Download 4-5 - Midland ISD

LESSON 4–5
Proving Triangles
Congruent – ASA, AAS
Five-Minute Check (over Lesson 4–4)
TEKS
Then/Now
New Vocabulary
Postulate 4.3: Angle-Side-Angle (ASA) Congruence
Example 1: Use ASA to Prove Triangles Congruent
Theorem 4.5: Angle-Angle-Side (AAS) Congruence
Proof: Angle-Angle-Side Theorem
Example 2: Use AAS to Prove Triangles Congruent
Example 3: Real-World Example: Apply Triangle Congruence
Concept Summary: Proving Triangles Congruent
Over Lesson 4–4
Determine which postulate
can be used to prove that
the triangles are congruent.
If it is not possible to prove
congruence, choose not
possible.
A. SSS
B. ASA
C. SAS
D. not possible
Over Lesson 4–4
Determine which postulate
can be used to prove that
the triangles are congruent.
If it is not possible to prove
congruence, choose not
possible.
A. SSS
B. ASA
C. SAS
D. not possible
Over Lesson 4–4
Determine which postulate
can be used to prove that the
triangles are congruent. If it is
not possible to prove
congruence, choose not
possible.
A. SAS
B. AAS
C. SSS
D. not possible
Over Lesson 4–4
Determine which
postulate can be used to
prove that the triangles
are congruent. If it is not
possible to prove
congruence, choose not
possible.
A. SSA
B. ASA
C. SSS
D. not possible
Over Lesson 4–4
Determine which postulate
can be used to prove that
the triangles are
congruent. If it is not
possible to prove
congruence, choose not
possible.
A. AAA
B. SAS
C. SSS
D. not possible
Over Lesson 4–4
Given A  R, what sides must you know to be
congruent to prove ΔABC  ΔRST by SAS?
A.
B.
C.
D.
Targeted TEKS
G.6(B) Prove two triangles are congruent by applying
the Side-Angle-Side, Angle-Side-Angle, Side-Side-Side,
Angle-Angle-Side, and Hypotenuse-Leg congruence
conditions.
G.6(D) Verify theorems about the relationships in
triangles, including proof of the Pythagorean Theorem, the sum
of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to
solve problems. Also addresses G.5(C).
Mathematical Processes
G.1(B), Also addresses G.1(G)
You proved triangles congruent using SSS and
SAS.
• Use the ASA Postulate to test for
congruence.
• Use the AAS Theorem to test for
congruence.
• included side
Use ASA to Prove Triangles Congruent
Write a two-column proof.
Use ASA to Prove Triangles Congruent
Proof:
Statements
Reasons
1. L____
is the midpoint of
WE.
1. Given
2.
2. Midpoint Theorem
3.
3. Given
4. W  E
4. Alternate Interior Angles
5. WLR  ELD
5. Vertical Angles Theorem
6. ΔWRL  ΔEDL
6. ASA
Fill in the blank in the following
paragraph proof.
A. SSS
B. SAS
C. ASA
D. AAS
Use AAS to Prove Triangles Congruent
Write a paragraph proof.
__ ___
Proof: NKL  NJM, KL  MN, and N  N by the
Reflexive property. Therefore,
__ ___ΔJNM  ΔKNL by
AAS. By CPCTC, LN  MN.
Complete the following flow proof.
A. SSS
B. SAS
C. ASA
D. AAS
Apply Triangle Congruence
MANUFACTURING Barbara designs a paper
template for a certain envelope. She designs the top
and bottom flaps to be isosceles triangles that have
congruent bases and
base angles. If EV = 8 cm
and the height of the
isosceles triangle is 3 cm,
find PO.
Apply Triangle Congruence
____
In order to determine the length of PO, we must first
prove that the two triangles are congruent.
• ΔENV  ΔPOL by ASA.
____
____
____
____
____
____
• NV  EN by definition of isosceles triangle
• EN  PO by CPCTC.
• NV  PO by the Transitive Property of Congruence.
Since the height is 3 centimeters, we can use the
Pythagorean theorem to calculate PO. The altitude of the
triangle connects to the midpoint of the base, so each
half is 4. Therefore, the measure of PO is 5 centimeters.
Answer: PO = 5 cm
The curtain decorating the window forms 2 triangles
at the top. B is the midpoint of AC. AE = 13 inches
and CD = 13 inches. BE and BD each use the same
amount of material, 17 inches. Which method would
you use to prove ΔABE  ΔCBD?
A. SSS
B. SAS
C. ASA
D. AAS
LESSON 4–5
Proving Triangles
Congruent – ASA, AAS