Congruence and Similarity Congruent means same shape and same size. Angles, segments and shapes can be congruent. Similar means same shape.

What is graphic design https://www.youtube.com/watc h?v=rUeiZ6c6EBw

Movement on the coordinate plane Preimage is the original image. Points on the Pre-image are indicated by capital letters. A,B,C Image is the image that came from the pre-image. Points on the image are indicated by primes. A’ B’ C’

Ways to transform a shape on a coordinate plane. Translation (Slide all points of a shape same distance and direction.) Reflection (flip all points on a shape over a line) Rotation (turn all points on a shape about a point)

Describing a Translation Description in words The preimage was translated 6 units to the left. Description using coordinate notation. (x,y)→(x-6,y)

Describing a Translation Description in words. Coordinate notation

Describing a Translation Description in words. Coordinate notation

Describing a Translation Description in words. Coordinate notation

Important concepts about a translation Translation is a movement of all points of a shape in the same ________and the same _______ A translated shape maintains its shape and size. In other words the image and the pre-image are ___________. All segments drawn from corresponding points on the image and pre-image are ______ All corresponding line segments are ____________

Translating a linear function Sue makes 5 dollars an hour. Sue makes 5 dollars an hour with a one time bonus of 10 dollars.

Write a new equation based on the translation. Translate 3 units down Y=2x Translate 4 units up y = -x Translate 2 units up y = 2x+1 Translate 5 units down y= 3x+2

Reflection.

Describing a reflection. Description in words Description as coordinate notation.

Describing a reflection Description in words Description as coordinate notation.

Describing a reflection Description in word Coordinate notation

Describing a reflection Description in words Coordinate notation

Describing a reflection Description in words

Describing a reflection Description in words

Important concepts about reflection A point on the pre-image is the same distance to the line of reflection as a point on the image. Line segments connecting corresponding points are bisected at the line of reflection.

Rotation 7.3 Rotation in real life Car parts/ Mechanics Clock Body motion Astronomy Weapons Fans Amusement park rides Rotation vocabulary Point of rotation Angle of rotation

Rotation Describing a rotation in words I am rotating about the origin. 90 degrees counter clockwise. Describing a rotation with coordinate notation. Preimage Image A (2,3) (-3,2) B (4,3) (-3,4) C (4,0) (0,-4) Rule (x,y) ( , )

Rotation and circles What do you notice about rotation when circles are drawn about the point of rotation. How do you know this is a 90 degrees rotation counterclockwise? Does a rotation change the shape or size of the triangle?

90 degree rotation and the hook The hook method. Vertical lines become horizontal lines. Horizontal lines become vertical lines.

Rotate 90 degrees counterclockwise about the origin using the hook method. Give the new coordinates for A’,B’ and C’ to your neighbor.

Rotate 90 degrees counterclockwise about the origin using the hook method. Give the new coordinates for A’,B’ and C’ to your neighbor.

Rotating objects a different number of degrees. I can rotate this object 120 degrees about the origin using technology.

Rotating objects in intervals of 90 degrees around points other than the origin.

Rotating objects in intervals of 90 degrees around points other than the origin. Rotate 90 degrees cc about the point C. of changing your vertical lines to horizontal lines and your horizontal lines to vertical lines.

Activity Draw a pre-image on a coordinate plane. Label the vertices. Then take the image through a translation, a reflection and a rotation in any order. Write rules for each transformation. Write coordinates for each transformation.

Activity with Paddy Paper Draw a shape on the coordinate plane. Trace the shape on paddy paper.

I can map one object onto another through multiple transformations.

Describe the transformations that would map one triangle onto another Compare the coordinates

Describe the transformations that would map one triangle onto another

Describe the transformations that would map one triangle onto another

Activity with geogebra See if you can use geogebra to replicate the transformations that took place from the pre-image to the image. Then

Triangle congruence Shortcuts

All I need are three pairs of congruent side lengths and a compass to prove sss congruency.

If SSS works for triangle congruency does AAA work If SSS works for triangle congruency does AAA work for triangle congruency If SSS works for triangle congruency does SSA work?

Two triangles are congruent if all corresponding sides are the same length SSS How can I tell if the corresponding sides are the same length?

http://nlvm. usu. edu/en/nav/fra mes_asid_165_g_1_t_3. html http://nlvm.usu.edu/en/nav/fra mes_asid_165_g_1_t_3.html?op en=instructions&from=topic_t_3 .html

Two triangles are congruent if all corresponding sides are the same length SSS What tool am I using here to show that all the corresponding side lengths are congruent? Explain to your friend how to use this tool to determine congruent side lengths.

Two triangles are congruent if all corresponding sides are the same length SSS What kind of transformations took place from the preimage ABC to the image EDF? Did the side lengths change through transformations?

Two triangles are congruent if all corresponding sides are the same length SSS Write a congruency statement for the two triangles in the graph at the right? How do you know the two triangles are congruent? Can you come up with a second way to determine triangle congruence?

Two triangles are congruent if all corresponding sides are the same length SSS Given AB and DE are congruent BC and EF are congruent D is the midpoint of AC C is the midpoint of DF Are the two triangles congruent? Explain your reasoning.

Two triangles are congruent if all corresponding sides are the same length SSS BD is a perpendicular bisector? Are the triangles congruent? Explain your reasoning.

Two triangles are congruent if two pairs of corresponding sides and the included angle are congruent. SAS You may remember the triangle inequality theorem which states that the largest side of a triangle is opposite the largest angle. If you completed both triangles to the right AC would be the same length as DF. Likewise < B would be congruent to <E

Two triangles are congruent if two pairs of corresponding sides and the included angle are congruent. SAS If BE is Congruent to CE and AD is Congruent to DE determine if the two triangles are congruent. List the ways that you could determine congruence. 1. 2. 3. 4.

Two triangles are congruent if two pairs of corresponding sides and the included angle are congruent. SAS If BD and AC are perpendicular and E is the midpoint of BD prove that BC is Congruent to CD.

Two triangles are congruent if two pairs of corresponding sides and the included angle are congruent. SAS Can you solve for missing sides and angles of the two triangles?

How could you show that all the triangles in the regular hexagon are congruent?

13.5 angle side angle (ASA) IF two triangles have 2 pairs of congruent angles and an included side then they are congruent. Using the diagram, can you explain why only one unique triangle can be created with 2 angles and an included side?

13.6 Angle Angle Side (AAS) Two triangles are shown to be congruent using the ASA postulate. How can we conclude that the AAS postulate would also work based on the ASA postulate?

Postulate SU + UT = ST Postulate is a common sense idea Examples A linear pair contains 180 degrees 2 line segments can be added together to get one large segment SU + UT = ST

Theorem A theorem is a rule in math that has been proven by using postulates, definitions, and other theorems. Prove the vertical angles theorem. In other words show that x and z are congruent Statement Reason <X+<Y = 180 Linear pairs are supplements <Z + <Y = 180 Linear pairs are supplements <Z+<Y= <X+<Y Substitution <Z = < X Subtraction property of equality

Practice with triangle congruency proofs Statement Reason AD is parallel to BC given AD = CB <DAC = <BCA Alternate interior angles <BEC = <AEC Vertical angles ∆𝐴𝐸𝐷=∆𝐶𝐸𝐵 AAS

Practice with triangle congruency proofs Statement Reason Given <JKM =90 and <LKM=90 <JKM and <LKM are congruent ∆𝐽𝐾𝑀≅∆𝐿𝐾𝑀

Practice with triangle congruency proofs Statement Reason given Reflexive property SAS

Practice with triangle congruency proofs Statement Reason B is midpoint of DC AB is perpendicular to DC DB = BC AB=AB <ABC and <ABD = 90 <ABC is congruent to <ABD ∆𝐴𝐵𝐷≅∆𝐴𝐵𝐶

Practice with triangle congruency proofs Statement Reason Given FG= GH EG=EG SSS <1 = <2 CPCTC

Practice with triangle congruency proofs Statement Reason