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Angles of a triangle

Here you will learn about angles of a triangle including what the sum of both interior and exterior angles of a triangle are, how to find missing angles, and how to use this alongside other angle facts to form and solve equations.

Students will first learn about angles of a triangle as a part of geometry in katex is not definedth grade and will extend their knowledge throughout high school geometry.

What are angles of a triangle?

Angles of a triangle refer to the sum of the angles of a triangle, found at each vertex in a triangle.

Angles of a triangle 1 US

A triangle has both interior angles and exterior angles.

[FREE] Angles Worksheet (Grade 4)

[FREE] Angles Worksheet (Grade 4)

[FREE] Angles Worksheet (Grade 4)

Use this quiz to check your grade 4 students’ understanding of angles. 10+ questions with answers covering a range of 4th grade angles topics to identify areas of strength and support!

DOWNLOAD FREE
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[FREE] Angles Worksheet (Grade 4)

[FREE] Angles Worksheet (Grade 4)

[FREE] Angles Worksheet (Grade 4)

Use this quiz to check your grade 4 students’ understanding of angles. 10+ questions with answers covering a range of 4th grade angles topics to identify areas of strength and support!

DOWNLOAD FREE

Interior angles of a triangle

Interior angles of a triangle are angles that are formed inside a triangle by its three sides. Each interior angle is formed by two adjacent sides of the triangle.

The sum of the interior angles of a triangle is katex is not defined

For example,

Angles of a triangle 2 US

Exterior angles of a triangle

The exterior angles of a triangle are angles that are formed on the outside of the triangle when its sides are extended.

The sum of exterior angles of a triangle is katex is not defined

Angles of a triangle 3 US

The exterior angle is also equal to the sum of the two opposite interior angles. Each exterior angle is supplementary to its adjacent interior angle.

Angles of a triangle 4 US

Other angle facts

Sometimes the problem will involve using other angle facts. Let’s recap some of the other important angle facts:

Angles of a triangle 5 US

What are angles of a triangle?

What are angles of a triangle?

Common Core State Standards

How does this relate to katex is not definedth grade math and high school math?

  • Grade 8: Geometry (8.G.A.5)
    Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

  • High school: Geometry (HS.G.CO.C.10)
    Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to katex is not defined base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

How to find a missing angle of a triangle

In order to find the measure of a missing angle of a triangle:

  1. Add up the angles that are given within the triangle.
  2. Subtract this total from katex is not defined

Angles of a triangle examples

Example 1: scalene triangle

Find the measure of the unknown angle labeled a in the following triangle:

Angles of a triangle 6 US

  1. Add up the angles that are given within the triangle.

The angles katex is not defined and katex is not defined are given. Add these together:

katex is not defined

2Subtract this total from katex is not defined

Subtract katex is not defined from katex is not defined

katex is not defined

Example 2: right triangle

Find the measure of the unknown angle labeled b in the following triangle:

Angles of a triangle 7 US

Add up the angles that are given within the triangle.

Show step

Subtract this total from katex is not defined

Show step

Example 3: isosceles triangle

Find the measure of the unknown angle labeled katex is not defined in the following triangle:

Angles of a triangle 8 US

When two sides of a triangle are equal, the angles at the ends of those sides will also be equal.

Add up the angles that are given within the triangle.

Show step

Subtract this total from katex is not defined

Show step

How to find a missing angle of a triangle using other angle facts

In order to find the missing angle of a triangle using other angle facts:

  1. Use angle facts to identify all possible angles.
  2. Calculate the missing angles in the triangle.

Example 4: using angles at a point

Find the measure of the angle labeled katex is not defined

Angles of a triangle 9 US

Use angle facts to identify all possible angles.

Show step

Calculate the missing angles in the triangle.

Show step

Example 5: using opposite angles

Find the measure of the angle labeled katex is not defined

Angles of a triangle 11 US

Use angle facts to identify all possible angles.

Show step

Calculate the missing angles in the triangle.

Show step

Example 6: two different triangles

Find the measure of the angle labeled katex is not defined

Angles of a triangle 13 US

Use angle facts to identify all possible angles.

Show step

Calculate the missing angles in the triangle.

Show step

Teaching tips for angles of a triangle

  • Visual aids, such as diagrams, and hands-on activities allow students to explore the properties of triangles in a different way.

  • Allow students to work in pairs or small groups while working through practice problems on worksheets. This encourages discussions about math between peers, which allows for students to share their personal insights and strategies used.

  • Provide real-time feedback to students as they are working through practice problems to clarify any misunderstandings as they arise. The use of a triangle calculator can support this, without having to take any additional time from you.

Easy mistakes to make

  • Mixing up the sum of interior angles and exterior angles
    Students may use katex is not defined instead of katex is not defined for the sum of the interior angles of the triangle and vice versa.

  • Equal angles in an isosceles triangle
    Selecting the wrong angles when identifying the equal angles in an isosceles triangle (particularly a problem when the equal angles are not at the bottom). The angle that is different in an isosceles triangle is the one between the two sides with equal length.

    Angles of a triangle 16 US

Angles in a triangle practice questions

1. Find the measure of the angle katex is not defined in the following triangle:

 

Angles of a triangle 18 US

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False
katex is not defined

2. Find the measure of the angle katex is not defined

 

Angles of a triangle 19 US

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

This is an isosceles triangle and the two angles at the bottom of the triangle are equal.

 

katex is not defined

3. Find the measure of angle katex is not defined in the following triangle:

 

Angles of a triangle 20 US

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

This is an isosceles triangle and the two angles on the right are equal.

 

katex is not defined

 

katex is not defined

4. What is the size of each angle in an equilateral triangle?

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

An equilateral triangle has three equal sides so

 

katex is not defined

5. Find the measure of the angle labeled katex is not defined in the following triangle:

 

Angles of a triangle 21 US

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

The angle opposite katex is not defined is also katex is not defined since vertically opposite angles are equal. The triangle is an isosceles triangle and the two angles on the left are the same size.

 

katex is not defined

 

katex is not defined

6. Find the measure of the angle labeled katex is not defined

Angles of a triangle 22 US

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz True

katex is not defined
GCSE Quiz False

katex is not defined
GCSE Quiz False

Let’s find the missing angle in the triangle on the left first,

 

katex is not defined

 

Then use the fact that angles on a straight line add up to katex is not defined to find the unlabeled angle in the right hand triangle.

 

katex is not defined

Angles of a triangle 23 US

Now, find angle katex is not defined

 

katex is not defined

Angles of a triangle FAQs

Do all triangles have the same interior angle measures?

No, the interior angles within a triangle can vary depending on the triangle. However, all of the measures of angles within a triangle will always equal katex is not defined degrees.

What is the relationship between interior and exterior angles of a triangle?

Each exterior angle of a triangle is supplementary to its adjacent interior angle. The sum of each exterior angle and its adjacent interior angle will equal katex is not defined degrees.

What is the Law of Sines?

The Law of Sines is a principle in trigonometry that relates the side lengths of a triangle to the sines of its angles. It states that in any triangle, the ration of the length of a side to the sine of its opposite angle is constant for all three sides. It can be expressed as:
katex is not defined

See also: Law of Sines

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