Triangle Angles Practice Problems - Sum Theorem Exercises

Master triangle angle calculations with step-by-step practice problems. Learn to find missing angles using the triangle sum theorem for all triangle types.

📚Master Triangle Angle Calculations with Interactive Practice
  • Apply the triangle sum theorem to find missing interior angles
  • Determine if three given angles can form a valid triangle
  • Calculate unknown angles in isosceles, equilateral, and scalene triangles
  • Solve complex angle problems involving parallel lines and triangles
  • Practice angle relationships in different triangle configurations
  • Build confidence with step-by-step solution methods

Understanding The Sum of the Interior Angles of a Triangle

Complete explanation with examples

The sum of the interior angles of a triangle is 180º 180º . If we add the three angles of any triangle we choose, the result will always be 180º 180º . This means that if we know the values of two angles of a triangle we can always calculate, with ease, the value of the third one: first we add the two angles we know and then we subtract from 180º 180º The result of this subtraction will give us the value of the third angle of the triangle.

For example, given a triangle with two known interior angles of 45º 45º and 60º 60º degrees, we are asked to discover the measure of the third angle. First we add 45º 45º plus 60º 60º resulting in 105º 105º degrees. Now we subtract 105º 105º from 180º 180º , yielding 75º 75º degrees. In other words, the third angle of the triangle equals 75º 75º degrees.

The above property is also called the triangle sum theorem, and can help us to solve problems involving the interior angles of a triangle, regardless of whether it is equilateral, isosceles or scalene.

Examples of different types of triangles and the sum of the interior angles in each

Isosceles triangle:

Isosceles triangle

Equilateral triangle:

Equilateral triangle

Scalene triangle:

Scalene triangle

Detailed explanation

Practice The Sum of the Interior Angles of a Triangle

Test your knowledge with 64 quizzes

Indicates which angle is greater

Examples with solutions for The Sum of the Interior Angles of a Triangle

Step-by-step solutions included
Exercise #1

In an isosceles triangle, the angle between ? and ? is the "base angle".

Step-by-Step Solution

An isosceles triangle is one that has at least two sides of equal length. The angles opposite these two sides are known as the "base angles."
The side that is not equal to the other two is referred to as the "base" of the triangle. Thus, the "base angles" are the angles between each of the sides that are equal in length and the base.
Therefore, when we specify the angle in terms of its location or position, it is the angle between a "side" and the "base." This leads to the conclusion that the angle between the side and the base is the "base angle."

Therefore, the correct choice is Side, base.

Answer:

Side, base.

Exercise #2

Indicates which angle is greater

Step-by-Step Solution

Answer B is correct because the more closed the angle is, the more acute it is (less than 90 degrees), meaning it's smaller.

The more open the angle is, the more obtuse it is (greater than 90 degrees), meaning it's larger.

Answer:

Video Solution
Exercise #3

Look at the two triangles below. Is EC a side of one of the triangles?

AAABBBCCCDDDEEEFFF

Step-by-Step Solution

Every triangle has 3 sides. First let's go over the triangle on the left side:

Its sides are: AB, BC, and CA.

This means that in this triangle, side EC does not exist.

Let's then look at the triangle on the right side:

Its sides are: ED, EF, and FD.

This means that in this triangle, side EC also does not exist.

Therefore, EC is not a side in either of the triangles.

Answer:

No

Video Solution
Exercise #4

Indicates which angle is greater

Step-by-Step Solution

In drawing A, we can see that the angle is an obtuse angle, meaning it is larger than 90 degrees:

While in drawing B, the angle is a right angle, meaning it equals 90 degrees:

Therefore, the larger angle appears in drawing A.

Answer:

Video Solution
Exercise #5

Which angle is greatest?

Step-by-Step Solution

In drawing A, we can see that the angle is more closed:

While in drawing B, the angle is more open:

In other words, in diagram (a) the angle is more acute, while in diagram (b) the angle is more obtuse.

Remember that the more obtuse an angle is, the larger it is.

Therefore, the larger of the two angles appears in diagram (b).

Answer:

Video Solution

Frequently Asked Questions

Everything you need to know about The Sum of the Interior Angles of a Triangle

How do you find a missing angle in a triangle?

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To find a missing angle in a triangle, add the two known angles and subtract the sum from 180°. For example, if two angles are 45° and 60°, the third angle is 180° - (45° + 60°) = 75°.

What is the triangle sum theorem?

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The triangle sum theorem states that the sum of all interior angles in any triangle always equals 180°. This applies to all triangles regardless of whether they are equilateral, isosceles, or scalene.

Can three angles of 90°, 60°, and 40° form a triangle?

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No, these angles cannot form a triangle because they sum to 190°, which exceeds the required 180°. For three angles to form a triangle, their sum must equal exactly 180°.

What are the angles in an equilateral triangle?

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In an equilateral triangle, all three angles are equal and measure 60° each. Since 60° + 60° + 60° = 180°, this satisfies the triangle sum theorem.

How do you solve triangle angle problems with parallel lines?

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When solving triangle problems with parallel lines, use properties like: 1) Corresponding angles are equal, 2) Alternate interior angles are equal, 3) Co-interior angles sum to 180°, then apply the triangle sum theorem.

What happens if triangle angles don't add up to 180°?

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If three angles don't add up to exactly 180°, they cannot form a valid triangle. The angles might be measurement errors or the figure might be a different polygon.

Are triangle angle problems the same for all triangle types?

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Yes, the triangle sum theorem applies equally to all triangle types - equilateral, isosceles, and scalene. However, some triangles have special angle relationships that can simplify calculations.

What's the easiest way to check triangle angle calculations?

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Always verify your answer by adding all three angles together. The sum should equal exactly 180°. If it doesn't, recheck your arithmetic or problem setup.

More The Sum of the Interior Angles of a Triangle Questions

Click on any question to see the complete solution with step-by-step explanations

Sum and Difference of Angles

Triangle Angle Analysis: Can 30°, 60°, and 90° Form a Valid Triangle? Triangle Angle Verification: Do 56°, 89°, and 17° Form a Valid Triangle? Calculate Angle X in Straight Line: Given 41° and 45° Angles Find the Missing Angle: Geometric Construction with 36° Arc Segment Find the Missing Angle in an 80° Angle Problem

Parts of a Triangle

Triangle Geometry: Verifying Line Segment DE's Position in Multiple Triangles Geometry Problem: Is Line Segment DE Part of the Triangle? Triangle ABC: Identifying the Side with Coinciding Median and Height Triangle Median and Height: Identifying Side Construction in Geometric Diagram Triangle Construction: Identifying Median and Altitude Locations

Continue Your Math Journey

Master The Sum of the Interior Angles of a Triangle first, then advance to these related topics that build on your fraction skills

Topics Learned in Later Sections

AreaThe sides or edges of a triangleTriangle HeightExterior angles of a triangleTypes of TrianglesObtuse TriangleEquilateral triangleIdentification of an Isosceles TriangleScalene triangleAcute triangleIsosceles triangleThe Area of a TriangleArea of a right triangleArea of Isosceles TrianglesArea of a Scalene TriangleArea of Equilateral TrianglesAreas of Polygons for 7th GradeRight TriangleMedian in a triangleCenter of a Triangle - The Centroid - The Intersection Point of MediansHow do we calculate the area of complex shapes?How to calculate the area of a triangle using trigonometry?All terms in triangle calculationPerimeterTrianglePerimeter of a triangleHow do we calculate the perimeter of polygons?

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