Content Area
Standard
Strand
Content Statement
Math
4.HS High School
G-SRT. Geometry – Similarity, Right Triangles, Trigonometry
CPI#
Cumulative Progress Indicator (CPI)
ACSSSD
Objectives
Understand similarity in terms of similarity transformations
4.HS.GSRT.1
Verify experimentally the properties of dilations given by a center and a scale
factor:
a.
A dilation takes a line not passing through 1. Demonstrate an understanding
the center of the dilation to a parallel line,
that dilation is a non-rigid
and leaves a line passing through the
transformation that maintains the
shape of a figure but not its size.
center unchanged.
2. Demonstrate that dilation of a
point in the coordinate plane is
found by multiplying the x- and
y- coordinates by the same
number n. Recognize that the
number n is referred to as the
scale factor.
3. Recognize that in dilation each
point and its image lie on a
straight line that passes through a
point known as the center of
dilation.
4. Identify the center of dilation in a
dilation.
b.
The dilation of a line segment is longer or
shorter in the ratio given by the scale
factor.
1. Demonstrate an understanding
that if the absolute value of the
scale factor is < 1 the dilation is a
contraction and if the absolute
value of the scale factor is >1 the
dilation is an expansion.
2. Demonstrate an understanding
that contraction results in a
smaller figure and expansion
results in a larger figure.
4.HS.GSRT.2
Given two figures, use the definition of
similarity in terms of similarity
transformations to decide if they are
similar; explain using similarity
transformations the meaning of similarity
for triangles as the equality of all
corresponding pairs of angles and the
proportionality of all corresponding pairs
of sides.
1. Demonstrate an understanding
that two figures are defined as
similar if and only if one is
congruent to the image of the
other through dilation.
2. Demonstrate an understanding
that proportion refers to he
equality of ratios of
corresponding sides in figures.
3. Demonstrate an understanding of
polygon similarity postulate that
two polygons are similar if and
only if each pair of
corresponding sides is
proportional and each pair of
corresponding angles is
congruent.
4. Demonstrate an understanding of
the properties of proportions:
cross-multiplication property,
reciprocal property, exchange
property, and “add one” property.
5. Demonstrate an understanding
that the ratio of any two sides in
a polygon is the same as the ratio
of the corresponding sides in a
similar polygon.
6. Demonstrate an understanding of
SSS (side-side-side) similarity of
triangle that two triangles are
similar if three sides of one
triangle are proportional to three
sides of another triangle.
7. Demonstrate an understanding of
SAS (side-angle-side) triangle
similarity that two triangles are
similar if two sides of one
triangle are proportional to two
sides of another triangle and the
included angles are congruent.
4.HS.GSRT.3
Use the properties of similarity
transformations to establish the AA
criterion for two triangles to be similar.
1. Demonstrate an understanding of
AA (angle-angle) triangle
similarity that two triangles are
similar if two angles of one
triangle are congruent to two
angles of another triangle.
4.HS.GSRT.4
Prove theorems about triangles. Theorems
include: a line parallel to one side of a
triangle divides the other two
proportionally, and conversely; the
Pythagorean Theorem proved using
triangle similarity.
1. Demonstrate an understanding of
Side Splitting Theorem, which
states that a line drawn parallel to
one side of a triangle divides the
other two sides proportionally.
2. Demonstrate an understanding of
the Triangle Midsegment
Theorem, which states that the
midsegment of a triangle is
parallel to one side of a triangle.
3. Demonstrate an understanding of
Two-Transversal Proportionality
Corollary, which states that three
or more parallel lines divide two
intersecting transversals
proportionally.
4. Demonstrate an understanding of
the converse of the Side Splitting
Theorem, that if a segment
divides two sides of a triangle
Prove theorems involving similarity
proportionally, then the segment
is parallel to the third side.
5. Demonstrate through the use of
SAS similarity, the crossmultiplication property of ratios,
and the substitution property of
equality that three right triangles
dissected from a rectangle can be
compared to one another in order
to prove the Pythagorean
Theorem.
4.HS.GSRT.5
Use congruence and similarity criteria for
triangles to solve problems and to prove
relationships in geometric figures.
1. Demonstrate an understanding of
Proportional Altitudes Theorem,
which states that if two triangles
are similar then the
corresponding altitudes have the
same ratio as the corresponding
sides.
2. Demonstrate an understanding of
Proportional Medians Theorem,
which states that if two triangles
are similar then the
corresponding medians have the
same ratio as the corresponding
sides.
3. Demonstrate an understanding of
Proportional Angle Bisectors
Theorem, which states that if two
triangles are similar then the
corresponding angle bisectors
have the same ratio as the
corresponding sides.
4. Demonstrate an understanding
that polygon and triangle
similarity and congruence are
used to determine unknown
measurements in geometric
figures.