Geometry 6-1 Proportions
A. Definitions
1. A ___________ is a comparison of two quantities.
2. The ratio of a to b can be expressed a/b where ___ is not zero.
Example 1: Last year, of the 270 high school students at CVC, 190 of them
participated in some kind of sport. Find the athlete to student ratio to the nearest
tenth.
Example 2: In a triangle, the ratio of the measures of the three sides is 5:12:13,
and the perimeter is 90 cm. Find the length of the shortest side.
B. Properties of proportions
1. An equation stating that two _________ are equal is called a _____________.
2. Equivalent fractions are set equal to each other to form a proportion.
Example: 1 = 2
3 6
3. Every proportion has two cross products.
4. _______________________________________ -For any numbers a and c,
and any nonzero number b and d, a/b = c/d if and only if_______________.
Example 3: Solve each proportion.
Example 4: A boxcar on a train has a length of 40 feet and a width of 9 feet. A
scale model made with a length of 16 inches, find the width of the model. scale
model made with a length of 16 inches, find the width of the model.
6-2 Exploring Similar Polygons
A.______________________ -When figures have the same shape but different
sizes, they are called similar figures.
1. Definition of ____________________ -Two polygons are similar if and only if
their corresponding angles are congruent and the measures of their corresponding
sides are proportional.
m<A = m<Q m<B = m<R
m<D = m<T m<E = m<U
m<C = m<S
2. The symbol ~ means similar to.
-We say polygon ABCDE ~ QRSTU
3. The ratio of the two corresponding sides is the __________________.
Example 1
-Polygon QUAD ~ to polygon FOUR.
a. Find the scale factor of polygon QUAD to polygon FOUR
b. Find the values of x and y.
Geometry 6-3 Similar Triangles
A. Identifying Similar Triangles
1. Postulate 6-1 ________________________ Similarity
If two angles of one _____________ are congruent to two angles of another
triangle, then the triangles are similar.
2. Theorem 6-2 ___________________________________Similarity
If the measures of the corresponding _________________of two triangles are
proportional, then the triangles are similar.
3. Theorem 7-2 ____________________________________ Similarity
If the measures of two sides of a triangle are proportional to the measures of two
corresponding sides of another triangle and the _____________ angles are
congruent, then the triangles are similar.
Example 1: In the figure AB || DC , CD
DE , AB BE , BE = 27, DE = 45,
AE = 21, and CE =35. Determine which triangles are similar.
Example 2: Given UT || RS, find SQ and QU.
Example 3: If you wanted to measure the height of the Sears tower in Chicago,
you could measure a 12-foot light pole and measure its shadow. If the length of
the shadow was 2 feet and the shadow of the Sears Tower was 242 feet, what is
the height of the Sears Tower?
Geometry 6-4 Parallel lines and Proportional Parts
A. Proportional Parts of Triangles
1. Theorem 6-4 ______________________________________________
If a line is parallel to one side of a triangle and intersects the other sides in two
distinct points, then it separates these sides into segments of proportional lengths.
Example:
Example 1: In
RST , RT VU , SV = 3, VR = 8, and UT = 12, find SU.
2. Theorem 6-5 Converse of the Triangle Proportionality Theorem
If a line intersects two sides of a triangle and separates the sides into
corresponding segments of proportional lengths, then the line is parallel to the
third side.
Example
Example 2: In
DEF, DH = 18, HE = 36, and DG = ½GF. Determine whether
GH || FE . Explain.
3. A________________________ of a triangle is a segment whose endpoints are
the ___________ of two sides of the triangle.
4. Theorem 6-6 Triangle ___________________ Theorem
A midsegment of a triangle is parallel to one side of the triangle, and its length is
one half the length of that side
Example 2: Triangle ABC has vertices A(-2, 2), B(2, 4) and C(4, -4). DE is a
midsegment of
ABC .
a.) Find the coordinates of D and E.
b.) Verify that BC || DE.
c.) Verify that DE = (1/2)BC.
B. Dividing Segments Proportionally
1. Corollary 6-1 - If three or more parallel lines intersect two transversals, then
they cut off the transversals proportionally.
SO - If DA ||EB|| FC, then AB = DE
BC
EF
2. Corollary 6-2 - If three or more parallel lines cut off congruent segments on
one transversal, then they cut off congruent segments on every transversal.
SO - If AB  BC, then DE  EF
Example 3: Find x and y.
Geometry 6-5 Parts of similar triangles
A. Perimeters
1. Theorem 6-7 _____________________________ Theorem
If two triangles are similar, then the perimeters are proportional to the measures
of corresponding sides.
Example 1: If ABC:
XYZ, AC = 32,
AB = 16, BC 16√5. and XY = 25. Find
the perimeter of XYZ
2. Theorem 6-8 - If two triangles are similar, then the measures of the
corresponding ____________ are proportional to the measures of the
corresponding sides.
3. Theorem 6-9 - If two triangles are similar, then the measures of the
corresponding ____________ __________ are proportional to the measures of
the corresponding sides.
4. Theorem 6-10 - If two triangles are similar, then the measures of the
corresponding ____________ are proportional to the measures of the
corresponding sides.
Example 2:
Example 3:
5. Theorem 6-11 ____________________________ Theorem
An angle bisector in a triangle separates the opposite side into segments that have
the same ratio as the other two sides.
Example:
Example 4: Solve for x.