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A triangle is uniquely defined by any one of these conditions:
SSS (knowing all 3 side lengths)
SAS (knowing 2 sides and the angle between them)
ASA (knowing 2 angles and the length of the side between them)
AAS (equivalent to ASA, since if we know 2 angles, we can compute the 3rd)
To show two triangles are congruent, we need to show that they have enough equivalent corresponding sides/angles, so that one of the above options is satisfied.
For this pair of triangles, we see that
¯¯¯¯¯¯
L
Y
≅
¯¯¯¯¯¯
U
G
,
since they both have length 7.2.
Next, both triangles have a side of length 6, those being
¯¯¯¯¯¯
Y
F
and
¯¯¯¯¯¯
G
B
,
meaning
¯¯¯¯¯¯
Y
F
≅
¯¯¯¯¯¯
G
B
.
Finally,
m
∠
Y
=
56
∘
,
and so is
m
∠
G
.
Thus,
∠
Y
≅
∠
G
.
Since all three of these congruent values appear in the same order in each triangle (i.e. in SAS order), this shows
△
L
Y
F
≅
△
U
G
B
.
It is important to list the vertices in this order, because it's the order in which they match up (or correspond). That is:
L
goes with
U
,
Y
goes with
G
, and
F
goes with
B
.
To determine the missing values, we look at the corresponding value in the other triangle. For instance,
x
=
B
U
,
and since
¯¯¯¯¯¯
B
U
≅
¯¯¯¯¯¯
F
L
, and
F
L
=
6.3
,
that gives us
x
=
6.3
.
Similarly,
z
∘
=
m
∠
B
=
m
∠
F
=
72
∘
,
and
y
∘
=
m
∠
U
=
m
∠
L
=
52
∘
.