Welcome to our Math lesson on Sum of the Interior Angles in a Triangle, this is the second lesson of our suite of math lessons covering the topic of Triangles, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
All triangles give the same value for the total measure of the interior angles. Let's write this assumption in the form of a theorem and eventually let's prove it.
Theorem:
The measure of the interior angles in every triangle sums up to 180°.
Proof
We will apply the properties of the angles formed by two parallel lines intersected by a third line explained in the previous chapters. In this case, we will consider six lines from L1 to L6 two by two parallel where L1 || L2, L3 || L4 and L5 || L6. If we express the angles of any of the triangles obtained by a, b, and c respectively (for example let's choose the leftmost triangle i.e. the yellow one as a reference). Thus, the first stage involves highlighting all corresponding angles of a, b, and c which - as known from the previous chapters - are congruent.
The next step involves completing the figure with the corresponding vertical angles of a, b, and c which - as known from the previous chapters - are also congruent.
Now, if you consider any three angles that form a straight line, you will see that a + b + c = 180° as it is known that the angle corresponding to a straight line is called a straight angle which has a measure of 180°. This value also corresponds to the sum of the interior angles of the triangle considered. Given that we didn't mention any restriction in the triangle chosen for analysis, it means the above sum is true for the interior angles of any triangle i.e. the theorem is confirmed as true.
Example
The angles x, y, and z of a triangle form a ratio of x : y : z = 2 : 5 : 8 between them. Find the measure of each angle.
Solution
We can write the above ratio in the following way:
x : y : z = 2 : 5 : 8
x/2 = y/5 = z/8 = k
where k is a constant of proportionality to be calculated first. Expressing all angles in terms of k yields:
x = 2k; y = 5k; z = 8k
Since the sum of the measures of the interior angles in a triangle is always 180° we write:
x + y + z = 180°
2k + 5k + 8k = 180°
15k = 180°
k = 180/15
k = 12
Therefore, the measures of the three angles x, y, and z are:
x = 2k = 2 × 12 = 24°
y = 5k = 5 × 12 = 60°
z = 8k = 8 × 12 = 96°
You have reached the end of Math lesson 18.3.2 Sum of the Interior Angles in a Triangle. There are 10 lessons in this physics tutorial covering Triangles, you can access all the lessons from this tutorial below.
More Triangles Lessons and Learning Resources
| Tutorial ID | Math Tutorial Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions |
|---|---|---|---|---|---|
| 18.3 | Triangles | ||||
| Lesson ID | Math Lesson Title | Lesson | Video Lesson | ||
| 18.3.1 | Definition of an Triangles | ||||
| 18.3.2 | Sum of the Interior Angles in a Triangle | ||||
| 18.3.3 | Classification of Triangles | ||||
| 18.3.4 | Exterior Angles of Triangles | ||||
| 18.3.5 | Interior and Exterior Regions of a Triangle | ||||
| 18.3.6 | The Intersection of a Triangle and a Line | ||||
| 18.3.7 | Auxiliary Elements of Triangles | ||||
| 18.3.8 | Definition of Concurrent Lines and their Applications in Triangles. Orthocentre, Incentre, Excentre and Centroid of a Triangle | ||||
| 18.3.9 | Perimeter of a Triangle | ||||
| 18.3.10 | Area of a Triangle | ||||
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Enjoy the "Sum of the Interior Angles in a Triangle" math lesson? People who liked the "Triangles lesson found the following resources useful:
- Interior Angles Feedback. Helps other - Leave a rating for this interior angles (see below)
- Angles and Geometrical Figures Math tutorial: Triangles. Read the Triangles math tutorial and build your math knowledge of Angles and Geometrical Figures
- Angles and Geometrical Figures Revision Notes: Triangles. Print the notes so you can revise the key points covered in the math tutorial for Triangles
- Angles and Geometrical Figures Practice Questions: Triangles. Test and improve your knowledge of Triangles with example questins and answers
- Check your calculations for Angles and Geometrical Figures questions with our excellent Angles and Geometrical Figures calculators which contain full equations and calculations clearly displayed line by line. See the Angles and Geometrical Figures Calculators by iCalculator™ below.
- Continuing learning angles and geometrical figures - read our next math tutorial: Triangles part Two. Congruence and Similarity
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