Unit 6 (Part II) – Triangle Similarity Download

Transcript
Cholkar MCHS MATH II
___/___/___
Name____________________________
U3L1INV2
What combinations of side or angle measures are sufficient to determine that two
triangles are similar?
HW #
Complete Handout & CYU pg. 172 [CYU, 2, 7, 8]
Do Now
ACB ~ AED . If AC = 60, CE = 40 and AB = 75, what is BD ?
A
C
E
B
D
INVESTIGATION: SUFFICIENT CONDITIONS FOR SIMILARITY OF TRIANGLES (pg. 168)
My role for this investigation _________________________
1.
a. Calculate
b. Are
your reasoning.
uniquely determined? That is, is there exactly one value possible for each? Explain
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c. In general, if you know the values of b, c, and m<A, can values of a, m<B, and m<C always be found?
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Could any values of a, m<B, or m<C have two or more values when b, c, and m<A are given? Explain.
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d. Summarize your work in Parts a-c in an if-then statement that begins as follows:
In a triangle, if the lengths of two sides and the measure of the angle included between those sides are known,
then _____________________________________________________________________________________.
2. Next, examine
shown below. In both cases, you have information given about two sides
and an included angle. In
, k is a constant. Test if this information is sufficient to conclude that
using Parts a-d as a guide.
a.
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b. Use the information given for
to write and then solve an equation to find length x. Based on your
work in Problem 1, how is x related to a?
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c.
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d.
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3.
a.
If so, explain how. If not, explain why not.
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b. Complete the following statement:
SIDE-ANGLE-SIDE (SAS) SIMILARITY THEOREM: If an angle of one triangle has the same measure
as an angle of a second triangle, and if the lengths of the corresponding side including these angles are
multiplied by the same scale factor k, then _________________________________.
4. Suppose you know that in
as in the diagram below.
the lengths of corresponding sides are related by a scale factor k
a. What additional relationship would you need to know in order to conclude that
possible strategy you could use to establish the relationship?
What is a
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b. Write a deductive argument proving the relationship you stated in Part a. What can you conclude?
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SIDE-SIDE-SIDE (SSS) SIMILARITY THEOREM: If the lengths of the three sides of one triangle are
multiplied by the same scale factor k to obtain the lengths of the three sides of another triangle, then
_______________________________.
5.
ANGLE-ANGLE (AA) SIMILARITY THEOREM: If the measures of two angles of one triangle are the
same as the measures of two angles of another triangle, then ________________________________________.
Lesson Summary
In this investigation, you established three sets of conditions, each of which is
sufficient to prove that two triangles are similar.
Math Toolkit Vocabulary: Angle-Angle Similarity Postulate, Side-Side-Side (SSS) Similarity Theorem,
Side-Angle-Side (SAS) Similarity Theorem
Write a proof for the problem below.
Given: JN || LM
Prove: JKN MKL
Cholkar
MCHS
MATH II
___/___/___
Name____________________________
HW #
CYU pg. 172
1. Use the diagram to the right to complete the statement.
a.
b.
c.
d.
Determine whether the triangles are similar. If they are, write a similarity statement.
2.
5.
3.
4.
6. The shortest side of a triangle similar to
RST is 12 units long. Find the other side
lengths of the triangle.
REVIEW:
7. Find the values of x and y that make
ABC DEF .