Download Topic 4 : Congruent Triangles

Topic 4 : Congruent Triangles
Exploring Congruent Triangles
I. Types of Triangles A. Acute - all interior angles are less than 90
B. Right - one interior angle equals 90
C. Obtuse - one interior angle is greater than 90
D. Equiangular - all interior angles are congruent
E. Equilateral - all sides are congruent
F. Scalene - none of the sides are congruent
G. Isosceles - at least two sides are congruent
II. Parts of a Triangle A.
B. Right Triangle
C. Isosceles Triangle
III. Theorem - Properties of Congruent Triangles
IV. Definition of Congruent Triangles
A. Two triangles are congruent IFF their corresponding parts are congruent
(CPCTC).
B. Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
V. Examples A. Example #1:
B. Example #2:
End of Sec. 4.1
Sec.4.2: Measuring Angles in Triangles
I. Theorem - The sum of the measures of the interior angles of a triangle equals 180.
A. Applications 1. Example #1:
2. Example #2:
II. Theorem - If two angles of one triangle are congruent to two angles of a second
triangle, then the third angles are also congruent.
III. Theorem - The measure of an exterior angle of a triangle equals the sum of the
measures of the remote interior angles.
IV. Theorem - The measure of an exterior angle of a triangle is greater than either remote
interior angle.
A. An exterior angle of a triangle forms a L.P. with the adjacent interior angle of
the triangle.
B. Every triangle has six exterior angles that form three pairs of V.A.'s.
V. Corollary - a statement that is proven using a given theorem (is called a corollary of
that theorem.)
VI. Corollary 4.1 - the acute angles of a right triangle are complementary.
VII. Corollary 4.2 - there can be at most, one right or obtuse angle in any given triangle.
VIII. Sec. 4.2 Proofs
A.
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B.
Proving Triangles Congruent
I. Postulate (SSS) - If each of the three sides of one triangle are congruent respectively, to
corresponding sides of a second triangle, then the two triangles are congruent.
II. Postulate (SAS) - If two sides and the included angle of one triangle are congruent to
two corresponding sides and the included corresponding angle of a second triangle, then
the two triangles are congruent.
III. Postulate (ASA) - If two angles and the included side of one triangle are congruent to
two corresponding angles and the included corresponding side of a second triangle, then
the two triangles are congruent.
IV. Examples A. Example #1:
B. Example #2:
More Congruent Triangles
I. Theorem (AAS) - If two angles and a non-included side of one triangle are congruent
to the corresponding two angles and a non-included side of a second triangle, then the
triangles are congruent.
II. Proof Using AAS A.
B.
III. Additional Proofs - deduce that segments and angles are congruent by first proving
that triangles are congruent.
A.
B.
IV. Overlapping Triangles - Congruent triangles are often overlapping triangles.
A. Example #1:
Isosceles and Right Triangles
II. Hypotenuse-Leg Theorem - If the hypotenuse and a leg of one right triangle are
congruent to the hypotenuse and corresponding leg of a second right triangle, then the
two triangles are congruent.
A. Proof Using H.L. -
III. Analyzing Isosceles Triangles A. Theorem - If two sides of a triangle are congruent, then the angles opposite
those sides are also congruent.
B. Theorem - If two angles or a triangle are congruent, then the sides opposite
those angles are also congruent.
C. Corollary - A triangle is equilateral IFF it is equiangular.
D. Corollary - Each interior angle of an equilateral triangle measures 60.
E. Corollary - The segment from the vertex angle of an isosceles triangle to the
midpoint of the base, bisects the vertex angle.
IV. Algebraic Examples A. Example #1:
B. Example #2:
V. Proof Example -