Download Mod 1 - Aim #24 - Manhasset Schools

Aim #24: How do we prove triangles congruent by AAS and HL?
CC Geometry H
Do Now:
Given: DE ≅ DG, EF ≅ GF
Prove: DF is the angle bisector of≮EDG
Statement
Reason
Let's examine threepossible triangle congruence criteria:
Angle-Angle-Side (AAS) and Side-Side-Angle (SSA), and Angle-Angle-Angle (AAA)
Angle-Angle-Side triangle congruence criteria (AAS): Given two Δs ABC and
A'B'C'. If m≮B = m≮B' (angle) and m≮C = m≮C' (angle) andAB = A'B' (side), then the
triangles are congruent.
Mark the triangles to show AAS:
In the diagram below, consider a pair of triangles that meet the
AAS criteria. If
you knew that two angles of one triangle corresponded to and were equal in measure
to two angles of the other triangle, what conclusion can you draw about the third
angles of each triangle? _________________________
Given this conclusion, which formerly learned triangle congruence criteria
can we use to
determine if the pair of triangles are congruent?
Therefore, the AAS
criterion is actually an extension of the
triangle congruence criterion.
Hypotenuse-Leg triangle congruence criteria (HL) : Given two right Δs ABC
and A'B'C' with right angles B and B'. If AB = A'B' (leg) OR BC = B'C' (leg) and
AC = A'C'(hypotenuse), then the triangles are congruent.
Below, ΔABC and ΔA'B'C' have brought together with corresponding angles
≮A = ≮A' and ≮C = ≮C'. The hypotenuse acts as a common side to the triangles.
B
B
A
C
A'
C'
B'
A
C
B
B'
We draw auxiliary line BB': A
Proof of the HL Theorem:
Statements
Reasons
1. AB = AB'
2. ≮______ = ≮_______
3. ≮ CBB' and ≮ ABB'are
complementary; ≮ CB'B and
≮ AB'B are complementary
4. ≮ CBB' = ≮ ______
5. ____ = _____
C
B'
1. Given
2. If two sides of a Δ are =, the angles opposite are =.
3. _________________________________________
4. Complements of equal angles are equal.
5. _________________________________________
6. ____ = _____
6. Reflexive Property
7. ΔABC ≅ ΔA'B'C'
7. ______________________
*When using HL in a proof, you must state as a reason the triangles are rightΔs.*
Criteria that do NOT determine two triangles as congruent: SSA and AAA
Side-Side-Angle (SSA): Observe the diagrams below. Each triangle has a set of
adjacent sides of measures 11 and 9, as well as the non-included angle of 23˚.
Yet, the triangles are not congruent
Examine the diagram below which is made of both triangles. The sides of lengths 9
each have been dashed to show their possible locations.
SSA cannot guarantee congruence criteria. T wo triangles under SSA criteria might
be congruent, but they might not be. W e cannot use SSA in proofs!
Angle-Angle-Angle (AAA):
ΔABC is an isosceles right triangle.
B
ΔDEF is also an isosceles right triangle.
E
D
F
C
What does this mean about the angles of the two triangles?
A
Why can‛t we categorize AAA as a congruence criteria?
Even though the angle measures may be the same, the sides can be proportionally
larger; you can have ___________triangles which may or may not be congruent.
1) Given: BC
Τ
CD, AB
Τ
AD , m≮1 = m≮2
Prove: ΔBCD ≅ ΔBAD
Statements
Reasons
3
4
Τ
Τ
2) Given: AD BD, BD BC, AB ≅ CD
Prove ΔABD ≅ ΔCDB
Reasons
Statements
B
3) Given: BD is an angle bisector of ≮ABC , ≮BAD ≅≮BCD.
Prove: ΔADC is isosceles.
Statements
D
A
Reasons
Let's Sum it Up!!
Criteria that can be used for triangle congruence:
SAS
ASA
SSS
AAS
Criteria that CANNOT be used for triangle congruence:
SSA
AAA
HL
C
Name_________________________
Date ________________
Τ
Τ
1) Given: AB BC, DE EG, BC ll EF, AF = DC
Prove: ΔABC ≅ ΔDEF
Statements
1)
Reasons
1) Givens
o
o
2) ≮B = 90 , ≮E = 90
3) ≮B = ≮E
4) ≮BCA = ≮EFD
5) FC = FC
6) AF + FC = DC + FC
7) AC = AF + FC
DF = DC + FC
8) AC = DF
9) ΔABC ≅ ΔDEF
2)
3)
4)
5)
6)
7)
8)
9)
Τ
Τ
2) Given: PA AR and PB BR and R is equidistant from PA and PB.
Prove: a) ΔPAR ≅ ΔPRB
b) PR bisects ≮APB
Statements
Reasons
CC Geometry H
HW #24
3) D
C
≅ EA, CE ≅ DE
Given: EB
Prove: Δ ACB ≅ Δ BDA
E
,≮CBA ≅ ≮DAB
B
A
Statements
Reasons
Review:
1) Parallel lines x and y are cut by transversal z. Ray w is perpendicular to line z.
o
If the m ≮1 = 58 , find the remaining numbered angles. State the geometric
z
w
reason for each step.
2
x
y
1
3
4
5