonparallel lines? What is the intersection of two nonparallel planes? Draw a figure to represent each of the following: Line a lies in plane P and contains point Z. Line b intersects plane Q at point Y. Planes Q and R intersect at line WX.

1-2: Linear Measure The students will… measure segments. calculate with measures.

A line segment is a portion of a line that has two endpoints. The measure of AB is written as AB. AB BA AB = 4.4 cm = 1¾ in A B

“Betweenness” of Points Point M is between points P and Q if and only if P, Q, and M are collinear and PM + MQ = PQ. P M Q

Example: Find JL. Assume that the figure is not drawn to scale. JK + KL = JL 2.3 + 8.4 = JL 10.7 cm = JL 2.3 cm K 8.4 cm L

Example: Find the value of x and ST if T is between S and U, ST = 7x, SU = 45, and TU = 5x - 3. ST + TU = SU 7x + 5x - 3 = 45 12x - 3 = 45 12x = 48 x = 4 ST = 7(4) = 28 S 7x T 5x - 3 U

Segments that have the same measure are called congruent segments. Example: Find the value of a and XY if Y is between X and Z, XY = 3a, XZ = 5a – 4, and YZ = 14. Segments that have the same measure are called congruent segments. Pg. 18-21 #10-30 even, 37 A Z B Y

Review Draw and label a figure for each relationship: FG lies in plane M and contains point H. Lines r and s intersect at point W. What is the length of CD if CE = 1.1 in, ED = 2.7 in, and E is between C and D. Find the value of x and BC if B is between C and D: CB = 2x, BD = 3x + 5, BD = 12.

1-3: Distance and Midpoint The students will… find the distance between two points. find the midpoint of a segment.

Distance Formula (on a number line) The distance between two points is the length of the segment with those points as its endpoints. Distance Formula (on a number line) d = |x2 – x1|

Distance Formula (in a coordinate plane) Distance can be rational or irrational. Rational – whole numbers, terminating decimals, repeating decimals Irrational – decimal that doesn’t repeat or terminate

On your calculator: When you square a negative number, put it in parenthesis (-12)2 When you square a number, it will ALWAYS be positive.

Example: Find the distance between (-4, 1) and (3, -1) d = √(x2 – x1)2 + (y2 – y1) 2 d = √(3 + 4)2 + (-1 – 1) 2 d = √(7)2 + (-2) 2 d = √49 + 4 d = √53 ≈ 7.28

Example: The United States Capitol is located 800 meters south and 2300 meters to the east of the White House. If the locations were placed on a coordinate grid, the White House would be at the origin. What is the distance between the Capitol and the White House?

The midpoint of a segment is the point halfway between the endpoints of the segment. If X is the midpoint of AB, then AX = XB and AX = XB. ~ A 2 in X 2 in B

Midpoint Formula (on a number line) Example: The temperature on a thermometer dropped from a reading of 28o to -8o. What is the average temperature?

Midpoint Formula (on a coordinate plane) Example: Find the coordinates of M, the midpoint of GH, for G(8, -6) and H(-14,12).

Example: Find the coordinates of M, the midpoint of ST, for S(-6, 3) and T(1,0). Example: Find the coordinates of J if K(-1, 2) is the midpoint of JL and L has coordinates (3, -5). (-5, 9)

Example: Find the measure of PQ if Q is the midpoint of PR. Example: Find the coordinates of D if E(-6, 4) is the midpoint of DF and F has coordinates (-5, -3). Example: Find the measure of PQ if Q is the midpoint of PR. PQ = QR 9y – 2 = 14 + 5y 4y – 2 = 14 4y = 16 y = 4 PQ = 9(4) – 2 = 34 P 9y - 2 Q 14 + 5y R

Example: Find the measure of YZ if Y is the midpoint of XZ and XY = 2x – 3 and YZ = 27 – 4x. Any segment, line, or plane that intersects a segment at its midpoint is called a segment bisector.

Constructions are figures that are created using only a compass and straightedge. Construct a segment bisector. Pg. 31-33 #14-48 even, 57, 63

Review Find the distance between the points (-1, -8) and (3,4). Find the value of a and ST is S is between R and T: RS = 7a, ST = 12a and RT = 76. What is the midpoint of a segment that has endpoints at (3, 4) and (-15, 2)?

1-4: Angle Measures The students will… measure and classify angles. identify and use congruent angles and the bisector of an angle.

A ray is part of a line that has one endpoint and extends forever in one direction. When you name a ray, list the endpoint first. ray XY XY Y X

An angle is formed by two noncollinear rays that have a common endpoint. The rays are called sides of the angle. The common endpoint is the vertex. Z side XZ Y X side XY vertex X

Naming an angle: Use THREE letters (the vertex must be the middle) Use a single letter (the vertex) ONLY when there is one angle at the vertex Use a number if the angle is labeled with one ZXY YXZ X 1 Z A Y 2 1 X

An angle divides the plane into three distinct parts: The angle The interior of the angle The exterior of the angle Z X Y

Example: Name all angles that have B as a vertex. Name the sides of 5. Write another name for 6. G A B 5 7 6 D E 4 3 F

Angles are measured in degrees. m ZXY = 78o Z X Y

Angles can be classified by their measures: Right – equals 90o Acute – less than 90o Obtuse – greater than 90o Example: Classify each angle. MJP LJP NJP M L N K J P

Angles that have the same measure are called congruent angles. Z A X C Y B

Construct an angle bisector. A ray that divides an angle into two congruent angles is called an angle bisector. Construct an angle bisector. Z A X Y

Example: KN bisects JKL Example: KN bisects JKL. If m JKN = 8x – 13 and m NKL = 6x + 11, find m JKN. J N L K

Line/Angle Art

Pg. 41-43 #12-44 even, 51 Example: Create your own line design: 1 9 2 8 3 7 4 6 5 5 6 4 7 3 8 2 60o 9 1 0.5 in 4.5 in sides

Review Use the figure below: Name a point that is collinear with points A and B. What is another name for plane P. Name three points that are noncollinear. Find the midpoint and length of a segment that has endpoints of (-2, -3) and (4, 1). F E B A s C D P r

Use the figure below: If m AXC = 8x – 7 and m AXB = 3x + 10, find m AXC. C B D A X E

1-5: Angle Relationships The students will… identify and use special pairs of angles. identify perpendicular lines.

Adjacent angles are two angles that lie in the same plane and have a common vertex and a common side, but no common interior points. 2 1

A linear pair is a pair of adjacent angles with noncommon sides that are opposite rays. 2 1

Vertical angles are two nonadjacent angles formed by two intersecting lines. Vertical angles are congruent. 2 1 3 4

Complementary angles are two angles with measures that have a sum of 90o. 1 70o 2 A B 20o

Supplementary angles are two angles with measures that have a sum of 180o. The angles of a linear pair are supplementary. 1 2 160o A B 20o

The two angles are 106o and 74o. Example: Find the measures of two supplementary angles if the difference in the measures of the two angles is 32o. m A + m B = 180o A + B = 180 m A - m B = 32o A - B = 32 2A = 212 A = 106 106 + B = 180 B = 74 The two angles are 106o and 74o. +

Lines, segments, or rays that form right angles are perpendicular. B C A D

Example: Find x and y so that KN and HM are perpendicular. KJH = 90o 9x + 3x + 6 = 90 12x + 6 = 90 12x = 84 x = 7 MJN = 90o 3y + 6 = 90 3y = 84 y = 28 L K M 9xo (3y + 6)o I (3x + 6)o J N H

In geometry, there are some things that you cannot assume based on a picture. Perpendicular lines Congruent angles or segments Look at Pg. 49 for a more detailed list. Pg. 51-53 #8-22, 29, 30, 49

Review Find the measure of each angle in the figure below: What is the distance between the points (2, 1) and (-3, 4)? Draw a diagram to represent each of the following: Two adjacent angles Vertical angles Supplementary nonadjacent angles. 7x + 17 3x - 20

1-6: Two-Dimensional Figures The students will… identify and name polygons. find perimeter, circumference, and area of 2D figures.

A polygon is a closed figure formed by a finite number of coplanar segments. Polygons Not Polygons

Each point is a vertex of the polygon. Each segment is a side of the polygon. A polygon is named using the letters of the vertices. B vertex C C A side AE Polygon ABCDE E D

Polygons can be convex or concave. If you draw lines along each side of the polygon, if any of the lines pass through the polygon, then it’s concave. Otherwise, it’s convex. convex concave

Classification of polygons: Number of Sides Name of Polygon 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 11 Hendecagon 12 Dodecagon n n-gon

An equilateral polygon is a polygon in which all sides are congruent. An equiangular polygon is a polygon in which all angles are congruent.

A convex polygon that is both equilateral and equiangular is a regular polygon. If a polygon is not regular, then it is irregular.

Example: Name each polygon by its number of sides and classify it as convex or concave and regular or irregular.

The perimeter of a polygon is the sum of the lengths of its sides. The circumference of a circle is the distance around the circle. The area of a figure is the number of square units needed to cover a surface.

Review of formulas for common polygons and circles (pg Review of formulas for common polygons and circles (pg. 58 in your book):

Perimeter and circumference are measured in units and area is measured in square units. Example: Find the perimeter or circumference and the area of each figure: 2.5 in 2.5 in 2.5 cm 1.5 in 3 in 4 in 5.6 cm

Example: Teri has 19 feet of tape to mark an area in the classroom where the students may read. Which of these shapes has a perimeter or circumference that would use most or all of the tape? Square with side lengths of 5 feet Circle with radius of 3 feet Right triangle with each leg length of 6 feet Rectangle with a length of 8 feet and a width of 3 feet

Example: Find the perimeter and area of ∆PQR with vertices P(-1, 3), Q(-3, -1), and R(4, -1). Pg. 61-63 #11-22, 24, 25, 33, 44

Review Name each polygon and classify it as convex or concave and regular or irregular. Find the value of each variable: The intersection of two planes is a ____. What is the midpoint of a segment that has endpoints at (9,1) and (-8,5). 12x + 7 5x x - 6 14x - 3

1-7: Three-Dimensional Figures The students will… identify and name 3D figures. find surface area and volume.

A solid with all flat surfaces that enclose a region of space is called a polyhedron. Each flat surface, or face, is a polygon. The line segments where the faces intersect are called edges. The point where three or more edges intersect is called a vertex.

Types of Solids: Polyhedrons A prism is a polyhedron with two parallel congruent faces called bases connected by parallelogram faces. A pyramid is a polyhedron that has a polygonal base and three or more triangular faces that meet at a common vertex.

Not Polyhedrons A cylinder is a solid with congruent parallel circular bases connected by a curved surface. A cone is a solid with a circular base connected by a curved surface to a single vertex. A sphere is a set of points in space that are the same distance from a given point.

Polyhedra are named by the shape of their bases. Triangular Prism Rectangular Prism Pentagonal Prism Pentagonal Pyramid Triangular Pyramid Rectangular Pyramid

A polyhedron is a regular polyhedron if all of its faces are regular congruent polygons and all of the edges are congruent. There are five types of regular polyhedrons, called Platonic solids.

Surface area is a 2D measurement of the surface of a solid figure. Volume is the measure of the amount of space enclosed by a solid. Surface area and volume formulas (Pg. 69)

Example: Find the surface area and volume of each solid to the nearest tenth.

Example: The diameter of the pool Joe purchased is 8 feet Example: The diameter of the pool Joe purchased is 8 feet. The height of the pool is 20 inches. What is the surface area of the pool? What is the volume of water needed to fill the pool to a depth of 16 inches? Pg. 71-73 #6-13, 18-24, 37