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Chapter 4 Congruent Triangles
In this chapter, you will:classify triangles by their parts,apply the Angle Sum Theorem and the
Exterior Angle Theorem,recognize SSS, SAS, ASA, AAS, and HL
triangle congruence,use properties of isosceles and
equilateral triangles.
4-1: Classifying Triangles
A triangle is a figure formed by three segments joining three noncollinear points. A triangle can be classified by its sides and by its angles.
sides: AB, BC, ACvertices: A, B, CAngles:<A, <B, <Cm<A+ m<B + m<C = 180°
A
B C
Each of the three points joining the sides of a triangle is a vertex. (The plural of vertex is vertices.) For example, in ∆ ABC, points A, B, and C are vertices.
In a triangle, two sides sharing a common vertex are adjacent sides. In ∆ABC, CA and BA are adjacent sides. The third side, BC, is the side opposite <A.
Side opposite <A
C
B A
Right and Isosceles Triangles: The sides of right triangles and isosceles triangles have special names. In a right triangle, the sides that form the right angle are the legs of the triangle. The side opposite the right angle is the hypotenuse of the triangle.
An isosceles triangle can have three congruent sides, in which case it is equilateral. When an isosceles triangle has only two congruent sides, then these two sides are the legs of the isosceles triangle. The third side is the base of the isosceles triangle. Vertex angle
hypotenuse leg
leg
leg leg
base
Theorem 4-2 Triangle Sum Theorem
The sum of the measures of the interior angles of a triangle is 180º.
m<A + m<B + m<C = 180
A
B
C
In this section, you will learn:
To apply the Angle Sum Theorem.To apply the Exterior Angle Theorem.
Triangle Angle-Sum Theorem
The sum of the measures of the angles of a triangle is 180°.
M<A + m<B + m<C =180°.
A
BC
Corollary – A statement that can be proved easily using a theorem or a definition.
Corollary to the Triangle Sum Theorem
The acute angles of a right triangle are complementary.
m<A + m<B = 90º
A
C
B
Example 3
Example 4: Finding angle measures
The measure of one acute angle of a right triangle is two times the measure of the other acute angle. Find the measure of each acute angle.
x + 2x = 90
3x = 90
x = 30
M<A = 30º
M<B = 60º
2x
xA
B
Theorem 4-3 Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.
m<1 = m<A + m<B
A
B
C
1
Example 6
Example 3: Finding an Angle Measure Find the measure of the exterior angle. First write and solve an
equation to find the value of x.
2x + 10 = x + 65
2x = x + 55
x = 55
So, the measure of the exterior angle is (2 ∙ 55 + 10)º, or 120
x
65
2x + 10
Example 7
Find the value of x, then find the measure of the exterior angle.
2x – 11 = 72 + x
2x = 83 + x
x = 83
2(83) – 11
166 – 11
155
So, the measure of the exterior angle is 155.
x
72 2x-11