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Geometry: Part IAAngles & Triangles ByDick 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 classify each of the following triangles by angle type.

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, and the smallest side is opposite the smallest angle. In the 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

B

C

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.

120o + x + x = 180o

2x = 180o -120o

2x = 60o

x = 30o

A

C

B

The sum of the angles of a triangle equal 180 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.o.

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 another is 72 degrees, find the third angle.

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

Your Turn

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

Complete Solution another is 72 degrees, find the third angle.

Complete Solution

Practice Problems Solutions another is 72 degrees, find the third angle.

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 = 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

X = second angle 72 degrees, find the third angle.

2x = first angle

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

9x - 6 = 180

9x = 186

x = 20 2/3

Return to Problems

2x = 40 4/3 = 41 1/3

6x - 6 = 120 12/3 - 6= 124-6=118

Check: 20 2/3 + 41 1/3 + 118 = 180

6 72 degrees, find the third angle.. 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

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