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Geometry

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Geometry: Part IA Angles & Triangles By Dick Gill, and Julia Arnold Elementary Algebra Math 03 online

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**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

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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.

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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).

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**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.

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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

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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

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**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.

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**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 = 70 degrees, the largest angle Do these angles satisfy the conditions of the problem? Do they add to 180?

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**Example 2: In a right triangle one angle is ninety degrees**

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 = 4x = 90 x = 22.5 degrees 3x = 3(22.5) = 67.5 degrees Check to see if the three angles add to 180.

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**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 = 180 x = 180 x = x = 28 degrees

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**Practice Problems Your Turn Complete Solution Complete Solution**

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.

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Complete Solution Complete Solution

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**Practice Problems Solutions**

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 x + 2x - 20 = 180 4x = 180 x = 45 smallest angle 20 + x = 65 middle angle 2x - 20 = 70 = largest angle Return to Problems

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**Let x = one of the two equal angles x = the other equal angle **

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

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**Let x = the third angle x + 80 + 72 = 180 x + 152= 180**

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 = 180 x + 152= 180 x = 28 the third angle Return to Problems

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**You must pay close attention to the name of the angles. **

X = second 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 = /3 - 6= 124-6=118 Check: 20 2/ / = 180

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**Let x = one of the two equal angles and x = the other equal angle **

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

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Go to Geometry Part 1B: Perimeter Use Lesson Index

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