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Part 1: Transformations/Rigid Motions
Rigid Motions and transformations
-Rigid Motions produce congruent figures
-Translation, Rotation, Reflections are all rigid motions
-Rigid Motions preserve size, shape and angle measure, they only change the
position of a figure
Translations
Ta,b
a →how to move your pre-image left/right
b →how to move your pre-image up/down
Vectors are drawn from pre-image to image and
show distance and direction of the slide
Reflections Notation: rline
1. Graph line of reflection
2. Count how far away each point is on the
line and count the opposite going the
other way
Special reflections
Point Reflections
YOU MUST MEMORIZE
Special compositions
-Perform Compositions from right to left!
-Glide Reflections: SHOW MATHEMATICALLY the slopes are the same for the
vector and line of reflection. This justifies that they are parallel.
- Composition of reflections over parallel lines are the same as one translation
Rotations and more special compositions Notation: R
Either know your rules, or Rotate paper!
Rotate counterclockwise for positive angles, and clockwise for negative angles!
- Composition of reflections over perpendicular lines are the same as one
rotation
Additional Notes
Rotational Symmetry
 Order
360

𝑛
Regents-Style Questions
1. In the diagram below, congruent figures 1, 2, and 3 are drawn.
Which sequence of transformations maps figure 1 onto figure 2 and
then figure 2 onto figure 3?
1) a reflection followed by a translation
2) a rotation followed by a translation
3) a translation followed by a reflection
4) a translation followed by a rotation
2. In the diagram below,
and
are graphed.
Use the properties of rigid motions to explain why
.
More Transformations Practice Problems Continued in the back
Part 2: Triangle Theorems/Properties
Name of Theorem or
In words/ Symbols
relationship
1. Side angle relationship
The longest side is across from the largest angle.
The medium length side is across from the medium-sized
angle.
The shortest side is across from the smallest angle
2. Triangle inequality Theorem
The sum of the lengths of the two smaller sides of a triangle is
greater than the length of the largest side.
3. Pythagorean Theorem
a) c2 = a2 + b2
a) To find a missing side
2
b) to Classify triangles
4.Isosceles triangle Theorem
2
b) If c < a + b
C is longest side ( hypotenuse)
2
If c2 > a2 + b2
it is acute
it is obtuse
If c2 = a2 + b2 it is right
The base angles of an isosceles triangle are equal in measure.
The sides opposite the base angles in an isosceles triangle
(called legs) are equal in length.
Diagrams/ Hints/ Techniques
Draw arrows!
Add up the two smaller sides and compare to the largest
side. If the sum is greater, it’s a triangle!
2,3,4
(2+3) > 4 ? yes!
If you see a right angle, it’s a right triangle!  use P.T to
solve for a missing side.
WATCH OUT! If asked “does this make a triangle” you
must use Theorem # 2- NOT PYTHAGOREAN.
Angles opposite are congruent! If you see expressions,
make them equal to each other!
5. Exterior angle theorem
The measure of an exterior angle of a triangle is equal to the
sum of the measures of the two remote interior angles of the
triangle.
IN + IN =OUT
6. Exterior angles in a polygon.
a). ONE exterior
a) 360/n
Exterior angles and formed by
extending a side of the triangle.
b) Sum is AlWAYS 360 degrees
b) Sum of Exterior
7. Interior angles in a polygon.
a) The supplement of on Exterior angle-they are linear pairs!
Exterior + Interior = 180
Remember! One exterior and one interior angle add up to
180 degrees!
a) ONE interior
b) Sum of Interior.
8. Segments in a triangle:
9. Points of concurrence.
10.Centroid and Ratios
b) Number of △ ′𝑠 times 180.
( n-2)180
Medians- Goes to the midpoint of the opposite side creating
two equal segments
Altitudes-Are perpendicular to the opposite side creating right
angles
Perpendicular bisectors- Goes to the midpoint of opposite
side and is perpendicular to it.
Angle bisectors- bisects the angle at the vertex it goes through
making 2 congruent angles.
** In Isosceles and Equilateral triangles these segments
coincide!
2 or more medians Centroid : Always inside the triangle.
Cuts each median into a 2:1 ratio
2 or more Altitudes Orthocenter: Inside for acute triangles,
on the triangle for right triangles and outside for obtuse
triangles.
2 or more angle bisectors Incenter: Always inside the
triangle.
2 or more perpendicular bisectors Circumcenter: Inside for
acute triangles, on the triangle for right triangles and outside
for obtuse triangles.
Centroid cuts every median into a 2:1 ratio. Use this ratio to
set up equation 2X + 1x= whole length of median.
ALL OF MY CHILDREN ARE BRING IN PEANUT BUTTER
COOKIES.
Read carefully- What is the segment they want?
Sometimes you need to substitute back in!
Regents-Style Questions
*Note: The Regents will occasionally address triangle questions by combining them with other
core units of study. The following are a few problems that exemplify this. We will cover
these additional topics in future Regents Review Sessions.
1. The coordinates of the vertices of
1) right
2) acute
3) obtuse
4) equiangular
are
,
, and
. Which type of triangle is
2. The following is a 4-point question. This student only earned two points.
Reason Why Student Only Earned 2/4 Points:
More Triangle Practice Problems Continued in the back
?
Part 3: Congruency
Concept
Key Ideas/Tips
Beginning a
Proof
*mark your picture
*annotate question (givens)
*use the tools from your tool box
*1st write your givens
*last step = what you're trying to prove
*make a plan
*use a checklist for your shortcuts!
*# your steps!
*All of your givens and markings should be a step in your proof!
Triangle
Congruency
Shortcuts
*WE CANNOT USE AAA or SSA!!! *
Proving Parts
*To prove triangles are congruent, you need to use a shortcut.
are Congruent
*To prove parts (sides and angles) of triangles are congruent, we:
1st: prove triangles are congruent
2nd: use CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
Proving
Definitions
Addition
Postulate
With
IF AB = EF
CD = CD
Then, AB + CD = EF + CD
AD = ED
Substitution
(Reflexive Property)
(Addition Postulate)
(Substitution Postulate)
=
Subtraction
Postulate
With
Substitution
Transitive
Property
IF AC = BD,
BC = BC
Then, AC- BC = BD - BC
AB = CD
(Reflexive Property)
(Subtraction Postulate)
(Substitution Postulate)
̅̅̅̅̅ ; ̅̅̅̅
AB = 5; ̅̅̅̅
𝐴𝐵 ≅ 𝐷𝐸
𝐷𝐸 ≅ ̅̅̅̅
𝐺𝐻
So, AB = GH
(by the TRANSITIVE PROPERTY)
<1 ≅ < 4
Using
Supplements
So, <2 ≅ <3
(since 2 is the supplement of <1
and <3 is the supplement of < 4)
Indirect Proof
*Use when proving something is not true or ≇ or to show something is not true.
Steps:
1. Assume the opposite of the prove statement
2. Prove normally
3. Contradict something in the given.
***Proof Pieces are Available on the website. You may
use them to help remember key details. Access at any
time
Example Regents Problem:
Given: Quadrilateral ABCD is a parallelogram with diagonals
and
intersecting at E
Prove:
Describe a single rigid motion that maps
onto
.
More Proof/Congruency Practice Problems Continued in the back
Review Session 1 Practice Problems
1. The vertices of
have coordinates
,
, and
). Under which transformation is the
image
not congruent to
?
1) a translation of two units to the right and
two units down
2) a counterclockwise rotation of 180 degrees
around the origin
3) a reflection over the x-axis
4) a dilation with a scale factor of 2 and
centered at the origin
2. Triangle ABC and triangle DEF are graphed on the set of axes below.
Which sequence of transformations maps triangle ABC onto triangle DEF?
1) a reflection over the x-axis followed by a
reflection over the y-axis
2) a 180° rotation about the origin followed by
a reflection over the line
3) a 90° clockwise rotation about the origin
followed by a reflection over the y-axis
4) a translation 8 units to the right and 1 unit up
followed by a 90° counterclockwise rotation
about the origin
3. Quadrilateral ABCD is graphed on the set of axes below.
When ABCD is rotated 90° in a counterclockwise direction about the
origin, its image is quadrilateral A'B'C'D'. Is distance preserved under
this rotation, and which coordinates are correct for the given vertex?
1) no and
2) no and
3) yes and
4) yes and
4. If
is the image of
, under which transformation will the triangles not be congruent?
1) reflection over the x-axis
2) translation to the left 5 and down 4
3) dilation centered at the origin with scale
factor 2
4) rotation of 270° counterclockwise about the
origin
5. A sequence of transformations maps rectangle ABCD onto rectangle
A"B"C"D", as shown in the diagram below.
Which sequence of transformations maps ABCD onto A'B'C'D' and then
maps A'B'C'D' onto A"B"C"D"?
1) a reflection followed by a rotation
2) a reflection followed by a translation
3) a translation followed by a rotation
4) a translation followed by a reflection
6. In the diagram of parallelogram FRED shown below,
.
If
1)
2)
3)
4)
, what is
is extended to A, and
is drawn such that
?
124°
112°
68°
56°
7. In the diagram below, which single transformation was used to map triangle A onto triangle B?
1)
2)
3)
4)
line reflection
rotation
dilation
translation
8. Which statement is sufficient evidence that
1)
and
2)
,
,
3) There is a sequence of rigid motions that
maps
onto
,
onto
, and
onto
.
4) There is a sequence of rigid motions that
maps point A onto point D,
onto
, and
onto
.
is congruent to
?
9. Triangle ABC is graphed on the set of axes below. Graph and label
reflection over the line
.
10. As graphed on the set of axes below,
Is
congruent to
is the image of
, the image of
after a sequence of transformations.
? Use the properties of rigid motion to explain your answer.
11. In parallelogram ABCD shown below, diagonals
Prove:
Statements
1. In parallelogram ABCD shown below, diagonals
and
intersect at E.
and
intersect at E.
Reasons
1. Given
after a
Guided Practice
Right Triangles:
12. To find the sides of a right triangle use the Pythagorean Theorem.
a2 + b2 = c2
{Remember that the a and b make up right angle}
13) Theorem: The measure of an exterior angle of a triangle is
equal to the sum of the measures of the two remote interior
angles of the triangle.
14) Isosceles Triangles:
The base angles of an isosceles triangle are equal in measure.
The sides opposite the base angles in an isosceles triangle (called legs) are equal in length.
15) Centroid: 2:1 Ratio for segments
̅̅̅̅̅ and ̅̅̅̅
In triangle ABC, ̅̅̅̅̅
𝐴𝐷, 𝐶𝐹,
𝐵𝐸 are medians. If ̅̅̅̅
𝐶𝐹 = 33, find CG and FG.
16) Side-Angle Relationship
The longest side is across from the largest angle.
The medium length side is across from the medium-sized angle.
The shortest side is across from the smallest angle
In triangle DOG, m<D = 40, m<O 60, and m<G = 80.

State the longest side of the triangle __________________

State the shortest side of the triangle _________________
17. In the accompanying diagram,
bisects
and
.
Prove: 𝐻𝐼 ≅ 𝐾𝐿
Statements
1.
bisects
and
.
Reasons
1. Given
18)
Given:
Prove:
19) Graph and state the coordinates of ∆𝐴′𝐵′𝐶′, the image of ∆𝐴𝐵𝐶 with A(1,2), B(3,0) and C(6,8)after the
composition 𝑇2,0o 𝑅180° .
20. Graph triangle ABC. A(1, 1), B(4, 5), C(3, 2) and reflect it through point (-2, 1). Label your image.
Use rigid motions to explain why the pre-image and image are congruent.
21. In the diagram of trapezoid ABCD below, diagonals
and
Which statement is true based on the given information?
1)
3)
2)
4)
intersect at E and
.