Day 7: Reviewing Geometry Concepts
Title
Goals
Reviewing Geometry Concepts
The teachers will create a mural that summarizes and
organizes the geometry concepts we have explored over the
first six days of the SMI
Materials for
Teacher
Materials for
Students
Description
Handout, color pencils, document camera (or overhead
projector)
Handout, color pencils.
Reflection
Looking Ahead
Over the first six days, the SMI has focused on (a) classifying
polygons based on their number of sides, (b) similarity and
congruence, (c), parallel and perpendicular line properties, and
(d) measurement (area and perimeter). In this session we are
going to organize this information in a way that makes sense
and makes retrieval easy.
Concept mapping is a useful tool for helping students to “see”
relationships among the various concepts they explore. This
session utilizes concept maps to organize the concepts that
have been encountered in the first six weeks.
Link to text
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Classifying 2D figures (Polygons) based on the number of sides:

Triangles: 3 sided polygon
— Any three lengths can be the three lengths of the sides of a triangle as
long as the sum of the two shorter sides is more than the longest
length.
— Triangles can be classified using side lengths (scalene, isosceles, or
equilateral).
— Triangles can be classified using angle measures.
— The sum of the interior angles of any triangle is 180 o
— If a right triangle then the Pythagorean theorem applies
— If the triangle is regular, then the measure of each interior angle is

n  2180 o . What does this turn out to be?
What do we call these
n
triangles?
 Quadrilaterals: 4 sided polygon
—We broadly classified them into convex and concave quadrilaterals.
—We focused on “special quadrilaterals: parallelograms, trapezoids,
rectangles, kites, squares, Rhombuses, right trapezoids, and isosceles
trapezoids)
—We classified these special quadrilaterals based on parallelism,
perpendicularity, and side congruencies.
—The sum of all interior angles is 360 o
— If the polygon is regular, then the measure of each interior angle is
n  2180 o . What does this turn out to be? What do we call these
n

quadrilaterals?
 Polygons with more than 4 sides:
— The sum of interior angles is n  2180 o where n is the number of
sides.
— The sum of exterior angles in one direction is 360 o
— If the polygon is regular,
then the measure of each interior angle is

n  2180 o
n


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Triangle Concept Map
3
Special Quadrilaterals
—The classifications of these quadrilaterals gave us the hierarchy and the properties to
go with it.
4
5
Similarity and Congruence
Similarity: Two figures are similar if:
a. All corresponding angles are congruent and
b. All corresponding sides are proportional
Example:
As a consequence of the proportionality of corresponding side, the ratio of the areas of
similar figures depend on the ratio of corresponding sides.
Congruence: Two figures are congruent if:
a. All corresponding angles are congruent and
b. All corresponding sides are congruent
As a consequence of the congruence of corresponding side, the areas of congruent figures
are equal.
Based on the information above, what is the relationship between congruences and
similarities?
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Special Case: Similarity and Congruence of Triangles
Triangle Congruence Theorems
Triangle Similarity Theorems
Theorem: AA Similarity Theorem:
If two triangles have two pairs of corresponding angles that are congruent, then the
triangles are similar. In particular, if A   A' and B   B' , then ABC ~ A' B'C' .
Theorem: SAS Similarity Theorem
 
 

If two triangles have two pairs of corresponding sides in the same ratio, and if the
included angles determined by the pairs of sides are congruent, then the triangles are
A' B' A'C'
similar. In particular, if
and A   A' , then ABC ~ A' B'C' .

AB
AC
 

Theorem: SSS Similarity Theorem

If two triangles have all three pairs of corresponding sides in the same ratio, then the
A' B' A'C' B'C'
triangles are similar. In particular, if
, then ABC ~ A' B'C' .


AB
AC
BC


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Parallel and Perpendicular Lines
Parallel lines:
— Parallel lines produce several pairs of supplementary angles.
 All linear pairs are supplementary (examples
 All pairs of interior angles on the same side of a transverse are
supplementary (examples
— Parallel lines produce several pairs of congruent angles

All pairs of alternate interior angles are congruent (examples

All pairs of alternate exterior angles are congruent (examples

All pairs of corresponding angles are congruent (examples
— If these lines are located on the coordinate plane, they have the same slope
— It’s from these properties of parallel lines that we get many properties of
parallelograms!
*Pedagogical implication:
Perpendicular lines:
— Perpendicular lines intersect to produce four right angles.
— Any pair of the four angles formed by perpendicular lines is supplementary
example (
— If these lines are located on the coordinate plane, their slopes are negative
reciprocals.
— It’s from these properties of parallel lines that we get many properties of
rectangles.
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