Transcript Document

6.4 Prove Triangles
Similar by AA
Before: You used the AAS Congruence Theorem
Now: You will use the AA Similarity Postulate
Why? So you can use similar triangles to understand
aerial photography
AA Similarity
Angle-Angle similarity . When two triangles have
corresponding angles that are congruent as shown
below, the triangles are similar.
EXAMPLE 1
Use the AA Similarity Postulate
Determine whether the triangles are similar. If they
are, write a similarity statement. Explain your
reasoning.
EXAMPLE 1
Use the AA Similarity Postulate
SOLUTION
Because they are both right angles,
congruent.
D and
G are
By the Triangle Sum Theorem, 26° + 90° + m E = 180°,
so m E = 64°. Therefore, E and H are congruent.
ANSWER
So, ∆CDE ~ ∆KGH by the AA Similarity Postulate.
EXAMPLE 2
Show that triangles are similar
Show that the two triangles are similar.
a.
b.
∆ABE and ∆ACD
∆SVR and ∆UVT
EXAMPLE 2
Show that triangles are similar
SOLUTION
a.
You may find it helpful to redraw the triangles
separately.
Because m ABE and m C both equal 52°,
ABE
C. By the Reflexive Property, A
ANSWER
So, ∆ ABE ~ ∆ ACD by the AA Similarity Postulate.
A.
EXAMPLE 2
Show that triangles are similar
SOLUTION
b. You know SVR
UVT by the Vertical Angles
Congruence Theorem. The diagram shows RS ||UT
so
S
U by the Alternate Interior Angles
Theorem.
ANSWER
So, ∆SVR ~ ∆UVT by the AA Similarity Postulate.
GUIDED PRACTICE
for Examples 1 and 2
Show that the triangles are similar. Write a similarity
statement.
1.
∆FGH and ∆RQS
ANSWER
In each triangle all three angles measure 60°, so by
the AA similarity postulate, the triangles are similar
∆FGH ~ ∆QRS.
GUIDED PRACTICE
for Examples 1 and 2
Show that the triangles are similar. Write a similarity
statement.
2.
∆CDF and ∆DEF
ANSWER
Since m CDF = 58° by the Triangle Sum Theorem
and m DFE = 90° by the Linear Pair Postulate the
two triangles are similar by the AA Similarity
Postulate; ∆CDF ~ ∆DEF.
GUIDED PRACTICE
3.
for Examples 1 and 2
Reasoning
Suppose in Example 2, part (b), SR
triangles still be similar? Explain.
TU . Could the
ANSWER
Yes; if S
T, the triangles are similar by the AA
Similarity Postulate.
EXAMPLE 3
Standardized Test Practice
EXAMPLE 3
Standardized Test Practice
SOLUTION
The flagpole and the woman form sides of two right
triangles with the ground, as shown below. The sun’s
rays hit the flagpole and the woman at the same
angle. You have two pairs of congruent angles, so the
triangles are similar by the AA Similarity Postulate.
EXAMPLE 3
Standardized Test Practice
You can use a proportion to find the height x. Write 5
feet 4 inches as 64 inches so that you can form two
ratios of feet to inches.
x ft = 50 ft
40 in.
64 in.
40x = 64(50)
x = 80
Write proportion of side lengths.
Cross Products Property
Solve for x.
ANSWER
The flagpole is 80 feet tall. The correct answer is C.
GUIDED PRACTICE
for Example 3
4. What If ? A child who is 58 inches tall is standing next
to the woman in Example 3. How long is the child’s
shadow?
ANSWER
36.25 in.
for Example 3
GUIDED PRACTICE
5.
You are standing in your backyard, and you
measure the lengths of the shadows cast by both
you and a tree. Write a proportion showing how
you could find the height of the tree.
SAMPLE ANSWER
tree height
your height
=
length of shadow
length of your shadow