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Triangle Congruence by ASA
and AAS
February 27, 2012
Warm-up
 Practice 4-2: Workbook p. 42, #1-15
Warm-up
Warm-up
Questions on Homework?
Questions on Homework?
#33:
Statements
Reasons
Questions on Homework?
41.Prove ΔFGK ≅ ΔKLF
42. Prove ΔACB ≅ ΔECD
AE and BD bisect each other
43. Prove ΔGJK ≅ ΔGMK
GK bisects ∠JGM
44. Prove ΔAMC ≅ ΔMBD
AB ┴ CM; AB ┴ DB; M is midpoint of AB
Section 4-3 Triangle Congruence
by ASA and AAS
 Objectives: Today you will learn to
 prove triangles congruent using the ASA
Postulate and AAS Theorem
 Remember: Quiz on Wednesday on
4-1, 4-2, and 4-3
Investigation
 Congruence Applet:
http://illuminations.nctm.org/ActivityDe
tail.aspx?ID=4
 Each Triangle Congruence Postulate
uses three elements (sides and
angles) to prove congruence.
 Today let’s investigate using Angles
(2 or 3) and/or one Side
Conclusions
 In which configuration(s) were the
triangles always congruent?
 Angle-Side-Angle (ASA) Postulate
 Angle-Angle-Side (AAS) Theorem
 In which configuration(s) were the
triangles sometimes congruent?
 Angle-Angle-Angle (AAA) – can’t
use!
Angle-Side-Angle (ASA) Postulate
If two angles and the included side of one
triangle are congruent to two angles
and the included side of another
triangle, then the two
triangles are congruent.
ΔTUV
≅ ΔWXY
Angle-Angle-Side (AAS) Theorem
If two angles and a nonincluded side of
one triangle are congruent to two
angles and the corresponding
nonincluded side of another triangle,
then the triangles are congruent.
ΔABC ≅ ΔDEF
Angle-Angle-Side (AAS) Theorem
Proof:
∠C ≅ ∠F
∠B and ∠E are right ∠’s
∠B ≅ ∠E
∠A ≅ ∠D
ΔABC ≅ ΔDEF
Given
Given
Rt ∠’s ≅
Thm 4.1
ASA Postulate
ΔABC ≅ ΔDEF
Four Ways to Prove Δ’s ≅
1. SSS
2. SAS
3. ASA
4. AAS
Postulate
Postulate
Postulate
Theorem
Which Postulate/Thm? (if possible)
and write Congruency Statement
Example 1: Prove: ΔAXP ≅ ΔBYP
Given: ≅ segments and angles as marked
Example 2: Prove ΔABC ≅ ΔCDA
Given: ≅ angles and || segments as marked
Example 3: Prove ΔABE ≅ ΔCDE
Example 4: Prove ΔJKL ≅ ΔPML
Example 5: Prove ΔQRT ≅ ΔSTR
Example 6: Find the values for x and y
Given: ΔABD ≅ ΔACD
Example 7: Find the values for x and y
Given: ΔABD ≅ ΔACD
If BC = 6, BD = ____
If m∠C = 55, m∠B = _____
and m∠BAC = _____
If m∠BAD = 40, m∠B = ____
Wrap-up
 Today you learned to prove triangles
congruent using the ASA and AAS
Postulates
 Tomorrow you’ll learn about the HL
Theorem and about and how to use CPCTC.
Homework:
pp. 197 – 199: 1 – 29,
31-34: write the proof if you can deduce
the conclusion from the given
 Quiz on Wednesday on 4-1, 4-2, and 4-3