Download Chapter 11 Notes – Congruence, Similarity, and Transformations

Chapter 11 Notes – Congruence, Similarity, and Transformations –
PART 1 – Angles and Polygons
Lesson 11-1: Angle and Line Relationships
Line and Angle Relationships
Names of Special Angles
Interior angles lie inside the parallel lines.
∠3, ∠4, ∠5, ∠6
Exterior angles lie outside the parallel lines.
∠1, ∠2, ∠7, ∠8
Alternate interior angles are on opposite sides
of the transversal and inside the parallel lines.
Alternate exterior angles are on opposite sides
of the transversal and outside the parallel lines.
Corresponding angles are in the same position
on the parallel lines in relation to the transversal.
∠3 and ∠5, ∠4 and ∠6
∠1 and ∠7, ∠2 and ∠8
∠1 and ∠5, ∠2 and ∠6,
∠3 and ∠7, ∠4 and ∠8
Try It
Jun is cutting the tile below as shown.
a) Classify the relationship between
∠ a and ∠ b.
∠ 13.
b) If m∠ a = 53o, what is the measure of ∠ b?
b
a
**See Example 2 on Page 496
a) Classify the relationship
between ∠ 9 and
b) If m∠ 13 is 75o, find m∠ 11
and m∠ 15.
Angles DEF and WXY are complementary angles, with m∠ DEF = 2xo and
m∠ WXY = (3x – 20)o. Find the value of x, and then find m∠ WXY.
Lesson 11-2: Triangles
**SUM of Interior Angles of ANY Triangle is ___________
Try It
Find m∠ F in △DEF
The measures of the angles of △XYZ are in
the ratio of 2:3:5. What are the measures of
D
E
the angles?
F
Lesson 11-2: Triangles (Continued)
** Exterior Angles: are made by _________________ one side of the triangle. The exterior
angle is _____________________________ to its adjacent interior angle.
**To Find Exterior Angles: __________ the two _________________ angles it isn’t touching (REMOTE)
Try It
K
L
5
2
In the figure at left, m∠ 4 = 135o. Find m∠ 2.
1
4
M
**Triangles can be classified using their _____________________ and _____________________.
By Angles
By Sides
Acute Scalene –
Obtuse -
Isosceles –
Right -
Equilateral –
Try It
Classify each triangle by its angles and its sides.
106o
45o
45o
38o
36o
Lesson 11-3: Polygons
**POLYGON - Simple, ____________________ figure
3 or more line ____________________
CANNOT have a ________________________ or ________________________ sides
Try It
Determine whether the figure is a polygon. If it is, classify the polygon. If it is not a
polygon, explain why.
**Sum of Interior Angles of a Polygon: _________________, where n is _________________________
- Found by counting the triangles formed by the diagonals from one vertex
Example:
**To Find Degrees of ONE Interior Angle: Find the Total Degrees and ____________ by sides
**REGULAR polygons: All ______________ are congruent and all _________________ are congruent
Try It
Find the sum of the measures of the interior angles of a heptagon.
A stop sign is a regular octagon. What is the measure of one interior angle in a stop sign?
**Tessellation: Repetitive pattern of polygons that fit together without ________ or _____________
- Sum of Angles where vertices (corners) meet in tessellation = ______________
Try It: Determine if a tessellation can be made using only regular decagons. Explain.
Chapter 11 Notes – Congruence, Similarity, and Transformations –
PART 2 – Transformations
Lesson 11-4: Translations and Reflections on the Coordinate Plane
Original figure is called the _____________________ and the new figure is called the _________
Images will be label with the _________________ symbol (point A becomes A’)
**TRANSLATION – is a _______________________
- Image is same ________________ and same ________________ as pre-image (CONGRUENT)
- Images face the same ____________________ (ORIENTATION)
- To find Coordinates of a Translated Image:
1) Look at the x-axis change:
- If it is Left - _________________ the change from the original x-value
- If it is Right - _______________ the change to the original x-value
2) Look at the y-axis change:
- If it is Down - _______________ the change from the original y-value
- If it is Up - ______________ the change from the original y-value
Try It
The vertices of triangle ABC are A(-3, 7), B(-1, 0), and C(5, 5).
Suppose it is translated 4 units right and 5 units down. Find
the coordinates of the vertices of its image, and then graph
the image and the original on the axes.
**REFLECTION – is a __________________ or _______________________
- Image is same ________________ and same ________________ as pre-image (CONGRUENT)
- Images face the opposite ____________________ (OPPOSITE ORIENTATION)
- To find Coordinates of a Reflected Image:
** Find the Line of Reflection (axis that you reflect over):
- If it is the X-AXIS:
x-coordinate is the __________________, y-coordinate is the ______________
- If it is the Y-AXIS:
x-coordinate is the __________________, y-coordinate is the ______________
Try It
The vertices of figure MNOP are M(-8, 6), N(5, 9), O(2, 1),
and P(-10, 3). Graph the figure and the image of the figure
after a reflection over the y-axis. List the new coordinates
of the image.
Lesson 11-5: Rotations on the Coordinate Plane
**ROTATION – is a _______________________
- Image is same ________________ and same ________________ as pre-image (CONGRUENT)
- Figures move around a FIXED point called the ______________________
- Can be a __________________ of the figure (then that point doesn’t move)
- Can be the ______________________ (0,0)
- Figures can Rotate _____________________________ or __________________________________
**Rotational Symmetry: If the figure can be rotated less than ___________ around its
center and it looks like the __________________________ figure
**To figure out degree : Divide ____________ by number of times the pattern _____________
Try It
Draw the figure shown after
a 90o clockwise rotation about
point A.
A
Triangle EFG has vertices
E(2, 1), F(1, -1), and G(4, -1).
Graph the figure and its image
after a clockwise rotation of
90o about vertex F. Then give
the coordinates of the vertices
for triangle E’F’G’.
Parallelogram ABCD has vertices
A(-3, -1), B(1, -2), C(-1, -4), and D(-5, -3).
Graph the parallelogram and its image
after a rotation of 180o about the origin.
Give the coordinates of the vertices for
parallelogram A’B’C’D’.
Determine whether or not the star shown has
rotational symmetry. If it does, what is the angle of rotation?
Lesson 11-6: Congruence and Transformations
**Two figures are congruent if you can make the _______________ after a series of
transformations (Translations, Reflections, Rotations) of the ______________
**Helpful hints: USE _________________________
Translation – Orientation is the ___________________
Reflection – Orientation is _____________________
Rotation – Orientation is _______________________
Try It
Determine if the two figures are congruent by using transformations. Explain.
Chapter 11 Notes – Congruence, Similarity, and Transformations –
PART 3 – Similarity
Lesson 11-7: Dilations on the Coordinate Plane
**DILATION - ____________________ or ____________________ a figure by a __________________________
- Think about __________________ and _______________________ figures
- Similar figures are the same ____________________ but different ___________________
- To find the Coordinates after a Dilation
1) Multiply each x-coordinate by the ______________________________
2) Multiply each y-coordinate by the ______________________________
- To find the Scale Factor
1) Find the lengths of 2 ____________________________ sides
- Subtract the ______________________
2) Make a __________________ comparing the length of the _________________
to the length of the __________________
3) Simplify the ratio
Try It
A figure has vertices D(-4, 4), E(4, 8),
F(8, 8), and G(4,0). Graph the figure
and the image of the polygon after a
dilation with a scale factor of ¼ .
A triangle has vertices H(-1, -2), I(1,4), and J(2,1). Find the coordinates of the
triangle after a dilation with a scale factor of 4.
Peri is enlarging a page from a book on the copy machine. The original page
measured 14 centimeters by 21 centimeters. If the enlarged image measures 17.5
centimeters by 26.25 centimeters, what is the scale factor of the dilation?
Lesson 11-8: Similarity and Transformations
**SIMILAR FIGURES – Same __________________ but different __________________
- Two figures are similar if the _____________________ can be obtained by a
sequence of transformations and dilations from the ___________________
- Compare _________________________ to decide if figures are similar
- Multiply Lengths of the original figure By _____________________________ to find the
lengths of the image
** SCALE FACTOR
– If it is less than one: Dilation is a ___________________________
- If it is greater than one: Dilation is an ________________________________
Try It
Determine if the two triangles are
Determine if the two rectangles are
similar by using transformations.
similar by using transformations.
Explain your reasoning.
Explain your reasoning.
H
K
H
I
J
K
J
E
F
G
L
M
O
N
Triangle RST has sides that are 20 mm, 10 mm, and 25 mm in length. It is related to
△FGH by a scale factor of 0.8. What are the lengths of the sides of △FGH?
A bakery transfers a photo image onto a sheet cake. The original photo, which is 8
inches by 10 inches, is reduced by a scale factor of 0.8. The baker decides the image
is still too large, and reduces the image by a scale factor of 0.9. What are the
dimensions of the final image that will appear on the cake? Is the reduced image
similar to the original?