Download Lesson 124: Conditions of Congruence, Proofs of Congruence

Lesson 124:
Conditions of
Congruence, Proofs of
Congruence, Isosceles
Triangles
A triangle has three sides and
three angles. Two triangles are
congruent if any one of the four
conditions state next are met.
1. SSS: if the lengths of the sides
in one triangle are equal to the
lengths of the sides in a second
triangle, the triangles are
congruent. We call this
condition side-side-side (SSS).
2. AAAS: if the angles in one triangle
have the same measures as the
angles in a second triangle, the
triangles are similar. If a side in one of
a pair of similar triangles has the same
length as the corresponding side in
the other triangle, the scale factor is 1
and the triangles are congruent. We
call this condition angle-angle-angleside (AAAS).
3. SAS: if two sides and the
included angle in one triangle
have the same measures as two
sides and the included angle in
a second triangle, the triangles
are congruent. We call this
condition side-angle-side (SAS).
4. HL: if the lengths of the
hypotenuse and a leg in one
right triangle equal the lengths
of the hypotenuse and a leg in
a second right triangle, the right
triangles are congruent. We call
this condition hypotenuse-leg
(HL).
These are also the conditions for
triangle similarity. If the lengths in
SSS, SAS, and HL are not equal but
are proportional, the triangles are
similar. As a statement of similarity,
the “side” in AAAS is unnecessary
because AAA is sufficient to
designate similarity.
We remember that when we write
a statement of similarity or
congruence we must list
corresponding vertices in the same
order. Here we show two
congruent triangles.
Many proofs require that we show that two
segments have equal lengths or that two
angles have equal measures. This is often
accomplished by showing that two triangles
are congruent. Thus the components have
equal measures, because corresponding
parts of congruent triangles are congruent
(CPCTC). If we can outline a proof first,
writing the formal proof is easy. To outline a
proof, we first sketch the figure and use tick
marks to record the given information on the
figure. Then we write the statement of
congruence, being careful to list the vertices
in corresponding order. Then we write AAAS,
SSS, SAS or HL to show why the triangles are
congruent.
Example:
______
______
Given: AD ≅ DB
_______
______
DC AB
Outline ______
a proof
that
shows:
______
AC ≅ BC
Answer:
ΔADC
≅ ΔBDC by SAS
_____
______
AC ≅ BC by CPCTC
Open books to Page 506.
HW: Lesson 124 #1-30