Why study Chapter 4?

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Why study Chapter 4?

• Roads and Bridges
• Lighting
• Arrangement of Sports Equipment
• Biology
• Kiteboarding
• Rates of Speed
• Amusement Park Rides
• Placement of Ladders
• Art

2

• Crossing Streams
• Picnic tables
• Platforms
• Marketing
• Archaeology
• Distribution
• Engineering

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Section 4.1More Proofs and Midpoints
F
Given
B
B
A
C
D
E
Prove
G

• Detour Proof A proof involving more than one
  pair of triangles. A two part proof, just like
  youve been doing!

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Section 4.1

• Procedure for Detour Proofs
• Determine which triangles you must prove to be
  congruent to reach the required conclusion
• Attempt to prove these triangles congruent, if
  you dont have enough info, proceed to step 3.
• Identify the parts of the triangle that you are
  missing.
• Find another pair of triangles that
• You can prove congruent
• Contain the pair of parts needed
• Prove the triangles in Step 4 congruent
• Use CPCTC and prove the triangles in Step 1
  congruent

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Section 4.1Detours and Midpoints
Midpoint formula!! M (xm, ym) ?
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Section 4.2Missing Diagrams

• if .givens, then.prove
• 1. If two altitudes of a triangle are
  congruent,
• then the triangle is isosceles
• 2. The medians of a triangle are
  congruent if
• the triangle is equilateral
• No if, then
• Givens usually in the beginning of the
  statement
• 1. The altitude to the base of an isosceles
  
• triangle bisects the vertex angle

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Section 4.3A Right-Angle Theorem

• Theorem If two angles are both supplementary
  and congruent, then they are right angles

D
2
1
A
B
C
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Section 4.4The Equidistance Theorems

• Distance The distance between two objects is
  the length of the shortest path between them.
• Postulate A line segment is the shortest path
  between two points.

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Section 4.4The Equidistance Theorems

• Perpendicular Bisector the line that bisects
  and is perpendicular to a segment.
• Theorem If 2 points are each equidistant from
  the endpoints of a segment, then the two points
  determine the perpendicular bisector of the
  segment.
• Theorem If a point is on the perpendicular
  bisector of a segment, then it is equidistant
  from the endpoints of that segment

A
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Section 4.5Intro to Parallel Lines

• Plane A surface such that if any 2 points on
  the surface are connected by a line, all points
  of the line are also on the surface
• Coplanar If points, lines, segments, and so
  forth, lie in the same plane.
• Noncoplanar If points, lines, segments do NOT
  lie on the same plane

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Section 4.5Intro to Parallel Lines

• Transversal A line that intersects two coplanar
  lines in two distinct points.

Interior Region
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Section 4.5Intro to Parallel Lines

• Alternate Interior angles A pair of angles
  formed by two lines in a transversal. The
  angles must
• Lie in the interior of the figure
• Lie on alternate sides of the transversal
• Have different vertices

1 2 3 4
5 6 7 8
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Section 4.5Intro to Parallel Lines

• Alternate Exterior angles A pair of angles
  formed by two lines in a transversal. The
  angles must
• Lie in the exterior of the figure
• Lie on alternate sides of the transversal
• Have different vertices

1 2 3 4
5 6 7 8
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Section 4.5Intro to Parallel Lines

• Corresponding angles A pair of angles formed by
  two lines in a transversal. The angles must
• One angle must lie on the interior and the other
  on the exterior
• Lie on same sides of the transversal
• Have different vertices

1 2 3 4
5 6 7 8
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Section 4.5Intro to Parallel Lines
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Section 4.5Intro to Parallel Lines

• Parallel Lines Two coplanar lines that do not
  intersect

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Section 4.6Slope

• Slope The slope m of a nonvertical line,
  segment, or ray containing and
  is defined by the formula

• What is the slope of a horizontal line?
• What is the slope of a vertical line?

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Section 4.6Slope
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Section 4.6Slope

• 4 More Theorems!!!!
• If two nonvertical lines are parallel, then their
  slopes are equal.
• If the slopes of two nonvertical lines are equal,
  then the lines are parallel (the converse)
• If two lines are perpendicular and neither is
  vertical, each lines slope is the opposite
  reciprocal of the others.
• If a lines slope is the opposite reciprocal of
  another lines slope, then two lines are
  perpendicular (the converse)