Chapter 1 – Tools of Geometry

Section 1 – Points, Lines, and Planes

1.1 - I can describe and name the undefined notions of points, lines and planes.

1.2 - I can identify intersecting lines and planes.

  • Points – a location that has neither shape nor size. It is represented by a capital letter. (ex: point A)
  • Lines – made up of at least 2 points and has no thickness or width. There is exactly one line through any two points. It is represented by either 2 points or a lowercase script letter. (ex: line p or line )
  • Planes – a flat surface made up of points that extends infinitely in all directions. There is exactly one plane through any 3 points not on the same line. (plane ABC)
  • Collinear– points that lie on the same line. (If they do not lie on the same line they are called noncollinear)
  • Coplanar – points that lie in the same plane. (if they do not lie in the same plane they are called noncoplanar)
  • Intersection – the place where 2 objects meet. Two lines intersect at a point and two planes intersect at a line.
  1. Use the figure to name each of the following:
  1. A line containing point Kc. A plane containing points T and Z
  1. A plane containing point Ld. A line containing point T
  1. Name the geometric shape modeled by each:
  2. A 10 x 12 patioc. Stripes on a sweater
  1. A button on a tabled. The corner of a box
  1. Draw and label a figure for each of the following:
  2. Plane R contains lines and , which intersect a point P. Add point C on plane R so that it is not collinear with or .

b. Line p lies in plane N and contains point L.

c. Line on a coordinate plane contains Q(-2,4) and R(4, -4). Add point T so that T is colliear with these points.

  1. Use the figure for parts a-c:
  1. How many planes appear in the figure? Name them.
  1. Name three points that are collinear.
  1. Are points A, B, C, and D coplanar? Explain.
  1. At what point do and intersect?

5.

  1. How many planes appear in this figure? Name them.
  1. Name the intersection of plane HDG with plane X.
  1. At what point do lines and intersect? Explain.
  1. Are points E, D, F and G coplanar?
  1. At what point or in what line does planes JDH, JDE, and EDF intersect?

Homework: Page 8 – 11 (13-31 ALL, 32, 34, 35, 36, 37, 40-49 ALL and 58, 61)

Section 2 – Linear Measure

2.1 - I can precisely define and measure line segments and describe their characteristics.

2.2 - I can use a variety of tools to perform the following constructions: I can bisect a segment.

  • Line Segment – unlike a line (that extends infinitely), a line segment CAN be measured because it has 2 endpoints. You name a line segment by its endpoints (ex: line segment AB or )
  • Segment Addition Postulate– for example: if M is between A and B, then + = . (little + little = big)
  • Congruent Segments – segments that have the same measure – they are equal. The symbol for congruence is = and it is represented by red slashes on the segments. (ex: )
  1. Find . Assume that the figure is not drawn to scale.

a. b.

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

a.b.

  1. a. Find the value of x and if T is between S and U, = 7x, = 45, and = 5x – 3.

b. Find the value of x and if B is between A and C, = 4x – 12, = x, and = 2x + 3.

4. a. The Arial font is often used because it is easy to read. Study the word time shown in Arial type. Each letter can be broken into individual segments. The letter T has two segments, a short horizontal segment and a longer vertical segment. Assume that all segments overlap where they meet. Which segments are congruent?

TIME

  1. Name the congruent segments in the sign shown on page 17.

*We will learn how to construct a line using a compass here!!!!

Homework – Page 19 – 21 (15 -33 ODD, 34, 37, 40, 43)

Section 3 – Distance and Midpoints

3.1 - I can use the Distance Formula or Pythagorean Theorem to find the length of a segment on the coordinate plane.

3.2 - I can use the Midpoint Formula to find the midpoint of a segment on the coordinate plane.

3.3 - I can find the point on a line segment, given two endpoints, that divides the segment into a given ratio.

  • Distance – the length between 2 points.

Formulas:

On a number line:On a coordinate plane:

  1. Use the number line on page 25 to find the measure of each:

a. b. c.

  1. Find the distance between each pair of points:
  1. E(-5, 6) and F(8, -4) b. J(4, 3) and K(-3, -7)

c.E(-4, 1) and F(3, -1)d. C(-4, -6) and D (5, -1)

  • Midpoint – a point that is halfway between the endpoints of a segment. (ex: if X is the midpoint of AB, then AX = XB)

Formulas:

On a number line:On a coordinate plane:

  1. a. The temperature on a thermometer dropped from a reading of 25 to -8. Find the midpoint of these temperatures.

b. Jacinta hangs a picture 15 inches from the left side of a wall. How far from the edge of the wall should she mark the location for the nail the picture will hang on if the right edge is 37.5 inches from the wall’s left side?

c. Marco places a couch so that its end is perpendicular and 2.5 feet away form the wall. The couch is 90” wide. How far is the midpoint of the couch back from the wall?

  1. Find the coordinates of the midpoint M of each problem:

a. G(8, -6) and H(-14, 12)b. A (5, 12) and B(-4, 8)

c. C (-8, -2) and D (5, 1)d. S (-6, 3) and T (1,0)

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

b. Find the coordinates of G if P is the midpoint. P(-5, 10) and E(-8,6).

6. a. Find the measure of if Q is the midpoint of .

b. Find the measure of if Q is the midpoint of .

c. Find the measure of if Y is the midpoint of and = 2x – 3 and = 27 – 4x.

d. Find the value of x if C is the midpoint of , = 4x + 5, and = 78.

  • Segment Bisector – any segment, line or plane that intersects a segment at its midpoint. On page 29, M is the midpoint of . Plane A, and and point M are all bisectors of .

*We will construct a bisector of a segment here!!!!

Homework – Page 31 – 34 (13, 17, 21, 23, 27, 29, 31, 33, 35, 39, 43, 45, 47, 51, 53 – 57 ALL, and68, 70)

Section 4 – Angle Measure

4.1 - I can precisely define and classify angles and rays and describe their characteristics.

4.2 - I can identify and use congruent angles and the bisector of an angle.

4.3 - I can use a variety of tools to perform the following constructions: I can bisect an angle.

  • Ray – part of a line. It has one endpoint and extends indefinitely in the other direction. They are named by stating the endpoint first and then the other point on the ray. (ex: )
  • Opposite Rays – a line is an example of opposite rays.
  • Angle– formed by 2 noncollinear rays that have a common endpoint. The rays are called SIDES of the angle and the common endpoint is called the VERTEX. You can name an angle in 3 ways:

1. By the vertex (ex: X)

2. By 3 letter where the vertex is the middle letter always (ex: YXZ or ZXY)

3. By a number that is at the vertex of the angle (ex: 3)

-interior points are points that lie inside the angle

-exterior points are points that lie outside the angle.

1.

  1. Name all the angles that have B as a vertex.
  1. Name the sides of 5
  1. Write another name for 6
  • Right angle – an angle that measures exactly 90 degrees. It is denoted by a box at the angle
  • Acute angle - an angle that measures less than 90 degrees.
  • Obtuse angle – an angle that measures more than 90 degrees.
  • Straight angle– an angle that measures exactly 180 degrees – a straight line!

2. Classify each angle as right, acute, or obtuse. Then use a protractor to measure the angle to the nearest degree.

a. TYVb. WYTc. TYU

  • Congruent angles – represented by red arcs of the same number – pg.39
  • Angle Bisector – a ray that divides an angle into 2 congruent angles. (ex: if is an angle bisector of XYZ, then point Q lies in the interior of XYZ and XYW = WYZ.)
  • Angle – Addition Postulate – little + little = big (same as segment addition) (WYZ + XYW + XYZ)

*We will use a ruler/protractor and compass to copy an angle here!!!!

3. Find mGBH and mHCI if GBH HCI, m GBH = 2x + 5 and m HCI = 3x – 10.

4. a. Suppose m JKL = 9y + 15 and m JKN = 5y + 2. Find m JKL. Also know that ray KN bisects JKL.

b. Find the m JKN if m JKN = 8x – 13 and m NKL = 6x + 11.

*We will use a ruler/protractor and compass to construct bisecting an angle!!

Homework – Page 41 – 44 (13-35 ODD, 38, 39, 41, 42, 49, 51, 54, 56)

Section 5 – Angle Relationships

5.1 - I can precisely define perpendicular lines and describe their characteristics.

5.2 - I can precisely define angles, including supplementary, complementary, adjacent, vertical, and linear pairs and describe their characteristics.

5.3 - I can use a variety of tools to perform the following constructions: I can construct perpendicular lines including the perpendicular bisector of a segment.

  • Adjacent angles – 2 angles that lie in the same plane and have a common vertex and a common side, but no common interior points. Pg. 46
  • Linear Pair – a pair of adjacent angles with noncommon sides that are opposite rays. A linear pair is a straight line and measures 180 degrees always. Pg. 46
  • Vertical angles – 2 nonadjacent angles formed by 2 intersecting lines. Vertical angles are always congruent. Pg. 46
  • Complementary angles – 2 angles with measures that have a sum of 90. Complementary angles do no have to be touching, they just have to equal 90 together! Pg. 47
  • Supplementary angles – 2 angles with measures that have a sum of 180. Supplementary angles do not have to be touching, they just have to equal 180 together!! Pg. 47
  • Perpendicular – lines, segments, or rays that form right angles. Intersect to form 4 right angles, form congruent adjacent angles, and the symbol looks like an upside down T. (ex:). Each angle equals 90.
  1. Name an angle pair that satisfies each condition.

a. two angles that form a linear pairb. two acute vertical angles

  1. a. Find the measures of two supplementary angles if the measure of one angle is 6 less than five times the measure of the other angle.

b. Find the measures of two complementary angles if the measure of the larger angle is 12 more than twice the measure of the smaller angle.

  1. a. Find x and y so that KO and HM are perpendicular.

b. Suppose mD = 3x – 12. Find the value of x so that D is a right angle.

c. Find the value of x and y.

  1. Determine whether each statement can be assumed from the figure below. Explain.

a. mVYT = 90b. TYW and TYU are supplementary.

c. VYW and TYS are adjacent angles.

***Construct perpendiculars…Page 55

Homework – Page 51 -53 (8, 9, 12, 13, 14, 15, 17, 19, 20, 21, 23, 26, 27, 29, 30, 31, 33, 34, 35, 49, 52)

Section 6 – Two-Dimensional Figures

6.1 - I can identify and name polygons.

6.2 - I can find perimeter, circumference, and area of two-dimensional figures.

6.3 - I can use coordinate geometry and the distance formula to find the area and perimeters of polygons on the coordinate plane.

  • Polygon – a closed figure formed by a finite number of coplanar segments called sides such that the sides that have a common endpoint are noncollinear and each side intersects exactly two other sides, but only at their endpoints.
  • Concave – some of the lines pass through the interior of a figure.
  • Convex – no points of the lines are in the interior of a figure.
  • Regular – a convex polygon that is both equilateral and equiangular.
  • Irregular – a polygon that is either not convex, not equilateral or not equiangular.
  • Equilateral - a polygon in which all sides are congruent.
  • Equiangular– a polygon in which all angles are congruent.

**Page 57 gives you the list of names for specific polygons – learn this chart!!

***Any figure not on this list is named by its number of sides –gon (ex: 15-gon)

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

  1. Find the perimeter or circumference and area of each figure.

3. Terri 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 of all of the tape?

A. Square with the side length of 5 feet

B. Circle with the radius of 3 feet

C. Right triangle with each leg length of 6 feet

D. Rectangle with a length of 8 feet and a width of 3 feet

4. a. Find the perimeter and area of ABC with vertices A(-1, 4), B(-1, -1) and C(6, -1).

b. Find the perimeter and area of pentagon ABCDE with A(0,4), B(4,0), C(3,-4), D(-3,-4), and E(-3,1).

Homework – Page 61 – 64 (11, 13, 15, 17, 18, 21, 23, 24, 25, 27, 29 (a-d), 31, 25, 37, 40, 44)

Section 7 – Three-Dimensional Figures

7.1 - I can identify and name three-dimensional figures.

  • Polyhedron – a solid with all flat surfaces that enclose a single region of space.
  • Regular Polyhedron – a polyhedron where all the faces are regular congruent polygons and all of the edges are congruent. These are also known as Platonic Solids!
  • Face– the flat surface of a polygon.
  • Edges – the line segments where the faces intersect.
  • Vertex – the point where three or more edges intersect.

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

a.

b.

c.

d.

Homework – Page 71 (6 – 17)ALL