Include: All inclusive or examples?

[tyates]

We are currently unpacking 9-12 CCSS Math Standards. Several of our teachers were concerned with the wording of some of the standards, and we need clarification.

For Example:

The Standard G-CO.9 states:

Prove theorems about lines and angles.  Theorems include:  vertical angles are congruent;  when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent;  Points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

 

Our Questions:

This standard states “theorems include.”

Is this an all-inclusive, finite list of theorems?
If not, should we list the theorems that are given in our current textbook?
Will our students be tested on just these theorems or additional related theorems?

 

 

One final question:

Is the development of the students’ understanding of congruence and triangle congruence based on and/or developed through “TRANSFORMATION”?

[Bill McCallum]

Curricula are not limited to the list, but it’s possible that assessments might limit themselves to the list; that’s really a question for the two assessment consortia, to which I do not know the answer.

As for your final question, yes, congruence is defined in terms of rigid motions—translations, rotations, and reflections. Two figures are defined to be congruent if there is a rigid motion that transforms one to the other. The standard congruence criteria for triangles—ASA, SAS, SSS—can be derived from this (taking as an axiom that they preserve distance and angle).

[jamiechaikin]

Will you please provide an example using rigid motions to show that vertical angles are congruent? Thanks.

[Bill McCallum]

Take two intersecting lines. Rotate them through 180 degrees around the point of intersection. Each line is taken to itself, and the point of intersection stays put, so each one of the pair of vertical angles is taken onto the other one. Therefore they are congruent (because there is a rigid motion which takes the one to the other).

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