Complementary and Supplementary Angles: Videos & Practice Problems

In geometry, understanding the concepts of complementary and supplementary angles is essential for working with angles and triangles. Complementary angles are defined as two angles that add up to 90 degrees, while supplementary angles are those that sum to 180 degrees. These terms serve as shorthand for these specific angle relationships.

To find the complement of a given angle, you can use the equation:

θ₁ + θ₂ = 90°

For example, if you have an angle of 30 degrees, the complement can be calculated as:

90° - 30° = 60°

This means that the complement of 30 degrees is 60 degrees. Similarly, for supplementary angles, the equation is:

θ₁ + θ₂ = 180°

Using the same angle of 30 degrees, the supplement is found by:

180° - 30° = 150°

Thus, the angles 30 degrees and 150 degrees are supplementary.

A helpful mnemonic to remember these concepts is that complementary angles form corners (both start with 'C'), while supplementary angles form straight lines (both start with 'S').

When working with angles greater than 90 degrees, such as 100 degrees, it’s important to note that a complementary angle does not exist because you would need to add a negative angle to reach 90 degrees. However, you can still find the supplement:

180° - 100° = 80°

In this case, the angle of 100 degrees has no complement but does have a supplement of 80 degrees.

Overall, mastering these concepts allows for a better understanding of angle relationships in various geometric contexts.

In geometry, understanding the relationships between angles is crucial, especially when dealing with triangles. Complementary angles are defined as two angles that sum up to 90 degrees, while supplementary angles add up to 180 degrees. In the context of right triangles, where one angle is always 90 degrees, the other two angles must be complementary, meaning they also add up to 90 degrees.

To illustrate this, consider a right triangle where two angles are expressed in terms of a variable, such as \( x \). For example, if one angle is \( x \) and the other is \( x + 10 \), we can set up the equation based on the complementary angle relationship:

\( x + (x + 10) = 90 \)

By simplifying this equation, we combine like terms:

\( 2x + 10 = 90 \)

Next, we isolate \( x \) by subtracting 10 from both sides:

\( 2x = 80 \)

Dividing both sides by 2 gives us:

\( x = 40 \)

Now that we have the value of \( x \), we can find the measures of the angles in the triangle. The first angle is simply \( x \), which is 40 degrees. The second angle, expressed as \( x + 10 \), becomes:

\( 40 + 10 = 50 \) degrees.

Thus, the angles in the triangle are 40 degrees, 50 degrees, and the right angle of 90 degrees. It’s important to verify that the sum of all angles in the triangle equals 180 degrees:

\( 40 + 50 + 90 = 180 \)

This confirms that our calculations are correct and highlights the utility of understanding complementary angles in solving for unknowns in geometric figures.

In this example, we explore the concepts of supplementary angles using a straight line and variables. Supplementary angles are defined as two angles that add up to 180 degrees. Given two angles expressed in terms of the variable \( x \), we can set up an equation to find the value of \( x \).

Consider the angles represented as \( 50 - 2x \) and \( 17x - 20 \). To find \( x \), we combine these angles and set their sum equal to 180 degrees:

\[(50 - 2x) + (17x - 20) = 180\]

By simplifying the equation, we first combine like terms:

\[50 - 20 + 17x - 2x = 180\]

This simplifies to:

\[30 + 15x = 180\]

Next, we isolate \( x \) by subtracting 30 from both sides:

\[15x = 150\]

Dividing both sides by 15 gives us:

\[x = 10\]

However, finding \( x \) is just the first step. We must substitute \( x \) back into the original angle expressions to determine the actual angle measures. For the first angle:

\[17x - 20 = 17(10) - 20 = 170 - 20 = 150 \text{ degrees}\]

For the second angle, we can either substitute \( x \) into the second expression or use the property of supplementary angles. Since the angles must add up to 180 degrees, we can calculate:

\[180 - 150 = 30 \text{ degrees}\]

Alternatively, substituting \( x \) into the first angle expression confirms this:

\[50 - 2x = 50 - 2(10) = 50 - 20 = 30 \text{ degrees}\]

Thus, the two angles are 150 degrees and 30 degrees, demonstrating the application of supplementary angles in solving for unknowns using algebraic expressions.