Geometry A Exam Review

1
Geometry A Exam Review

• Noreen Habana
• North Huron Schools

2
Sect. 1.2Points, Lines, Planes

• Undefined terms- points, lines, planes,
  space(p.11)
• Postulate or axiom accepted statement of fact
  (p.12).
• Postulates 1.1-1.4 (p.12-13)

3
Undefined Geometric Terms

• Point - no dimension location.
• Line 1 dimensional series of point going in
  opposite directions infinitely.
• Plane - 2 dimensional flat surface made up of
  points and lines that extend infinitely within
  this surface.
• Space - 3 dimensional set of all points, lines,
  and planes extending in all directions.

3
4
Axiom or Postulate

• A self-evident principle or one that is accepted
  as true without proof as the basis for argument
  a postulate.

5
Postulate 1.1

• Through any two points there is exactly one line.

6
Postulate 1.2

• If two lines intersect, then they intersect in
  exactly one point.

7
Postulate 1.3

• If two planes intersect, then they intersect in
  exactly one line.

8
Postulate 1.4

• Through any three non-collinear points there is
  exactly one plane.

9
Naming Geometric Objects

• Point - by 1 capitalized letter
• Line - either by 1 italicized small cased letter
  or by 2 capitalized letters of endpoints
  connecting to the line
• Plane - either 1 italicized capitalized letter,
  or by capitalized letters for the vertices, or by
  capitalized letters for the points contained in
  it.
• Space - capitalized letters for the vertices of
  the 3dimensional space.

9
10
Expanding the Geometric Terms

• Collinear - a set of points on one line.
• Coplanar - a set of points and lines on one plane.

A? B? C? D?
10
11
Sect.1.3Segment, Rays, Parallel Lines, and Planes

• Lines, segments, and rays (p.17)
• Parallel lines and skew lines (p.18)

12
Line Notations

• Line - extends infinitely in opposite directions
  
• Segment - line with 2 endpoints, so it does not
  extend infinitely

A B
A B
12
1.3
13
Line Notations

• Ray - line with one 1 endpoint, so one part of it
  extends infinitely
• Opposite rays have the same endpoints and extend
  in opposite directions to create a line.
  http//www.mathopenref.com/oppositerays.html

A B
B A C
13
14
Parallel Lines

• Set of coplanar lines that extend in the same
  direction they do not intersect.
• Notation for parallel lines is two slanted lines
  //. Ex.

14
15
Skew Lines

• Skew Lines set of lines that do not intersect.
  They are also non-coplanar so they are not
  parallel. (p.18)

15
16
Parallel and Skew

• Parallel planes opposite faces of prisms.
• There is no such thing as skew planes because if
  theyre extended, they either intersect or prove
  to be parallel.

16
17
Sect.1.4Measuring Segments and Angles

• Ruler Postulate (p.25)
• Congruent vs. equal (p.25)
• Segment addition postulate (p.26)
• Definition of Midpoint (pp.26-27)
• Types of angles (pp.27-28)
• Angle addition postulate (p.28,29)

18
Ruler postulate

• AB ?a - b ?, where a is the ruler coordinate of
  point A, and b is the ruler coordinate of point
  B.
• Because distance is absolute value, it is only
  positive and the order of the coordinates
  subtracted does not matter.

18
19
Congruent or Equal

• Congruent compares the physical aspect of
  geometric figures. The notation is ?. Example
• Equal compares the measured values of geometric
  figures. The notation is .
• Example

19
20
Segment Addition Postulate

• If three points A, B, and C are collinear and B
  is between A and C, then AB BC AC.
• AB BC AC
• (2x 8) (3x -12) AC
• 5x 20 AC

2x -8 3x - 12
A B C
20
21
Definition of a Midpoint

• A midpoint of a segment is a point that divides a
  segment into two congruent segments. A midpoint,
  or any line, ray, or other segment through a
  midpoint bisects the segment.
• Ex. If C is the midpoint of , then

21
22
Definition of bisector

• Bisect to divide into two congruent figures.
• A bisector is any geometric figure which divide
  another geometric figure into two congruent
  figures.

22
23
Angle

• Definition Two rays with the same endpoint.
• Symbol ?
• Notation With a single number (?1) with a
  single capital letter for its vertex (?B) if it
  doesnt share its vertex with another angle with
  3 capital letters for its endpoints and vertex if
  another angle shares it vertex (?ABC or ?CBA).

24
Angles

• m? refers to the measure of an angle with
• a unit of degrees (?), while ? refers to its
  physical attributes.
• Example ?A ? ?B
• If both ?A and ?B have the same measure, then
• m ?A m?B

25
Protractor postulate

• If an endpoint of an angle corresponds with 135?
  and its other endpoint corresponds with 86, then

26
Angle Addition Postulate

• Two or more angles are put together to create a
  larger angle.
• m?AOB m ?BOC m ?AOC
• 88 30 118

A
B
88º
30º
O
C
27
Definition of Linear Pair

• 2 angles add up to 180 or create a straight
  angle.
• m?AOB m ?BOC 180

B
150º
30º
O
C
A
28
Angle Bisector

• An angle bisector is a ray which divides an angle
  into two smaller but congruent angles.
• m?AOB m ?BOC
• ?AOB ? ?BOC
• Therefore,
• Ray OB is a bisector.

29
Sect.1.6 The Coordinate Plane

• Distance formula (p.43)
• Midpoint formula (p.45)

30
Notes Sect.1.6 Coordinate Plane

• A number line is 1-dimensional and takes only
  into account the length between 2 points.
• A coordinate plane is 2-dimensional. The
  distance between 2 points must take into
  consideration 2 variables, x (left-right) and y
  (forward-back).

31
Distance formula

• Given two endpoints
• and ,

32
Midpoint Formula

• The midpoint formula does NOT determine the
  length or distance from one endpoint to the
  midpoint.
• The midpoint formula gives the coordinate (x,y)
  of the midpoint between two endpoints.

33
Midpoint formula

• If the midpoint and one endpoint is given, find
  the other midpoint separating the coordinates.

34
Sect.1.7Perimeter, Circumference, and Area

• Perimeter of square and rectangle (p.52)
• Area of square and rectangle (p.52)
• Circumference and area of circle (p.52)

35
Perimeter and Area

• Square (having equal sides)
• P 4s, where s is the length of a side.
• A s²
• All 90 angles
• Rectangle (Opposite sides are equal)
• P 2L 2W, where L length and Wwidth
• A LW
• All 90 angles

36
Circumference and Area

• Circle (radius is equal throughout)
• C 2?r, where r radius, ??3.14
• A ?r²
• 2r d, where d diameter
• C ?d

37
Sect.3.1Properties of Parallel Lines

• Angles created by a transversal and coplanar
  lines (p.115)
• Corresponding angles, AIA, SSIA Theorems (p.115)

38
Transversal Line

• Line that intersects two or more coplanar lines
  at two or more distinct points. Line t is a
  transversal.

t
a
b
39
Alternate Interior Angles (AIA)

• Angles inside two coplanar lines and at opposite
  sides of the transversal. Angles 1 and 2 and
  angles 3 and 4 are 2 pairs of AIA.

40
Same-Side Interior Angles (SSIA)

• Angles inside two coplanar lines and on the same
  side of the transversal. Angles 1 and 4, and
  angles 3 and 2 are two pairs of SSIA.

t
a
1 3
b
4 2
41
Corresponding Angles

• Angles on the same side of the transversal, but
  skip an angle. Angles 1 and 7 are corresponding
  angles.

42
Properties of Parallel Lines

• If the coplanar lines intersected by a
  transversal are parallel ( ),
• Alternate interior angles are congruent
• ( ).
• Corresponding angles are congruent
• ( ).
• Same-side interior angles are supplementary
• ( ).

43
Vertical angles theorem

• When two lines intersect, they create two pairs
  of vertical angles, which are diagonally across
  from each other.
• vertical angles are congruent.
• Angles 1 and 4 are vertical angles,
• and angles 2 and 3 are vertical angles.

44
Sect.3.3Parallel Lines and the Triangle
Angle-Sum Theorem

• Triangle Angle Sum Theorem (p.131)
• Types of triangles (p.133)
• Triangle Exterior Angle Theorem (p.133)

45
Triangle Angle-Sum Theorem

• The sum of the measures of all the angles of a
  triangle is 180.
  

46
TYPES OF TRIANGLES

• Triangle any polygon having 3 sides and 3
  interior angles. (p.133)
• Named by angles
• Equiangular 3 equal angles
• Acute 3 acute angles
• Right 1 right angle
• Obtuse 1 obtuse angle

47
Types of Triangles

• Named by sides
• Equilateral 3 equal sides
• Isosceles 2 equal sides
• Scalene no equal sides

48
Triangle Exterior Angle Theorem

• The measure of each exterior angle of a triangle
  equals the sum of the measures of its two remote
  interior angles.

49
Sect.3.4The Polygon Angle-Sum Theorems

• Definition of a polygon (p.143)
• Convex vs. concave polygons (p.144)
• Types of polygons (p.144)
• Polygon Angle-Sum Theorem (p.145)
• Polygon Exterior Angle Sum Theorem (p.146)
• Definition of regular polygon (p.146)

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Definition of a polygon

• Polygon any closed 2-dimensional figure made of
  sides connecting at corners or vertices.
• open closed

51
Convex and concave polygons

• Convex Diagonals between 2 nonconsecutive
  points are within the polygon.
• Concave Contains a diagonal between 2
  nonconsecutive points which lies outside the
  polygon so it appears as if a part of the polygon
  is caving in.

52
Polygon Angle-Sum Theorem

• The sum of the measures of the angles of an n-gon
  (any number side of polygon) is
• (n-2)180 the sum of interior angles of any
  polygon.

53
Polygon Exterior Angle-Sum Theorem

• The sum of the exterior angles of any polygon is
  360.
• For a regular polygon (having equal sides and
  angle measure)

54
Sect.5.1Midsegments of Triangles

• Midsegments (pp.243-244)

55
Midsegments of a Triangle

• A midsegment of a triangle is a segment extend
  from a midpoint of a triangles side to a
  midpoint of another side of the same triangle.
• There are 3 midsegments in a triangle.

E
B
A
F
D
56
Midsegment Theorem

• Triangle Midsegment Theorem the midsegment of a
  triangle is parallel to the third side, and is ½
  its length.
• AB ½ EF
• AC ½ DF
• BC ½ AE

D
B
A
E
F
C
57
Midsegment of a Triangle

• Since a midsegment of a triangle is parallel to
  the 3rd line, properties of parallel lines apply.
  
• SSIA are supplemental
• AIA are congruent
• Corr. angles are congruent

D
B
1
2
4
3
F
E
C
58
Sect.5.2Bisectors in Triangles

• Perpendicular Bisector Theorem (p.249)
• Angle Bisector Theorem (p.250)

59
Perpendicular Bisector

• A segment, ray, or line which intersects the
  midpoint of another segment at 90º.

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Perpendicular Bisector Theorem

• Perpendicular Bisector Thm. a point on a
  perpendicular bisector of a segment is
  equidistant to the endpoints of the segment.
• Converse of the Perpendicular Bisector Thm. If
  the endpoints of a segment is equidistant to a
  point on a line, then the point is on a
  perpendicular bisector.

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Perpendicular Bisector Thm
62
Angle Bisector Theorem

• Angle Bisector Thm. If a point is on the
  bisector of an angle, then the point is
  equidistant from the sides of the angle.
• Converse of the Angle Bisector Thm.- If a point
  in the interior of an angle is equidistant from
  the sides of the angle, then the point is on the
  angle bisector.

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Angle Bisector Thm
64
Sect.5.3Concurrent Lines, Medians, and Altitudes

• Definition of concurrency (p.257)
• Circumscribe a triangle (p.257)
• Inscribe in the triangle (p.257)
• Median of a triangle (p.258)
• Altitude of a triangle (p.259)

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Concurrent Lines

• When 3 or more lines intersect at one point.
• The point of intersection is called the point of
  concurrency.

66
Concurrent Lines

• For any triangle, 4 different sets of
• lines are concurrent.

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Concurrent Lines of perpendicular bisectors

• The perpendicular bisectors of the sides of a
  triangle are concurrent at a point equidistant
  from the vertices.

68
Concurrent Lines of angle bisectors

• The angle bisectors of a triangle are concurrent
  at a point equidistant from the sides.

69
Circumscribe about a triangle

• The point of concurrency is also the circumcenter
  of the triangle.

70
Circumscribe about a polygon

• Since the distance between the point of
  concurrency of perpendicular bisectors and each
  vertices of a triangle are equal, a circle can be
  constructed around the triangle.
• The center of the circle is the point of
  concurrency.
• The radius length is the distance from the point
  of concurrency to any vertices.

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Circumscribe a triangle

• http//www.mathopenref.com/constcircumcircle.html

72
Inscribe in the triangle

• The point of concurrency is also the incenter of
  the triangle.

73
Inscribe in a polygon

• Since the distance between the point of
  concurrency of the angle bisectors and each
  midpoint of a side are equal, a circle can be
  constructed or inscribe within a polygon.
• The center is the point of concurrency.
• The radius length is the distance between the
  point of concurrency and each midpoint of a side
  of polygon.

74
Inscribe in a triangle

• http//www.mathopenref.com/constincircle.html

75
Median of a Triangle

• A segment whose endpoints are a vertex and the
  midpoint of an opposite side of a triangle.

A
Segment AD is the median of triangle ABC.
B
D
C
76
Medians of a Triangle Theorem

• The medians of a triangle are concurrent at a
  point that is 2/3 the distance from each vertex
  to the midpoint of the opposite side.

DC 2/3 DJ EC 2/3 EG FC 2/3 FH
77
Medians of a Triangle

• The point of concurrency of all medians in a
  triangle is called the centroid.
• It is also known as the center of gravity of a
  triangle because it is the point where a
  triangular shape will balance.

78
Altitude of a Triangle

• An altitude of a triangle is the perpendicular
  segment from a vertex to the line containing the
  opposite side.
• It can be inside, outside, or on a side of a
  triangle.

79
Orthocenter of a Triangle

• The point of intersection of the altitudes of a
  triangle
• In the figure, AD, BE, and CF are the altitudes
  drawn from the vertices A, B, and C respectively.
  The point of intersection of these altitudes is
  H. So, H is the orthocenter of triangle ABC.

80
Orthocenter of an Obtuse Triangle

• Orthocenter of an obtuse triangle
• lies outside the triangle.

81
Orthocenter of an Acute Triangle

• Orthocenter of an acute triangle lies inside the
  triangle.

82
Orthocenter of a Right Triangle

• Orthocenter of a right triangle lies on the
  triangle.

83
Sect.7.2The Pythagorean Theorem and its Converse

• Pythagorean Theorem (p.357)
• Pythagorean triple (p.357)
• Converse of Pythagorean Theorem (pp.359-360)

84
Pythagorean Theorem

• In a right triangle, a² b² c², where a and b
  are the lengths of the legs of the triangle and c
  is the length of the its hypotenuse.

85
Pythagorean Triple

• A Pythagorean triple is a set of nonzero whole
  numbers a,b, and c that satisfy the equation a²
  b² c².
• Example 3,4,5
• 3² 4² 5² 9 16 25
• Nonexample 2,3,4
• 2² 3² ? 4² 4 9 ? 16
• 2² 3² 13 0R

86
Converse of Pythagorean Theorem

• A) If a² b² c², then triangle is a right
  triangle.
• B) If a² b² lt c², then triangle is an obtuse
  triangle.
• C) If a² b² gt c², then triangle is an acute
  triangle.

87
Sect.7.3Special Right Triangles

• 45-45-90 Triangle Theorem (pp.366-367)
• 30-60-90 Triangle Theorem (pp.367-368)

88
Sect. 7.3 Special Right Triangles

• Draw an isosceles right triangle.
• Label the legs, x.
• Label the hypotenuse, y.
• Use Pythagorean Theorem to find the relationship
  between legs and hypotenuse of an isosceles right
  triangle (45?-45?-90?).

89
45?-45?-90? Triangle Theorem

• In a 45?-45?-90? triangle, the legs are congruent
  to each other, and the hypotenuse is times
  the length of a leg.
• Hypotenuse leg

90
30?-60?-90? Triangle Theorem

• In a 30?-60?-90? triangle,
• Hypotenuse 2 short leg
• Long leg short leg x

30?
2s
s
60?
s