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Chapter 4a: Congruent Triangles
By: Nate Hungate, Gary Russell, J.P. Lawrence, Kyle Stegman
4.1
Triangles and Angles
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Triangle- A shape formed by three segments joining three non collinear
points.
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Vertex- The three points that join the sides of a triangle.
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Adjacent Sides- Two sides that share a common vertex.
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Legs- Sides that form the right angle of a triangle.
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Hypotenuse- The side opposite of the right angle.
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Base- In an isosceles with two congruent sides, the third side is the base.
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Interior Angles- The three angles inside a triangle.
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Exterior Angles- Angles outside of a triangle.
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Corollary- A corollary to a theorem is a statement that is easily proven using
the theorem.
Names of Triangles
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Classification by sides
Three congruent sides=
equilateral
Two congruent sides=
isosceles
No congruent sides=
scalene
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Classification by angles
Three acute angles=
acute
Three congruent
angles= equiangular
One right angle= right
One obtuse angle=
obtuse
Classifying Triangles
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If triangle ABC has
one obtuse angle and
two congruent sides,
then it is a
________________.
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If triangle DEF has
three congruent
angles and three
congruent sides, then
it is a
________________.
Theorems
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Theorem 4.1:
Triangle Sum
Theorem= The sum
of the measures of
the interior angles
of a triangle is 180.
Corollary to the
Triangle theorem:
The acute angles of
a right triangle are
complementary
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Theorem 4.2:
Exterior Angle
Theorem= the
measure of an
exterior angle of a
triangle is equal to
the sum of the
measures of the two
nonadjacent interior
angles.
Finding Angle Measures
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Classify the triangle
by its angles and by
its sides:
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Find the measure of
the exterior angle
shown:
30
(3x+12)
4.2
Congruence and Triangles
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Congruent- Two figures are congruent if they
are the same size and shape.
Corresponding Angles- When two figures are
congruent, the corresponding angles are in
corresponding positions and are congruent.
Corresponding Sides- When two figures are
congruent, the corresponding sides are the
sides that are in corresponding positions, and
they are congruent
Naming Congruent Parts of a
Triangle
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Identify all pairs of congruent corresponding
parts:
Theorems
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Theorem 4.3: Third Angles Theorem= If two angles of
one triangle are congruent to two angles of another
triangle, then the third angles are also congruent.
Theorem 4.4: Properties of Congruent Triangles:
Reflexive Property of Congruent Triangles- Every
triangle is congruent to itself.
Symmetric Property of Congruent Triangles- If
triangle ABC is congruent to triangle DEF, then
triangle DEF is congruent to triangle ABC.
Transitive Property of Congruent Triangles- If
triangle ABC is congruent to triangle DEF and
triangle DEF is congruent to triangle JKL, then
triangle ABC is congruent to triangle JKL.
4.3
Proving Triangles are Congruent: SSS
and SAS
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SSS- If three sides
of one triangle are
congruent to three
sides of a second
triangle, then the
two triangles are
congruent.
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SAS- If two sides
and the included
angle of one triangle
are congruent to
two sides and the
included angle of a
second triangle,
then the two
triangles are
congruent.
Using the SSS Congruence
Postulate
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The marks on the triangle show that segment NM is
congruent to segment OL, segment NO is congruent
to segment ML, and segment MO is congruent to MO
(reflexive property). So by SSS triangle NMO is
congruent to triangle LOM.
Using the SAS Congruence
Postulate
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The marks on the triangles show that segment AB is
congruent to DE, angle B is congruent to angle E, and
that segment BC is congruent to EF. So by SAS
triangle ABC is congruent to triangle DEF.
E
A
F
D
F
B
C
Practice
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Prove the two triangles
are congruent.
A
B
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Prove the two triangles
are congruent.
4.4
Proving Triangles are Congruent: ASA
and AAS
• ASA- If two angles and • AAS- If two angles and
the side the included side a non-included side of
of one triangle are
one triangle are
congruent to two angles
congruent to two angles
and the included side of
and the corresponding
a second triangle, then
non-included side of a
the two triangles are
second triangle, then the
congruent.
two triangles are
congruent.
Using the ASA Congruence
Postulate
• Determine if the two triangles
can be proven congruent. If so
explain.
• Determine if the two triangles
can be proven congruent. If so
explain.
Using the AAS Congruence
Postulate
• Determine if the two triangles
can be proven congruent. If so
explain.
• Determine if the two triangles
can be proven congruent. If so
explain.