Download Triangle Congruence Unit

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TEKS Focus:
(7)(B) Apply the Angle-Angle criterion to verify
similar triangles and apply the proportionality of
the corresponding sides to solve problems.
(1)(E) Create and use representations to
organize, record, and communicate
mathematical ideas.
(1)(F) Analyze mathematical relationships to
connect and communicate mathematical ideas.
(1)(G) Display, explain, or justify mathematical
ideas and arguments using precise
mathematical language in written or oral
communication.
(8)(A) Prove theorems about similar triangles,
including the Triangle Proportionality theorem,
and apply these theorems to solve problems.
Figures that are similar (~) have the same shape but
not necessarily the same size.
There are several ways to prove certain triangles
are similar. The following postulate, as well as the
SSS and SAS Similarity Theorems, will be used in
proofs just as SSS, SAS, ASA, HL, and AAS were
used to prove triangles congruent.
This was accidently left off of the notes packet. Copy the words.
A)
B)
Yes; vertical angles are congruent.
The triangles are similar by AA~.
The third angle in ΔJKL = 80.
The third angle in ΔPQR = 25.
The triangles are not similar.
Example 2:
Yes; ratios of corresponding sides are
6/8, 6/8, and 9/12. Those simplify to
3/4, 3/4, and 3/4. The triangles are
similar by SSS~.
Yes; ratios of corresponding sides are
6/12 and 8/16. Those simplify to
1/2 and 1/2. The right angle is the
Included angle in both triangles.
The triangles are similar by SAS~.
A similarity ratio is the ratio of the lengths of
the corresponding sides of two similar polygons.
The similarity ratio of ∆ABC to ∆DEF is
, or
The similarity ratio of ∆DEF to ∆ABC is
, or 2.
.
Writing Math
Writing a similarity statement is
like writing a congruence
statement—be sure to list
corresponding vertices in the
same order.
Example: 3
Determine whether the
polygons are similar. If so,
write the similarity ratio
and a similarity statement.
∆PQR and ∆STW
P and R = 72. W = 62 and T = 56.
Corresponding angles are not congruent.
The triangles are not similar.
Example: 4
Determine if ∆JLM ~ ∆NPS.
If so, write the similarity ratio
and a similarity statement.
Step 1 Identify pairs of congruent angles. Step 2 Compare corresponding sides.
∡N  ∡M, ∡L  ∡P (both given),
∡S  ∡J by Third ∡s Theorem.
Thus the similarity ratio is
, and ∆LMJ ~ ∆PNS.
Helpful Hint
When you work with proportions,
be sure the ratios compare
corresponding measures.
Example: 5
Explain why the triangles
are similar and write a
similarity statement.
Statements
A→
1. ∡B  ∡E
Reasons
1. Right Angles Theorem
2. m∡C = 47⁰, m∡C= 47⁰ 2. Triangle Sum Theorem
A→
3. ∡C  ∡F
3. Transitive Prop. of 
4. ∆ABC ~ ∆DEF
4. AA~
Example: 6
Example: 7
X ft
5.5 ft
6 ft
34 ft
Example: 8
Verify that the triangles are similar.
∆PQR and ∆STU
S→
S→
S→
Therefore ∆PQR ~ ∆STU by SSS ~.
Example: 9
Verify that the triangles are similar.
∆DEF and ∆HJK
S→
A→
∡D  ∡H by the Definition of Congruent Angles.
S→
Therefore ∆DEF ~ ∆HJK by SAS ~.
Example: 10
Verify that ∆TXU ~ ∆VXW.
S→
A→
∡TXU  ∡VXW by the Vertical Angles
Theorem.
S→
Therefore ∆TXU ~ ∆VXW by SAS ~.
Example: 11
Explain why ∆ABE ~ ∆ACD, and then
find CD.
Step 1 Prove triangles are similar.
A→
A→
∡EBA  ∡C since they are both right angles.
∡A  ∡A by Reflexive Property of 
Therefore ∆ABE ~ ∆ACD by AA ~.
Example: 11: continued
Step 2 Find CD.
Corr. sides are proportional. Seg. Add.
Postulate.
Substitute x for CD, 5 for BE, 3 for CB,
and 9 for BA.
x(9) = 5(3 + 9)
9x = 60
Cross Products Prop.
Simplify.
Divide both sides by 9.
Example: 12
Explain why ∆RSV ~ ∆RTU and
then find RT.
Step 1 Prove triangles are similar.
A→
given that ∡RSV  ∡T
A→
∡R  ∡R by Reflexive Property of 
Therefore ∆RSV ~ ∆RTU by AA ~.
Example: 12 continued
Step 2 Find RT.
Corr. sides are proportional.
Substitute RS for 10, 12 for TU, 8 for SV.
RT(8) = 10(12)
8RT = 120
RT = 15
Cross Products Prop.
Simplify.
Divide both sides by 8.