Download Chapter 4 Lecture Notes

Chapter 4 Lecture Notes
Sec.4.1: 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 4.1 - 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 4.2 (Angle Sum 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 4.3 (Third Angle 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 4.5 (Exterior Angle Theorem) - The measure of an exterior angle of a
triangle equals the sum of the measures of the remote interior angles.
IV. Theorem 4.6 (Exterior Angle Inequality 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.
End of Sec. 4.2
Sec.4.3: Proving Triangles Congruent
I. Postulate 17 (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 18 (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 19 (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:
End of Sec. 4.3
Sec.4.4-4.5: More Congruent Triangles
I. Theorem 4.7 (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:
End of 4.4-4.5
Sec. 4.6: Isosceles and Right Triangles
I. Theo. 4.10 (Hypotenuse-Angle Theo.) - If the hypotenuse and an acute angle of one
right triangle are congruent to the hypotenuse and corresponding acute angle of a second
right triangle, then the two triangles are congruent.
II. Theo. 4.12 (Hypotenuse-Leg Theo.) - 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. Theo. 4.8 (Angles Opposite Theo.) - If two sides of a triangle are congruent,
then the angles opposite those sides are also congruent.
B. Theo. 4.9 (Sides Opposite Theo.) - If two angles or a triangle are congruent,
then the sides opposite those angles are also congruent.
C. Corollary 4.7 - a triangle is equilateral IFF it is equiangular.
D. Corollary 4.8 - Each interior angle of an equilateral triangle measures 60.
E. Corollary 4.10 - 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 -
End of Chpt. 4