GEOMETRY Concepts of Congruence
OBJECTIVE #: G.CO.6
OBJECTIVE
· Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
BIG IDEA (Why is this included in the curriculum?)
· Rigid motions can be used to determine if two figures are congruent.
PREVIOUS KNOWLEDGE (What skills do they need to have to succeed?)
· The student must have a thorough knowledge of isometric transformations.
· The student must understand one-to-one functions.
VOCABULARY USED IN THIS OBJECTIVE (What terms will be essential to understand?)
PREVIOUS VOCABULARY (Terms used but defined earlier)
· Image: The new figure that results from any transformation of a figure in the plane.
· Mapping: A correspondence between set of points, which pairs each member of the domain with an element of the range.
· One-to-One Function: A function in which every element in the range (output) corresponds to one and only one element in the domain (input). A one-to-one function must pass the horizontal line test.
· Pre-Image: The original figure in the transformation of a figure in the plane.
NEW VOCABULARY (New Terms and definitions introduced in this objective)
· Corresponding Parts of Congruent Figures: The sides or angles that are in corresponding positions when two figures are congruent. [G.CO.7p, G.CO.8p]
§ CPCFC: Corresponding Parts of Congruent Figures are Congruent. [G.CO.7p]
§ CPCTC: Corresponding Parts of Congruent Triangles are Congruent. [G.CO.8p]
Notation:
SKILLS (What will they be able to do after this objective?)
· Students will be able to show two figures are congruent if there is a sequence of rigid motions that map one figure to another.
· Students will be able to show that two figures are congruent if and only if they have the same size and shape.
· Students will be able to use composite transformations to map one figure onto another.
SHORT NOTES (A short summary of notes so that a teacher can get the basics of what is expected.)
· Students should be able to identify whether a given transformation will create an image that is congruent to its pre-image.
· This section emphasizes CPCFC (Congruent Parts of Congruent Figures are Congruent)
· Congruence should now be defined as one or more transformations that maps a pre-image onto its image.
· If CABD ≅ XYZW, list all of the congruent parts
o Congruent Sides : CA≅XY, AB≅YZ, BD≅ZW, CD≅XW
o Congruent Angles: ∠C≅∠X, ∠A≅∠Y, ∠B≅∠Z, ∠D≅∠W
· Describe the type of rigid motion that would map QRST onto Q’R’S’T’.
o Answers may vary.
§ Reflection
§ Reflect horizontally, rotate, translate
MISCONCEPTIONS (What are the typical errors or difficult areas? Also suggest ways to teach them.)
· The orientation of the pre-image and image may change, but the congruency statement would not change.
FUTURE CONNECTIONS (What will they use these skills for later?)
· Triangle congruency and transformations will be utilized to discover the characteristics of different quadrilaterals.
ADDITIONAL EXTENSIONS OR EXPLANATIONS (What needs greater explanation?)
· Sequences of rigid motions will be used to discover congruency postulates for triangles.
ASSESSMENTS (Questions that get to the heart of the objective – multiple choice, short answer, multi-step)
1. ABCD ≅ KJHL. Find the value of x and y.
x = 3
y = 25
2. Explain how to transform ∆ABC to ∆A'B'C'.
Answers may vary
Reflection
Translation, then reflection
From CCSD Geometry Honors Semester 1 Practice Exam 2012 – 2013
1. Look at the figure below.
Look at these three figures.
Which figures are congruent to the first figure?
(A) I only
(B) II only
(C) I and II only
(D) I, II, and III
For questions 2 - 4, evaluate whether the image of a figure under the described transformation is congruent to the figure.
2. A transformation T follows the rule. The image of a figure under T is congruent to the figure.
(A) True
(B) False
3. A transformation T follows the rule. The image of a figure under T is congruent to the figure.
(A) True
(B) False
4. A transformation T follows the rule. The image of a figure under T is congruent to the figure.
(A) True
(B) False