Part 1: Transformations/Rigid MotionsRigid 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

Additional Notes

TranslationsTa,b

a →how to move your pre-image left/rightb →how to move your pre-image up/downVectors are drawn from pre-image to image and show distance and direction of the slide

Reflections Notation: rline

1. Graph line of reflection2. 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: REither 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

Rotational Symmetry Order

360n

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 translation2) a rotation followed by a translation3) a translation followed by a reflection4) 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 backPart 2: Triangle Theorems/Properties

Name of Theorem or relationship

In words/ Symbols Diagrams/ Hints/ Techniques

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

Draw arrows!

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. 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!

3. Pythagorean Theorem

a) To find a missing side

b) to Classify triangles

a) c2 = a2 + b2 C is longest side ( hypotenuse)

b) If c2 ¿ a2 + b2 it is acute If c2 ¿ a2 + b2 it is obtuse If c2 ¿ a2 + b2 it is right

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.

4.Isosceles triangle Theorem 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.

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

b) Sum of Exterior

a) 360/n

b) Sum is AlWAYS 360 degrees

Exterior angles and formed by extending a side of the triangle.

7. Interior angles in a polygon.

a) ONE interior

b) Sum of Interior.

a) The supplement of on Exterior angle-they are linear pairs! Exterior + Interior = 180

b) Number of △ ' s times 180.

( n-2)180

Remember! One exterior and one interior angle add up to 180 degrees!

8. Segments in a triangle: Medians- Goes to the midpoint of the opposite side creating two equal segmentsAltitudes-Are perpendicular to the opposite side creating right anglesPerpendicular 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!

9. Points of concurrence. 2 or more medians Centroid : Always inside the triangle. Cuts each median into a 2:1 ratio2 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.

ALL OF MY CHILDREN ARE BRING IN PEANUT BUTTER COOKIES.

10.Centroid and Ratios Centroid cuts every median into a 2:1 ratio. Use this ratio to set up equation 2X + 1x= whole length of median.

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 are , , and . Which type of triangle is ?1) right2) acute3) obtuse4) equiangular

2. The following is a 4-point question. This student only earned two points.

More Triangle Practice Problems Continued in the back

Part 3: Congruency

Reason Why Student Only Earned 2/4 Points:

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 Congrue

ncy Shortcut

s

*WE CANNOT USE AAA or SSA!!! *

Proving Parts are

Congruent

*To prove triangles are congruent, you need to use a shortcut.*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 Definitio

ns

Addition Postulate

With Substituti

on

Subtraction

PostulateWith

Substitution

Transitive Property

AB = 5; AB ≅ DE ; DE ≅ GH

So, AB = GH(by the TRANSITIVE PROPERTY)

Using Suppleme

nts

<1 < 4≅So, <2 <3≅(since 2 is the supplement of <1and <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 statement2. Prove normally3. 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

IF AB = EFCD = CD (Reflexive Property)Then, AB + CD = EF + CD (Addition Postulate)AD = ED (Substitution Postulate)

=

IF AC = BD,BC = BC (Reflexive Property)Then, AC- BC = BD - BC (Subtraction Postulate)AB = CD (Substitution Postulate)

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 backReview 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 down2) a counterclockwise rotation of 180 degrees

around the origin3) a reflection over the x-axis4) 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-axis2) 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-axis4) 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-axis2) translation to the left 5 and down 43) dilation centered at the origin with scale

factor 24) 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 rotation2) a reflection followed by a translation3) a translation followed by a rotation4) a translation followed by a reflection

6. In the diagram of parallelogram FRED shown below, is extended to A, and is drawn such that .

If , what is ?1) 124°2) 112°3) 68°4) 56°

7. In the diagram below, which single transformation was used to map triangle A onto triangle B?

1) line reflection2) rotation3) dilation4) translation

8. Which statement is sufficient evidence that is congruent to ?

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 .

9. Triangle ABC is graphed on the set of axes below. Graph and label , the image of after a reflection over the line .

10. As graphed on the set of axes below, is the image of after a sequence of transformations.

Is congruent to ? Use the properties of rigid motion to explain your answer.

11. In parallelogram ABCD shown below, diagonals and intersect at E.

Prove: Statements Reasons1. In parallelogram ABCD shown below, diagonals

and intersect at E.1. Given

Guided PracticeRight 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 segmentsIn triangle ABC, AD, CF , and BE are medians. If CF = 33, find CG and FG.

16) Side-Angle RelationshipThe 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: H I ≅KLStatements Reasons1. bisects and . 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 T 2,0o 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 intersect at E and .

Which statement is true based on the given information?1) 3)2) 4)