Understanding Complimentary Angles and Ratios

The question asks us to find the greater of two complimentary angles that are in the ratio of 4 : 5. Let's first understand what complimentary angles are.

Complimentary angles are two angles whose measures add up to exactly 90 degrees. For example, an angle of 30 degrees and an angle of 60 degrees are complimentary because ${30^\circ} + {60^\circ} = {90^\circ}$.

Setting up the Angles based on Ratio

We are told the two complimentary angles are in the ratio of 4 : 5. When angles are given in a ratio, we can represent them using a common multiplier, let's call it ${x}$.

• The first angle can be represented as ${4x}$.
• The second angle can be represented as ${5x}$.

Forming an Equation for Complimentary Angles

Since these two angles are complimentary, their sum must be 90 degrees. So, we can write the equation:

${4x} + {5x} = {90^\circ}$

Solving for the Unknown Multiplier

Now, we need to solve this equation for ${x}$.

Combine the terms on the left side:

${(4 + 5)x} = {90^\circ}$

${9x} = {90^\circ}$

To find ${x}$, divide both sides of the equation by 9:

${x} = \frac{90^\circ}{9}$

${x} = {10^\circ}$

Calculating the Angles

Now that we know the value of ${x}$, we can find the measure of each angle:

• First angle: ${4x} = {4 \times 10^\circ} = {40^\circ}$
• Second angle: ${5x} = {5 \times 10^\circ} = {50^\circ}$

Let's check if these angles are indeed complimentary: ${40^\circ} + {50^\circ} = {90^\circ}$. Yes, they are.

Identifying the Greater Angle

The question asks for the greater angle. Comparing the two angles we found:

• First angle = ${40^\circ}$
• Second angle = ${50^\circ}$

Clearly, ${50^\circ}$ is greater than ${40^\circ}$.

Conclusion: The Greater Complimentary Angle

Therefore, the greater angle among the two complimentary angles in the ratio 4:5 is ${50^\circ}$.

Revision Table: Key Concepts

Additional Information: Related Angle Types

Besides complimentary angles, there are other important angle relationships in geometry:

• Supplementary Angles: Two angles whose measures add up to ${180^\circ}$. They often form a straight line. Example: ${120^\circ}$ and ${60^\circ}$ are supplementary.
• Adjacent Angles: Two angles that share a common vertex and a common side, but have no common interior points.
• Vertical Angles: A pair of opposite angles formed by the intersection of two lines. Vertical angles are always equal in measure.