Download Module 5 Lesson 1: Investigating Angles of Triangles Characteristics

Module 5 Lesson 1: Investigating Angles of Triangles
Characteristics of Triangles
Triangles may be characterized by their sides or by their angles.
Scalene – all sides of the triangle are of different lengths
Isosceles – two sides of the triangle are congruent.
Equilateral – all sides of the triangle are congruent.
Lengths of the sides of the triangle are calculated to determine if the triangle is scalene, isosceles or
equilateral.
Acute – all angles of the triangle measure less than 90 o.
Right – one angle of the triangle measures exactly 90 o. The other two angles are acute angles.
Obtuse – one angle of the triangle measures greater than 90 o. The two remaining angles are acute
angles.
Equiangular – all angles of the triangle are congruent.
Triangles in the Coordinate Plane
In Unit 1 you learned how to calculate the length of a segment. Let’s review the distance formula:
For any two points A(x1, y1) and B(x2, y2), the length of
AB =
is found by the formula
EXAMPLE: Triangle JAM has vertices at J(-3, 2), A(4, -5) and M(6, 3). What kind of triangle is JAM?
y
9
8
7
6
5
4
3
(-3,2)
2
1
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
-1
8
9
(8,-1)
-2
-3
-4
-5
(4,-5)
-6
-7
-8
-9
Let’s check the lengths of the sides of the triangle.
JA =
=
=
AM =
=
JM =
=
=7
=
.
=
=
=4
=
All three of the side lengths are different values, so we can conclude the triangle is a scalene triangle.
Two sides of any triangle cannot be parallel because triangles are formed by intersecting lines.
However, two sides of a triangle may be perpendicular. A triangle with a pair of perpendicular sides is
a right triangle. Let’s check next to see if the two sides that appear to be perpendicular are, in fact,
perpendicular.
’s slope is
=
= -1.
slope is
.
The product of the slopes is -1, therefore the sides of the triangle are perpendicular.
ANSWER: The final result is JAM is a right scalene triangle.
Part 2: The Sum of Three Angle Measures
Take a sheet of paper and draw a large triangle on it. Label the interior of the angles A, B and C.
Tear each of the angles off the triangle. Line up the angles so they are adjacent to each other.
What is your conjecture about the sum of the measures of the angles of a triangle? Do you see that
<A, <B and <C create a line?
We have an important proof related to this conjecture.
Triangle Angle Sum Theorem
The sum of the measures of the angles of a triangle is 180 o.
o
.
More notes – keep scrolling down…….
Examples of Triangle Angle Sum Theorem:
1. Find the value of y.
Solution: y+ 102+30 = 180
y + 132 = 180
y = 48
2. Find
if
Solution:
.
Now “You Try” these problems. (The solutions to the You Try problems are at the end of these notes.)
1. Solve for x.
2. Solve for x.
3. Find the measures of all 3 angle measures of
Part 3: Triangle Exterior Angle Theorem
The next important theorem relates an exterior angle of a triangle and the remote interior angles. See
the figure below that identifies an exterior angle and the remote angles.
Recall that <1 and < 4 create a linear pair and all linear pairs are supplementary.
So we have m<1 + m<4 = 180o. By the Triangle Angle Sum Theorem we have
m<2 + m<3+m<4 = 180 o. These two sums are equal to each other by the Transitive Property of
Equality. The resulting equation is
m<1 + m<4 = m<2 + m<3 + m<4.
Then m<1 = m<2 + m<3 by subtraction of m<4 from both sides. This result is the theorem.
Triangle Exterior Angle Theorem
The measure of any exterior angle of a triangle is the sum of the measures of its two remote interior
angles.
Now, “You Try” these problems. The solutions can be found at the end of these notes.
4. Solve for x:
5. Solve for x:
7. Find the m<X.
8. Find the m<QTS.
6. Find the measure of the exterior angle.
Part 4: Isosceles & Equilateral Triangles
Let’s focus now on two types of triangles in particular – the isosceles triangle and the equilateral
triangle. The definition of an isosceles triangle is one that has at least two congruent sides. These
congruent sides are called legs and the remaining side is called the base.
The angle formed by the legs is called the vertex angle. The two angles that are opposite the legs
are called the base angles.
There are three important theorems about isosceles triangles. All of these theorems can be proved
using one of the triangle congruence theorems you learned in Part I of this unit.
Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
A  C
Converse of Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite the angles are congruent.
from the above figure.
Theorem
The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.
There are two important corollaries that follow from these theorems.
Corollary 1: If a triangle is equilateral, then the triangle is equiangular.
Corollary 2: If a triangle is equiangular, then the triangle is equilateral.
Complete the following “You Try” problems. (The answers are at the end of these notes.)
9. Solve for x and find m<L.
10. Solve for x. (Hint: what kind of isosceles triangle is this?)
11. Solve for w, x and the length of one side of the triangle.
12. Find CB.
Answers to “You Try” problems (If you have any questions about these problems, please contact
your teacher and ask about them.)
1. x = 50
2. x = 28
3. m<P = 72 o, m<A= 82o, m<P=26o
4. x = 32
5. x = 16
6. x = 20, exterior angle measure = 151 o
7. x = 15, m<X = 26o
8. x = 19, m<QTS = 128o
9. Since both sides are equal, then both base angles are equal so x + 30 = 3x – 4. Solve and get x =
17, m<L = 86 o
10. Since the two sides are equal, the two angles are also so they are both 4x + 9. The one with the
box is 90 so the set up is 4x + 9 + 4x + 9 + 90 = 180. Answer: x = 9
11. Since this triangle is equilateral, all the sides are equal. Pick the two that have “w” in them and
set them = so w+14 = 4w – 7. Once you get w = 7, then plug it back into one of the sides to get the
side = 21. Finally, to solve for x, set that side 2x + 3 = 21. Answers: w = 7, x = 9, side = 21
12.Since <A = <B, then 5x-31 = 2x+8, so x = 13, CB = 34