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Angle Measures in Given Triangles

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Angle Measures in Given Triangles

Have you ever heard of the Bermuda Triangle?

Isaac and Marc are continuing their work on their skate park. They are both fascinated by the Bermuda Triangle and have decided to name one of the parts of their design after this triangle.

The Bermuda Triangle is located in an area of water right around Bermuda. There have been many mysteries surrounding this area of the ocean. Many ships have been lost there as well!!

Since they love the idea of building a challenging rail, they have decided to name it the Bermuda Triangle. The triangle will be connected to a ramp on each side of the triangle, so that students will come down the ramp onto the rails. There they will either succeed or be lost at sea!

Marc drew the following picture of the triangle.

The triangle has three angles and the boys want to reproduce these angles in their structure. The first angle B is equal to 62^\circ , the second angle C is equal to 63^\circ .

Marc can’t remember the measure of angle A . He thinks there is a way to figure this out, but he can’t remember what it is. Do you know?

Pay attention in this Concept so you can help Marc figure this out. There is way to do it without looking up the answer!!

Guidance

There are some problem solving aspects of working with triangles. Some of this consists of figuring out missing angles, and some of it concerns drawing specified triangles. Angle measures are important in both of these topics.

The sum of the angles of a triangle is equal to 180^\circ .

What happens if you know two but not all three of the measures of a triangle? How can you figure out the measure of the missing angle?

What does this look like?

Now we can tell that this is a right triangle and that one of the angles is equal to 90 degrees. To figure out the measure of the missing angle, we have used a variable to represent the unknown quantity. Here is our equation.

55 + 90 + x & = 180 \\145 + x & = 180 \\180 - 145 & = x \\x & = 35^\circ .

Our answer is 35^\circ .

How can we draw a specific triangle?

We can draw a specific type of triangle using a ruler and a protractor. Remember in an earlier Concept that a protractor is used to measure angles. We can use the protractor to figure out the measure of an angle and then draw in the rest of the triangle with the ruler.

Draw an obtuse triangle.

Remember that an obtuse triangle has one angle that is greater than 90 degrees.

To draw an obtuse triangle, begin by drawing the obtuse angle. Use your protractor to measure an angle that is greater than 90 degrees.

Here is a protractor where an angle that is 105 degrees has been drawn. We can call this an obtuse angle because it is greater than 90 degrees but less than 180 degrees. Next, we can draw in the rest of the triangle using our ruler.

Here is our obtuse triangle.

Next we can draw an acute triangle. An acute triangle has all three angles that are smaller than 90 degrees. We will need to be careful as we draw and measure this triangle. Let’s begin with one angle.

The first angle that I have drawn here is 30^\circ . I can draw in the other two angles by making sure that they are less than 90 degrees.

Here is an acute triangle.

You can draw any type of triangle that you wish by using a protractor and a ruler.

Practice finding the missing angle in the following triangles.

Example A

Solution: 70 degrees

Example B

Solution: 71 degrees

Example C

A triangle with the following angles.

90 + 45 + x = 180^\circ

Solution: 45 degrees

Now let's go back to the problem from the beginning of the Concept.

Marc drew the following picture of the triangle.

The triangle has three angles and the boys want to reproduce these angles in their structure. The first angle B is equal to 62^\circ , the second angle C is equal to 63^\circ .

Marc can’t remember the measure of angle A . He thinks there is a way to figure this out, but he can’t remember what it is.

The sum of the interior angles of a triangle is equal to 180^\circ .

Marc knows the measure of two of the angles of the triangle. Therefore, he can write an equation to figure out the measure of the third angle.

63 + 62 + x = 180

The variable x is used to represent the measure of angle A . Marc is working to find the measure of angle A .

125 + x & = 180 \\180 - 125 & = 55

The measure of Angle A is 55^\circ

Vocabulary

Acute Triangle
all three angles are less than 90 degrees.
Right Triangle
One angle is equal to 90 degrees and the other two are acute angles.
Obtuse Triangle
One angle is greater than 90 degrees and the other two are acute angles.
Equiangular Triangle
all three angles are equal
Scalene Triangle
all three side lengths are different
Isosceles Triangle
two side lengths are the same and one is different
Equilateral Triangle
all three side lengths are the same
Protractor
A tool used to measure angles
Interior angles
the angles inside a figure

Guided Practice

Here is one for you to try on your own.

Look at the following angle sums. Figure out the measure of the missing angle.

25 + 45 + x = 180^\circ

Answer

To figure out the measure of the missing angle, we must first add up the measures of the two given angles.

25 + 45 = 70

Now we can subtract that measure from 180.

180 - 70 = 130

The measure of the missing angle is 130 degrees.

Video Review

James Sousa: Animation: The Sum of the Interior Angles of a Triangle - This video will provide you with important information necessary for success in this Concept.

Practice

Directions: Critical Thinking - Each question combines information about the angles and side lengths. Answer each question carefully.

1. True or false. If a triangle is equiangular, it can also be equilateral.

2. True or false. A scalene triangle can not be an equilateral triangle.

3. True or false. The word “equiangular” applies to side lengths.

4. True or false. An isosceles triangle can be an obtuse or acute triangle.

5. A ___________________ is a tool used to measure angles.

6. A __________ angle is equal to 90 degrees.

7. A __________angle is equal to 180 degrees.

8. An ________ angle is less than 90 degrees.

9. An _________ angle is greater than 90 but less than 180 degrees.

10. The prefix “tri” means ______________.

11. How many angles are there in a triangle?

Directions: Look at each and determine the missing angle measure.

12. 20 + 70 + x = 180^\circ

13. 60 + 60 + x = 180^\circ

14. 90 + 15 + x = 180^\circ

15. 110 + 45 + x = 180^\circ

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