Embed Size (px)
DESCRIPTION
4.4 Proving Triangles are Congruent: ASA and AAS. Objectives/Assignment. Prove that triangles are congruent using the ASA Congruence Postulate and the AAS Congruence Theorem Use congruence postulates and theorems in real-life problems Assignment: 2-22 even, 32-37 all, quiz page 227. - PowerPoint PPT Presentation
Citation preview
4.4 Proving Triangles are 4.4 Proving Triangles are Congruent: ASA and Congruent: ASA and
AASAAS
Postulate 21: Angle-Side-Angle Postulate 21: Angle-Side-Angle (ASA) Congruence Postulate(ASA) Congruence Postulate• If two angles and the
included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent.
B
C
A
F
D
E
Goal 1: Using the ASA and AAS Congruence Methods
Theorem 4.5: Angle-Angle-Side Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem(AAS) Congruence Theorem• If two angles and a non-
included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent.
B
C
A
F
D
E
Theorem 4.5: Angle-Angle-Side Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem(AAS) Congruence Theorem
Given: A F, C D, BA EF
Prove: ∆ABC ∆DEF
B
C
A
F
D
E
Theorem 4.5: Angle-Angle-Side Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem(AAS) Congruence Theorem
You are given that two angles of ∆ABC are congruent to two angles of ∆DEF. By the Third Angles Theorem, the third angles are also congruent. That is, B E. Notice that BC is the side included between B and C, and EF is the side included between E and F. You can apply the AAS Congruence Postulate to conclude that ∆ABC ∆DEF.
B
C
A
F
D
E
Example 1: Developing ProofExample 1: Developing Proof
Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.
G
E
JF
H
Example 1: Developing ProofExample 1: Developing Proof In addition to the angles
and segments that are marked, EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. You can use the AAS Congruence Theorem to prove that ∆EFG ∆JHG.
G
E
JF
H
Example 1: Developing ProofExample 1: Developing Proof
Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.
N
M
Q
P
Example 1: Developing ProofExample 1: Developing Proof In addition to the
congruent segments that are marked, NP NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent.
N
M
Q
P
Example 1: Developing ProofExample 1: Developing Proof Is it possible to prove
the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.
UZ ║WX AND UW ║ ZX
U
W
Z
X
12
34
Example 1: Developing ProofExample 1: Developing Proof
The two pairs of parallel sides can be used to show 1 3 and 2 4. Because the included side WZ is congruent to itself, ∆WUZ ∆ZXW by the ASA Congruence Postulate.
U
W
Z
X
12
34
Example 2: Proving Triangles Example 2: Proving Triangles are Congruentare Congruent
Given: AD ║CE, BD BC
Prove: ∆ABD ∆EBC
Plan for proof: Notice that ABD and EBC are congruent. You are given that BD BC
. Use the fact that AD ║EC to identify a pair of congruent angles.
B
A
ED
C
ProofProof
Statements:1. BD BC2. AD ║ EC3. D C4. ABD EBC5. ∆ABD ∆EBC
Reasons:1. Given2. Given3. Alternate Interior
Angles4. Vertical Angles
Theorem5. ASA Congruence
Theorem
B
A
ED
C
NoteNote
• You can often use more than one method to prove a statement.
