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.

πŸ“šMaster 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 180ΒΊ 180ΒΊ . If we add the three angles of any triangle we choose, the result will always be 180ΒΊ 180ΒΊ . 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 180ΒΊ 180ΒΊ 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 45ΒΊ 45ΒΊ and 60ΒΊ 60ΒΊ degrees, we are asked to discover the measure of the third angle. First we add 45ΒΊ 45ΒΊ plus 60ΒΊ 60ΒΊ resulting in 105ΒΊ 105ΒΊ degrees. Now we subtract 105ΒΊ 105ΒΊ from 180ΒΊ 180ΒΊ , yielding 75ΒΊ 75ΒΊ degrees. In other words, the third angle of the triangle equals 75ΒΊ 75ΒΊ 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 63 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

Is the straight line in the figure the height of the triangle?

Step-by-Step Solution

The triangle's altitude is a line drawn from a vertex perpendicular to the opposite side. The vertical line in the diagram extends from the triangle's top vertex straight down to its base. By definition of altitude, this line is the height if it forms a right angle with the base.

To solve this problem, we'll verify that the line in question satisfies the altitude condition:

  • Step 1: Identify the triangle's vertices and base. From the diagram, the base appears horizontal, and the vertex lies directly above it.
  • Step 2: Check the nature of the line. The line is vertical when the base is horizontal, indicating perpendicularity.
  • Conclusion: The vertical line forms right angles with the base, thus acting as the altitude or height.

Therefore, the straight line depicted is indeed the height of the triangle. The answer is Yes.

Answer:

Yes

Video Solution
Exercise #2

Can a triangle have a right angle?

Step-by-Step Solution

To determine if a triangle can have a right angle, consider the following explanation:

  • Definition of a Right Angle: An angle is classified as a right angle if it measures exactly 90∘90^\circ.
  • Definition of a Right Triangle: A right triangle is a type of triangle that contains exactly one right angle.
  • According to the definition, a right triangle specifically includes a right angle. This is a well-established classification of triangles in geometry.

Thus, a triangle can indeed have a right angle and is referred to as a right triangle.

Therefore, the solution to the problem is Yes.

Answer:

Yes

Video Solution
Exercise #3

Is the straight line in the figure the height of the triangle?

Step-by-Step Solution

To determine if the straight line is the height of the triangle, we'll analyze its role within the triangle:

  • Step 1: Observe the triangle and the given line. The triangle seems to be made of three sides and a vertical line within it.
  • Step 2: Recall that the height of a triangle, in geometry, is defined as a perpendicular dropped from a vertex to the opposite side.
  • Step 3: Examine the positioning of the line: The vertical line starts at one vertex of the triangle and intersects the base, appearing to be perpendicular.
  • Step 4: Verify perpendicularity: Given that the line is shown as a clear vertical (and a small perpendicular indicator suggests perpendicularity), we accept this line as the height.
  • Step 5: Conclude with verification that the line is effectively meeting the definition of height for the triangle in the diagram.

Therefore, the vertical line in the figure is indeed the height of the triangle.

Yes

Answer:

Yes

Video Solution
Exercise #4

Is the straight line in the figure the height of the triangle?

Step-by-Step Solution

In the given problem, we have a triangle depicted with a specific line drawn inside it. The question asks if this line represents the height of the triangle. To resolve this question, we need to discern whether the line is perpendicular to one of the sides of the triangle when extended, as only a line that is perpendicular from a vertex to its opposite side can be considered the height.

The line in question is shown intersecting one of the sides within the triangle but does not form a perpendicular angle with any side shown or the ground (as is required for it to be the height of the triangle). A proper height would typically intersect perpendicularly either at or along the extended line of the opposite side from a vertex.

Therefore, based on the visual clues provided and the typical geometric definition of a height (or altitude) in a triangle, this specific line does not fit the criteria for being a height.

Thus, we conclude that the line depicted is not the height of the triangle. The correct answer is No.

Answer:

No

Video Solution
Exercise #5

Is the straight line in the figure the height of the triangle?

Step-by-Step Solution

The height of a triangle is defined as the perpendicular distance from a vertex to the line containing the opposite side (base). In this problem, we observe a vertical line segment drawn from a point on the base (horizontal line at the bottom of the triangle) to some level above the base. To determine if this line is a height, it must be perpendicular to the base and also reach to the opposite vertex of the triangle.

In the provided figure, the vertical line extends vertically from the base but does not connect to the opposite vertex of the triangle (at the top). Instead, it terminates at some intermediate point above the base. Since the line does not satisfy the full condition of being perpendicular and reaching an opposite vertex, it cannot be considered the height of this triangle.

Therefore, the given straight line is not the height of the triangle.

The correct and final answer is: No.

Answer:

No

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 180Β°. For example, if two angles are 45Β° and 60Β°, the third angle is 180Β° - (45Β° + 60Β°) = 75Β°.

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 180Β°. This applies to all triangles regardless of whether they are equilateral, isosceles, or scalene.

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

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

What are the angles in an equilateral triangle?

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

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

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If three angles don't add up to exactly 180Β°, 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 180Β°. If it doesn't, recheck your arithmetic or problem setup.

More The Sum of the Interior Angles of a Triangle Questions

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