Download 3-3 Parallel Lines and the Triangle Angle

3-3 PARALLEL LINES AND THE TRIANGLE ANGLE-SUM THEOREM
(p. 131-139)
Do the Investigation: The Sum of the Three Angle Measures on p. 131. Each person
could do this individually.
Theorem 3-7 Triangle Angle-Sum Theorem
The sum of the measures of the angles of a triangle is 180.
See the proof for Theorem 3-7 on p. 132
Why does m 1  m 2  m 3  180 ?
What property allows you to replace m 1 with
Example: Using the following diagram, find
m
m
A and m 2 with
m
B?
Z.
Z
54
65
Y
X
Do 1a and b on p. 132.
Example: In the following diagram of ABC, ACB is a right angle and CD  AB.
Find the values of x, y, and z.
C
z 64
y
x
A
D
B
Triangles can be classified according to their angle measures and their side lengths.
An acute triangle is a triangle where all three angles are acute.
Example: Sketch an acute triangle where all three angle measures are different. Place
angle measures next to the angles.
A right triangle is a triangle with a right angle.
Example: Sketch a right triangle where all three angle measures are different. Place
angle measures next to the angles. Can a right triangle have two congruent angles?
An obtuse triangle is a triangle with one obtuse angle.
Example: Sketch an obtuse triangle where all three angle measures are different. Place
angle measures next to the angles. Can an obtuse triangle have two congruent angles?
An equiangular triangle is a triangle where all three angles are congruent.
Example: Sketch an equiangular triangle and show the measure of each angle of the
triangle.
A scalene triangle is a triangle with no congruent sides.
Example: Sketch a scalene triangle and put side lengths next to the three sides. Make
sure that the sum of the lengths of any two sides is greater than the length of the third side
(this is called the triangle inequality relationship). Can a scalene triangle have side
lengths of 5 cm, 11 cm, and 6 cm?
An isosceles triangle is a triangle with at least two congruent sides.
Example: Sketch an isosceles triangle with exactly two congruent sides. Put side
lengths next to the three sides of the triangle. Make sure you satisfy the triangle
inequality relationship. Can an isosceles triangle have three congruent sides?
An equilateral triangle is a triangle with all three sides congruent.
Example: Sketch an equilateral triangle. Use tick marks to indicate congruent sides. Is
an equilateral triangle always isosceles? Is an isosceles triangle always equilateral?
Example: Classify the following triangle by its side lengths and angle measures.
H
10 dm
7 dm
F
107
5 dm
G
Do 3 on p. 133.
An exterior angle of a triangle is an angle formed by a side and an extension of an
adjacent side of the triangle.
Example: Sketch a triangle and form an exterior angle at the obtuse angle of the
triangle.
What is the relationship between the exterior angle of the triangle and its adjacent interior
angle?
The remote interior angles of an exterior angle are the two nonadjacent interior angles.
What does the term remote mean? In your previous sketch, identify the two remote
interior angles.
There is a special relationship between an exterior angle of a triangle and its two remote
interior angles.
Theorem 3-8 Triangle Exterior Angle Theorem
The measure of each exterior angle of a triangle equals the sum of the measures of
its two remote interior angles.
Example: Sketch a triangle and one of its exterior angles. Place angle measures next to
the three interior angles of the triangle and the exterior angle of the triangle so that your
choice of numbers satisfies our two new theorems in this section.
You will prove Theorem 3-8 in a homework problem.
Example: In the following diagram, BC  AC. Find the measure of A by using the
Triangle Exterior Angle Theorem. Is there another way that you can determine A?
D
117
B
C
A
Do 4 a and b on p. 134.
Examine the diagram in Ex. 5 on p. 134. Then, do 5 a and b.
Homework p. 134-138: 2,9,10,14,22,23,25,28,29,33,35,40,43,46,49,55,64,66,74
10. Solve 3x  90
55. Make some sketches. 90 