1. 1.6 Motion in Geometry
Objective
Identify and draw the three basic rigid transformations: translation, rotation, and reflection.

2. Glossary Terms image preimage reflection rigid transformation rotation
1.6 Motion in Geometry
Glossary Terms
image
preimage
reflection
rigid transformation
rotation
translation

3. Key Skills Identify translations, rotations, and reflections.
1.6 Motion in Geometry
Key Skills
Identify translations, rotations, and
reflections.

4. Key Skills Translate a figure along a line. B A C B’ A’ C’
1.6 Motion in Geometry
Key Skills
Translate a figure along a line. B A C B’ A’ C’

5. 1.6 Motion in Geometry
Key Skills
Rotate a figure about a point.

6. 1.6 Motion in Geometry
Key Skills
Reflect a figure across a line.
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7. 1.7 Motion in the Coordinate Plane
Objectives
Review the algebraic concepts of coordinate plane, origin, x- and y-coordinates, and ordered pair.
Construct translations, reflections across axes, and rotations about the origin on a coordinate plane.

8. Theorems, Postulates, & Definitions
1.7 Motion in the Coordinate Plane
Theorems, Postulates, & Definitions
Horizontal and Vertical Coordinate
Translations
Horizontal translation of h units:
H(x, y) = (x + h, y)
Vertical translation of v units:
H(x, y) = (x, y + v)

9. Theorems, Postulates, & Definitions
1.7 Motion in the Coordinate Plane
Theorems, Postulates, & Definitions
Reflections Across the x- or y-Axes
Reflection across the x-axis:
M(x, y) = (x, –y)
Reflection across the y-axis:
N(x, y) = (–x, y)
180° Rotation About the Origin
R(x, y) = (–x, –y)

10. Key Skills Use the coordinate plane to transform geometric figures.
1.7 Motions in the Coordinate Plane
Key Skills
Use the coordinate plane to transform
geometric figures.
x +6
To translate the triangle 6 units right, use the rule
H(x, y) = (x + 6, y).
x +6
x +6

11. Key Skills Use the coordinate plane to transform geometric figures.
1.7 Motions in the Coordinate Plane
Key Skills
Use the coordinate plane to transform
geometric figures. y
To reflect the triangle across the x-axis,
use the rule
M(x, y) = (x, –y). y –y –y
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12. Key Skills Write a definition of an object. These objects are
2.3 Definitions
Key Skills
Write a definition of an object.
These objects are
rhombuses.
These objects are
not rhombuses.

13. Key Skills a. b. c. a d d. Which of these objects are rhombuses?
2.3 Definitions
Key Skills a. b. c. a d d.
Which of these objects are rhombuses?
Write a definition of a rhombus.
Objects a and d are rhombuses.
Definition: A rhombus is a four-sided figure
in which all four sides have the same length.

14. Objectives Define polygon.
3.1 Symmetry in Polygons
Objectives
Define polygon.
Define and use reflectional symmetry and rotational symmetry.
Define regular polygon, center of a regular polygon, central angle of a regular polygon, and axis of symmetry.

15. Glossary Terms axis of symmetry center of a regular polygon
3.1 Symmetry in Polygons
Glossary Terms
axis of symmetry
center of a regular polygon
central angle of a regular polygon
equiangular polygon
equilateral polygon
polygon
reflectional symmetry
regular polygon
rotational symmetry

16. Theorems, Postulates, & Definitions
3.1 Symmetry in Polygons
Theorems, Postulates, & Definitions
Triangles Classified by Number of Congruent Sides
Three congruent sides
equilateral
At least two congruent sides
isosceles
No congruent sides
scalene

17. Theorems, Postulates, & Definitions
3.1 Symmetry in Polygons
Theorems, Postulates, & Definitions
Reflectional Symmetry: A figure has reflectional symmetry if and only if its reflected image across a line coincides exactly with the preimage. The line is called an axis of symmetry.
Rotational Symmetry: A figure has rotational symmetry if and only if it has at least one rotation image, not counting rotation images
of 0° or multiples of 360°, that coincides with
the original image.

18. Key Skills Identify reflectional symmetry of figures.
3.1 Symmetry in Polygons
Key Skills
Identify reflectional symmetry of figures.
Draw all the axes
of symmetry of
this figure.

19. Key Skills Identify rotational symmetry of figures.
3.1 Symmetry in Polygons
Key Skills
Identify rotational symmetry of figures.
The figure has 4-fold
rotational symmetry.
The image will coincide
with the original figure
after rotations of
90°, 180°, 270° and 360°.
At 360°, the figure returns to its original position.
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20. 3.3 Parallel Lines and Transversals
Objectives
Define transversal, alternate interior angles, alternate exterior angles, same-side interior angles, and corresponding angles.
Make conjectures and prove theorems by using postulates and properties of parallel lines and transversals.

21. Glossary Terms alternate exterior angles alternate interior angles
3.3 Parallel Lines and Transversals
Glossary Terms
alternate exterior angles
alternate interior angles
corresponding angles
same-side interior angles
transversal

22. Theorems, Postulates, & Definitions
3.3 Parallel Lines and Transversals
Theorems, Postulates, & Definitions
Corresponding Angles Postulate 3.3.2: If two lines cut by a transversal are parallel, then corresponding angles are congruent.
corresponding angles
2  3

23. Theorems, Postulates, & Definitions
3.3 Parallel Lines and Transversals
Theorems, Postulates, & Definitions
Alternate Interior Angles Theorem 3.3.3:
If two lines cut by a transversal are parallel, then
alternate interior angles are congruent.
alternate interior angles
1  3

24. Theorems, Postulates, & Definitions
3.3 Parallel Lines and Transversals
Theorems, Postulates, & Definitions
Alternate Exterior Angles Theorem 3.3.4:
If two lines cut by a transversal are parallel,
then alternate exterior angles are congruent.
alternate exterior angles
2  5

25. Theorems, Postulates, & Definitions
3.3 Parallel Lines and Transversals
Theorems, Postulates, & Definitions
Same-Side Interior Angles Theorem 3.3.5:
If two lines cut by a transversal are parallel, then
same-side interior angles are supplementary.
same-side interior angles
1 + 4 = 180

26. Key Skills Identify special pairs of angles. Corresponding angles
3.3 Parallel Lines and Transversals
Key Skills
Identify special pairs of angles.
Corresponding angles
1 and 5
1 and 3
Alternate interior angles
Same-side interior angles
1 and 4
Alternate exterior angles
2 and 5

27. Key Skills Find angle measures formed by parallel lines
3.3 Parallel Lines and Transversals
Key Skills
Find angle measures formed by parallel lines
and transversals.
m || n and m1 = 135°.
Then m2 = m3 = m5 = 135° and m4 = 45°.
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28. 3.6 Angles in Polygons
Objective
Develop and use formulas for the sums of the measures of interior and exterior angles of a polygon.

29. 3.6 Angles in Polygons
Glossary Terms
concave polygon
convex polygon

30. Theorems, Postulates, & Definitions
3.6 Angles in Polygons
Theorems, Postulates, & Definitions
Sum of the Interior Angles of a Polygon 3.6.1: The sum, s, of the measures of the interior angles of a polygon with n sides is given by
s = (n – 2)180.
The Measure of an Interior Angle of a Regular
Polygon 3.6.2: The measure, m, of an interior
angle of a regular polygon with n sides is given by
m = 180 
360 n .

31. Theorems, Postulates, & Definitions
3.6 Angles in Polygons
Theorems, Postulates, & Definitions
Sum of the Exterior Angles of a Polygon
3.6.3: The sum of the measures of the exterior
angles of a polygon is 360.

32. Key Skills Find interior and exterior angle measures of polygons.
3.6 Angles in Polygons
Key Skills
Find interior and exterior angle measures of
polygons.
Find the sum of the measures of the interior angles and of the exterior angles of a polygon
with 15 sides.
Sum of the measures of the interior angles:
(15 – 2)180 = 2340
Sum of the measures of the exterior angles:
360

33. Key Skills Find interior and exterior angle measures of polygons.
3.6 Angles in Polygons
Key Skills
Find interior and exterior angle measures of
polygons.
Find the measure, m, of each interior angle of a regular polygon with 15 sides.
m = 180 
360 n
360 15
= 180 
= 156
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34. 3.7 Midsegments of Triangles and Trapezoids
Objectives
Define midsegment of a triangle and midsegment of a trapezoid.
Develop and use formulas based on the properties of triangle and trapezoid midsegments.

35. Glossary Terms midsegment of a trapezoid midsegment of a triangle
3.7 Midsegments of Triangles and Trapezoids
Glossary Terms
midsegment of a trapezoid
midsegment of a triangle

36. Theorems, Postulates, & Definitions
3.7 Midsegments of Triangles and Trapezoids
Theorems, Postulates, & Definitions
Midsegment of a Triangle: A midsegment of
a triangle is a segment whose endpoints are the midpoints of two sides.
Midsegment of a Trapezoid: A midsegment of
a trapezoid is a segment whose endpoints are
the midpoints of the nonparallel sides.

37. Key Skills Solve problems by using triangle and trapezoid midsegments.
3.7 Midsegments of Triangles and Trapezoids
Key Skills
Solve problems by using triangle and
trapezoid midsegments.
Find DE and FG.
DE is a midsegment
of ABC, so
DE = 80 2
= 40 meters
FG is a midsegment of trapezoid ABED, so
FG = 2
= 60 meters
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38. Theorems, Postulates, & Definitions
3.8 Analyzing Polygons With Coordinates
Theorems, Postulates, & Definitions
Definition of Slope: The slope of a nonvertical
line that contains the points (x1, y1) and (x2, y2)
is equal to the ratio:
y2  y1
x2  x1 .
Parallel Lines Theorem 3.8.2: Two nonvertical
lines are parallel if and only if they have the
same slope. Any two vertical lines are parallel.

39. Theorems, Postulates, & Definitions
3.8 Analyzing Polygons With Coordinates
Theorems, Postulates, & Definitions
Perpendicular Lines Theorem 3.8.3: Two non-
vertical lines are perpendicular if and only if
the product of their slopes is –1. Any vertical line is perpendicular to any horizontal line.
Midpoint Formula: The midpoint of a segment
with endpoints (x1, y1) and (x2, y2) has the
following coordinates:
x1 + x2 2 ,
y1 + y2 .

40. Key Skills Use slope to determine whether lines and
3.8 Analyzing Polygons With Coordinates
Key Skills
Use slope to determine whether lines and
segments are parallel or perpendicular.
A quadrilateral has vertices A(2, 3), B(6, 4),
C(8, 7), and D(4, 6). Is ABCD a parallelogram?
and
are parallel (same slope: ). 1 4 CD AB
and
are parallel (same slope: ). BC DA 3 2
ABCD is a parallelogram (opposite sides of
quadrilateral ABCD are parallel).
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