Geometry Section 4-5 1112

SECTION 4-5
Proving Congruence: ASA, AAS

ESSENTIAL QUESTIONS
How do you use the ASA Postulate to test for triangle
congruence?
How do you use the AAS Postulate to test for triangle
congruence?

VOCABULARY
1. Included Side:
Postulate 4.3 - Angle-Side-Angle (ASA) Congruence:
Theorem 4.5 - Angle-Angle-Side (AAS) Congruence:

VOCABULARY
1. Included Side: The side between two consecutive
angles in a triangle
Postulate 4.3 - Angle-Side-Angle (ASA) Congruence:

VOCABULARY
Postulate 4.3 - Angle-Side-Angle (ASA) Congruence: If
two angles and the included side of one triangle are
congruent to two angles and included side of a
second triangle, then the triangles are congruent

VOCABULARY
Postulate 4.3 - Angle-Side-Angle (ASA) Congruence: If
two angles and the included side of one triangle are
congruent to two angles and included side of a
second triangle, then the triangles are congruent
Theorem 4.5 - Angle-Angle-Side (AAS) Congruence: If
two angles and the nonincluded side of one triangle
are congruent to the corresponding angles and
nonincluded side of a second triangle, then the
triangles are congruent

EXAMPLE 1
Prove the following.
Prove: △WRL ≅△EDL
Given: L is the midpoint of WE, WR ! ED

EXAMPLE 1
1. L is the midpoint of WE, WR ! ED

EXAMPLE 1
1. Given

EXAMPLE 1
1. Given
2. WL ≅ EL

EXAMPLE 1
1. Given
2. WL ≅ EL 2. Midpoint Thm.

EXAMPLE 1
1. Given
3. ∠WLR ≅ ∠ELD

EXAMPLE 1
1. Given
3. ∠WLR ≅ ∠ELD 3. Vertical Angles Thm.

EXAMPLE 1
1. Given
4. ∠LWR ≅ ∠LED

EXAMPLE 1
1. Given
4. ∠LWR ≅ ∠LED 4. Alternate Interior Angles Thm

EXAMPLE 1
1. Given
5. △WRL ≅△EDL

EXAMPLE 1
1. Given
5. △WRL ≅△EDL 5. ASA

EXAMPLE 2
Prove: LN ≅ MN
Given: ∠NKL ≅ ∠NJM, KL ≅ JM

EXAMPLE 2
Prove: LN ≅ MN
1. ∠NKL ≅ ∠NJM, KL ≅ JM

EXAMPLE 2
Prove: LN ≅ MN
1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given

EXAMPLE 2
Prove: LN ≅ MN
2. ∠N ≅ ∠N

EXAMPLE 2
Prove: LN ≅ MN
2. ∠N ≅ ∠N 2. Reﬂexive

EXAMPLE 2
Prove: LN ≅ MN
3. △JNM ≅△KNL

EXAMPLE 2
Prove: LN ≅ MN
3. △JNM ≅△KNL 3. AAS

EXAMPLE 2
Prove: LN ≅ MN
4. LN ≅ MN

EXAMPLE 2
Prove: LN ≅ MN
4. LN ≅ MN 4. Corresponding Parts
of Congruent Triangles
are Congruent (CPCTC)

EXAMPLE 3
On a template design for a certain envelope, the top
and bottom ﬂaps are isosceles triangles with congruent
bases and base angles. If EV = 8 cm and the height of
the isosceles triangle is 3 cm, ﬁnd PO.

EXAMPLE 3
EV ≅ PL, so each segment has a measure
of 8 cm. If an auxiliary line is drawn
from point O perpendicular to PL, you
will have a right triangle formed.

EXAMPLE 3
EV ≅ PL, so each segment has a measure
of 8 cm. If an auxiliary line is drawn
from point O perpendicular to PL, you
will have a right triangle formed.
In the right triangle, we have one leg (the height) of 3 cm.
The auxiliary line will bisect PL, as point O is equidistant
from P and L.

EXAMPLE 3
a2
+ b2
= c2
42
+ 32
= (PO)2

EXAMPLE 3
a2
+ b2
= c2
42
+ 32
= (PO)2
16 + 9 = (PO)2

EXAMPLE 3
a2
+ b2
= c2
42
+ 32
= (PO)2
16 + 9 = (PO)2
25 = (PO)2

EXAMPLE 3
a2
+ b2
= c2
42
+ 32
= (PO)2
16 + 9 = (PO)2
25 = (PO)2
25 = (PO)2

EXAMPLE 3
a2
+ b2
= c2
42
+ 32
= (PO)2
16 + 9 = (PO)2
25 = (PO)2
25 = (PO)2
PO = 5 cm

PROBLEM SET
p. 276 #1-23 odd, 35
“There is only one you...Don’t you dare change just because
you’re outnumbered.” - Charles Swindoll

Geometry Section 4-5 1112

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Geometry Section 4-5 1112