Introduction:
Complementary angles are angles that add up to 90 degrees. In this problem, we are given that two complementary angles are in the ratio of 4:5. We need to find the measures of these angles.

Step 1: Set up the equation:
Let the measures of the angles be 4x and 5x, where x is a common factor. Since the angles are complementary, we can write the equation as:
4x + 5x = 90

Step 2: Solve the equation:
Combining like terms, we have:
9x = 90

Dividing both sides by 9, we get:
x = 10

Step 3: Find the measures of the angles:
Substituting the value of x back into the equation, we have:
4x = 4 * 10 = 40
5x = 5 * 10 = 50

Therefore, the two complementary angles are 40 degrees and 50 degrees.

Explanation of Angles made by a Transversal of Parallel Lines:
When a transversal intersects two parallel lines, it creates various angles. There are several types of angles formed by a transversal, including corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles.

Corresponding Angles:
Corresponding angles are located on the same side of the transversal and in the same position with respect to the parallel lines. They have equal measures. For example, if angle 1 is 50 degrees, then angle 5 will also be 50 degrees.

Alternate Interior Angles:
Alternate interior angles are located on opposite sides of the transversal and inside the parallel lines. They have equal measures. For example, if angle 3 is 70 degrees, then angle 6 will also be 70 degrees.

Alternate Exterior Angles:
Alternate exterior angles are located on opposite sides of the transversal and outside the parallel lines. They have equal measures. For example, if angle 2 is 60 degrees, then angle 7 will also be 60 degrees.

Consecutive Interior Angles:
Consecutive interior angles are located on the same side of the transversal and inside the parallel lines. They are supplementary, which means their measures add up to 180 degrees. For example, if angle 3 is 70 degrees, then angle 4 will be 110 degrees (180 - 70).

Summary:
- Two complementary angles are in the ratio of 4:5.
- We can set up an equation to find the measures of these angles.
- By solving the equation, we find that the angles are 40 degrees and 50 degrees.
- When a transversal intersects two parallel lines, it creates corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles.
- These angles have specific properties and relationships, such as equal measures or being supplementary.

Complementary Angles= 90
They're in the ratio of 4:5
4+5=9
90÷9=10
4:5
=4×10:5×10
=40:50