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 65 quizzes

Given the following triangle:

Write down the height of the triangle ABC.

AAABBBCCCDDD

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

Step-by-step solutions included
Exercise #1

True or false:

DE not a side in any of the triangles.
AAABBBCCCDDDEEE

Step-by-Step Solution

To solve the problem of determining whether DE is not a side in any of the triangles, we will methodically identify the triangles present in the diagram and examine their sides:

  • Identify triangles in the diagram. The diagram presented forms a right-angled triangle ABC with additional lines forming smaller triangles within.
  • Triangles formed: Triangle ABC (major triangle), Triangle ABD, Triangle BEC, and Triangle DBE.
  • Let's examine the sides of these triangles:
    • Triangle ABC has sides AB, BC, and CA.
    • Triangle ABD has sides AB, BD, and DA.
    • Triangle BEC has sides BE, EC, and CB.
    • Triangle DBE has sides DB, BE, and ED.
  • Notice that while point D is used, the segment DE is only part of line BE and isn't listed as a direct side of any triangle.

Therefore, the claim that DE is not a side in any of the triangles is indeed correct.

Hence, the answer is True.

Answer:

True

Video Solution
Exercise #2

Is DE side in one of the triangles?
AAABBBCCCDDDEEE

Step-by-Step Solution

Since line segment DE does not correspond to a full side of any of the triangles present within the given geometry, we conclude that the statement “DE is a side in one of the triangles” is Not true.

Answer:

Not true

Video Solution
Exercise #3

What type of angle is α \alpha ?

αα

Step-by-Step Solution

Remember that an acute angle is smaller than 90 degrees, an obtuse angle is larger than 90 degrees, and a straight angle equals 180 degrees.

Since the lines are perpendicular to each other, the marked angles are right angles each equal to 90 degrees.

Answer:

Straight

Exercise #4

True or false:

AB is a side of the triangle ABC.

AAABBBCCC

Step-by-Step Solution

To solve this problem, let's clarify the role of AB in the context of triangle ABC by analyzing its diagram:

  • Step 1: Identify the vertices of the triangle. According to the diagram, the vertices of the triangle are points labeled A, B, and C.
  • Step 2: Determine the sides of the triangle. In any triangle, the sides are the segments connecting pairs of distinct vertices.
  • Step 3: Identify AB as a line segment connecting vertex A and vertex B, labeled directly in the diagram.

Considering these steps, line segment AB connects vertex A with vertex B, and hence, forms one of the sides of the triangle ABC. Therefore, AB is indeed a side of triangle ABC as shown in the diagram.

The conclusion here is solidly supported by our observation of the given triangle. Thus, the statement that AB is a side of the triangle ABC is True.

Answer:

True

Video Solution
Exercise #5

True or false:

AD is a side of triangle ABC.

AAABBBCCC

Step-by-Step Solution

To determine if line segment AD is a side of triangle ABC, we need to agree on the definition of a triangle's side. A triangle consists of three sides, each connecting pairs of its vertices. In triangle ABC, these sides are AB, BC, and CA. Each side is composed of a direct line segment connecting the listed vertices.

In the diagram provided, there is no indication of a point D connected to point A or any other vertex of triangle ABC. To claim AD as a side, D would need to be one of the vertices B or C, or a commonly recognized point forming part of the triangle’s defined structure. The provided figure and description do not support that AD exists within the given triangle framework, as no point D is defined within or connecting any existing vertices.

Therefore, according to the problem's context and based on the definition of the sides of a triangle, AD cannot be considered a side of triangle ABC. It follows that the statement "AD is a side of triangle ABC" should be deemed not true.

Answer:

Not true

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 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 Triangle Height and Median: Identify the Corresponding Sides in ABC Triangle Median and Altitude: Identifying Lines in Triangle ABC

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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