Transcript Geometry - TCC: Tidewater Community College

Geometry
Geometry: Part IA
Angles & Triangles
By
Dick Gill, and Julia Arnold
Elementary Algebra Math 03 online
The Angles in Triangles
In every triangle the sum of the measures of the angles is 180o.
In the triangle below, sides AC and BC are perpendicular to
each other. Perpendicular lines form 90o angles which are
called right angles.
So if angle C is 90o and angle B is 20o what does that leave for
angle A?
Think before you click.
A
A = 180o – (B + C) = 180o – 110o = 70o
B
C
Types by Angle
An angle that is more than 90o but less than 180o is called an
obtuse angle. An angle that is less than 90o is called acute.
A triangle with an obtuse angle is called an obtuse triangle.
A triangle with a right angle is called a right triangle.
A triangle that has three acute angles is called an acute
triangle.
Why is it not possible for a triangle to have more than one
right angle?
Two right angles would total 180o. Each
triangle has only 180o so there would be no
room for the third angle.
Using the information in the previous slide, see if you can
classify each of the following triangles by angle type.
This is an
acute triangle
since all three
angles are
acute.
This is a right
triangle since it
has one right angle
(lower left).
This is an obtuse
triangle since it has
one obtuse angle
(upper).
Take Sides on Triangles
Triangles can also be classified by sides.
In an equilateral
triangle, all three
sides are equal.
In an isosceles
triangle, only two
sides are equal.
In a scalene
triangle, none of the
sides are equal.
In every triangle the largest side is opposite the largest angle,
. the smallest side is opposite the smallest angle. In the
and
triangle below, side AB is the longest side which makes angle
C the largest angle; side AC is the smallest side which makes
angle B the smallest angle.
A
C
B
If the largest side is opposite the largest angle then it stands to
reason that sides of equal length will be opposite angles of equal
measure. In the sketch below, sides AB and AC are equal which
means that angles B and C will be equal also.
.
Suppose the angle A is 120o. Take a minute to see if you can
find the measures of angles B and C. Do your work before you
click.
Let x = the measure of
angles B and C.
A
120o + x + x = 180o
2x = 180o -120o
2x = 60o
x = 30o
C
B
The sum of the angles of a triangle equal 180o.
Using this fact, you may encounter geometry problems
involving finding the angles of a triangle as in the
following examples.
Example 1. In a triangle the sum of the three angles is always
180 degrees.
If the middle angle is twenty degrees more than the smallest
and the large angle is twenty degrees less than twice the
smallest find the three angles.
Let x = the smallest angle
x + 20 = the middle angle
2x - 20 = the largest angle
Since the three angles have to add up to be 180 degrees:
x + (x + 20) + ( 2x - 20) = 180
4x = 180
x = 45 degrees, the smallest angle
x + 20 = 65 degrees, the middle angle
2x - 20 = 70 degrees, the largest angle
Do these angles satisfy the conditions of the problem?
Do they add to 180?
Example 2: In a right triangle one angle is ninety degrees. If
the middle angle is three times the smallest find the other two
angles of the right triangle.
Write down your guess now and we’ll see how close you come
at the end of the solution.
Let x = the smallest angle
3x = the middle angle
x + 3x + 90 = 180 since all three angles must add to be 180.
4x + 90 - 90 = 180 - 90
4x = 90
x = 22.5 degrees
3x = 3(22.5) = 67.5 degrees
Check to see if the three angles add to 180.
Example 3. If one angle of a triangle is 80 degrees and
another is 72 degrees, find the third angle.
Let x = the measure of the third angle
x + 80 + 72 = 180
x + 152 = 180
x + 152 - 152 = 180 - 152
x = 28 degrees
Practice Problems
Your Turn
1. .In a triangle the sum of the three angles is always 180
degrees. If the middle angle is twenty degrees more than the
smallest and the large angle is twenty degrees less than twice
the smallest find the three angles
Complete Solution
2. In an isosceles triangle, two of the sides are always equal and two
of the angles are always equal. If the third angle is forty degrees, find
the other two. Complete Solution
3. In a right triangle one angle is ninety degrees. If the middle angle
is three times the smallest find the other two angles of the right
triangle.
Complete Solution
4. If one angle of a triangle is 80 degrees and another is 72 degrees,
find the third angle.
Complete Solution
5.
Three angles of a triangle always add up to 180 degrees. If a certain
triangle has one angle that is twice the second angle and the third angle
is 6 less than 3 times the first, what are the measures of the three
angles?
Complete Solution
6.
Find the angles of a triangle if two angles are equal and the third is 3
times the others.
Complete Solution
Practice Problems Solutions
In a triangle the sum of the three angles is always 180 degrees.
If the middle angle is twenty degrees more than the smallest and
the large angle is twenty degrees less than twice the smallest find
the three angles?
1.
Let x = smallest angle
20 + x = middle angle
2x - 20 = largest angle
x + 20 + x + 2x - 20 = 180
4x = 180
x = 45 smallest angle
20 + x = 65 middle angle
2x - 20 = 70 = largest angle
Return to Problems
2. In an isosceles triangle, two of the sides are always equal
and two of the angles are always equal. If the third angle is
forty degrees, find the other two.
Let x = one of the two equal angles
x = the other equal angle
40 = third angle
x + x + 40 = 180
2x + 40 = 180
2x = 140
x = 70
The three angles are 70, 70, and 40
Return to Problems
3. In a right triangle one angle is ninety degrees.
If the middle angle is three times the smallest find the other two angles of the
right triangle.
90 = the right angle
x = smallest angle
3x = middle angle
90+x + 3x = 180
4x = 90
x = 22.5
4. If one angle of a triangle is 80 degrees and another is 72 degrees,
find the third angle.
Let x = the third angle
x + 80 + 72 = 180
x + 152= 180
x = 28 the third angle
Return to Problems
5.
Three angles of a triangle always add up to
180 degrees. If a certain triangle has one
angle that is twice the second angle and the
third angle is 6 less than 3 times the first,
what are the measures of the three angles?
X = second angle
2x = first angle
Return to Problems
3 (2x) - 6 = third angle
You must pay close attention to the name of the angles.
The third angle depends on the first which depends on
the second.
X + 2x + 6x - 6 = 180 2x = 40 4/3 = 41 1/3
6x - 6 = 120 12/3 - 6= 124-6=118
9x - 6 = 180
9x = 186
Check: 20 2/3 + 41 1/3 + 118 = 180
x = 20 2/3
6. Find the angles of a triangle if two angles
are equal and the third is 3 times the others.
Let x = one of the two equal angles and
x = the other equal angle
3x = the third angle
X + x + 3x = 180
5x = 180
X = 36
The angles are 36, 36 and 108
Return to Problems
End Slide Show
Go to
Geometry Part 1B: Perimeter
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