Name: Date:
Chapter 1: Foundations of Geometry
Essential Understandings: The characteristics and properties of 2 and 3 dimensional geometric shapes can be analyzed to develop mathematical arguments about geometric relationships.
Essential Question: How do algebraic concepts relate to geometric concepts? How do patterns and functions help us represent data and solve real-world problems?
Section 1: Understanding Points, Lines & PlanesObjectives of Lesson: Identify, name, and draw points, lines, segments, rays, and planes. Apply basic facts about points, lines, and planes.
Undefined Terms: ______________________________________________________
________________________________________________________________________
Undefined TermsTerm Name Diagram
A point names a location and has no size. It is represented by a dot.A line is a straight path that has no thickness and extends forever.
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A plane is a flat surface that has no thickness and extends forever.
Collinear: ______________________________________________________
______________________________________________________
Coplanar: ______________________________________________________
______________________________________________________
Examples: Using diagram below1) Name four coplanar points.
2) Name three lines.
3) Name two possible planes.
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Segments and RaysDefinition Name Diagram
Segment:
Endpoint:
Ray:
Opposite Rays:
Examples: Draw and label each of the following. 1) A segment with endpoints U and V.
2) Opposite rays with a common endpoint Q.
3) Draw and label a ray with endpoint M that contains N.
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Postulate: ______________________________________________________
______________________________________________________
Postulates: Points, Lines & Planes1-1-1 Through any two points, there is
exactly one line. 1-1-2 Through any three non-collinear
points there is exactly one plane containing them.
1-1-3 If two points lie in a plane, then the line containing those points lies in the plane.
Examples: Use diagram to answer questions. 1) Name a line that passes through two points.
2) Name a plane that contains three non-collinear points.
Postulates: Intersection of Lines and Planes1-1-4 If two lines intersect, then they intersect in exactly one
point. 1-1-5 If two planes intersect, then they intersect in exactly one
line.
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Homework: Page 9-10: # 1-12 & 25-27
Section 2: Measuring and Constructing Segments
Objectives of Lesson: Use length and midpoint of a segment. Construct midpoints and congruent segments.
Coordinate: ______________________________________________________
______________________________________________________
Distance: ______________________________________________________
______________________________________________________
Length: ______________________________________________________
______________________________________________________
Examples: Find the length of each segment.
1) DC
2) EF
3) FC
4) DE
Congruent Segments: ____________________________________________________________________________________________________________
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Construction: ____________________________________________________________________________________________________________
Examples: Construct a Congruent Segment to AB, call it CD.
A B
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Between: ____________________________________________________________________________________________________________
Postulate 1-2-2 (Segment Addition Postulate)
Examples: Use the segment addition postulate to solve the following problems.
1) B is between A and C, AC = 14 and BC = 11.4. Find AB.
2) S is between R and T. Find RT.
3) Y is between X and Z, XZ = 3, and XY = 1 . Find YZ.
4) E is between D and F. Find DF.
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Midpoint: ____________________________________________________________________________________________________________
Bisect: ____________________________________________________
5) The map shows the route for a race. You are 365m from drink station R and 2km from drink station S. The first-aid station is located at the midpoint of the two drink stations. How far are you from the first-aid station? What is the distance to a drink station located at the midpoint between your current location and the first aid station?
6) B is the midpoint of , AB = 5x, and BC = 3x + 4. Find AB,
BC and AC.
Segment Bisector: ______________________________________________________________________________________________________
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Construction of a Segment BisectorSTEPS
1) Draw line segment if not already given2) Put tip on end point and extend leg past midpoint3) Make arc above and below line segment 4) Repeat steps 2 & from other endpoint5) Draw line through intercepted arcs 6) You now know that:
A B
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Homework: Practice B (1.2)
Section 3: Measuring and Construction Angles
Objectives of Lesson: Name and classify angles. Measure and construct angles and angle bisectors.
Angle: ____________________________________________________________________________________________________________
Vertex: ____________________________________________________________________________________________________________
Examples: Name the following angles
Measure: ____________________________________________________________________________________________________________
Degree: ____________________________________________________________________________________________________________
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Using a Protractor
Types of Angles
Examples: Measuring the following angles1) <AOD
2) <COD
3) <BOA
4) <DOB
5) <EOC
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Congruent Angles: ____________________________________________________________________________________________________________
Practice Constructing Congruent Angles
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Postulate 1-3-2 (Angle Addition Postulate)
Examples: Use angle addition postulate to answer the following questions
1) m<ABD = 37° and m<ABC = 84°. Find m<DBC.
2) m<XWZ = 121° and m<XWY = 59°. Find m<YWZ.
Angle Bisector: ____________________________________________________________________________________________________________
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Practice Constructing Angle Bisectors
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Examples: Find the measure of the following angles1) bisects <ABC, m<ABD = (6x + 3)°, and m<DBC = (8x – 7)°.
Find m<ABD.
2) bisects <PQR, m<PQS = (5y – 1)°, and m<PQR = (8y + 12)°. Find m<PQS.
3) bisects <LJM, m<LJK = (-10x + 3)°, and m<KJM = (-x+ 21)°. Find m<LJM.
Section 4: Pairs of Angles
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Homework: Page 24-25: #4-10; #17-22 & #29-32
Objectives of Lesson: Identify adjacent, vertical, complementary, and supplementary angles. Find measures of pairs of angles.
Pair of AnglesAdjacent Angles:
Linear Pair:
Examples: Tell whether adjacent angles or linear pairs1) < 1 and < 2
2) < 2 and < 4
3) < 1 and < 3
4) < 5 and < 6
5) < 7 and < SPU
6) < 7 and < 8
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Complementary Angles: ____________________________________________________________________________________________________________
Supplementary Angles: ____________________________________________________________________________________________________________Examples: Find the measure of each of the following.
1) Complement of <M.
2) Supplement of <N.
3) Complement of <E.
4) Supplement of <F.
5) An angle is 3 less than twice the measure of its complement. Find the measure of its complement.
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6) An angle’s measure is 12° more than ½ the measure of its supplement. Find the measure of the angle.
7) Light passing through a fiber optic cable reflects off the walls in such a way that <1 <2. <1 and <3 are complementary, and <2 and <4 are complementary. If m<1 = 38°, find m<2, m<3, and m<4.
Vertical Angles: ______________________________________________________________________________________________________
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Homework: Page 31-32: # 14-22 & 26-31