ANGLES – COMPLEX FIGURES

Concept: A plane is a flat surface that has no boundaries. Two lines in the same plane either cross once or they to not cross at all. When two lines cross, we say that they intersect. They intersect at one point.

Two lines that do not intersect remain the same distance apart. These are called parallel lines. Parallel lines are lines that run side-by-side and do not cross at any point. A transversal is a third line that crosses two parallel lines.

When a transversal crosses two parallel lines, a set of four angles is formed at each point of intersection.

Since one set of angles () looks exactly like the other set of angles

(), the following relations are true.

1. Corresponding Angles are equal. If you cut the transversal in half and place one set of angles over the other, corresponding angles would sit on top of each other. For example, corresponds to , corresponds to .

2. Alternate Interior Angles Are Equal. Alternate interior angles are angles that are inside (interior) the parallel lines and that are on opposite (alternate) sides of the transversal. For example, and are equal and and are equal.

3. Alternate Exterior Angles are Equal. Alternate exterior angles are angles that are outside (exterior) the parallel lines and that are on opposite (alternate) sides of the transversal. For example, and are equal, and and are equal.

These relations also apply to squares, rectangles and other parallelograms, and are helpful when determining missing angle values. Squares, rectangles and parallelograms each contain a set of parallel lines which are intersected by a pair of parallel transversals. The relations which are true for a set of parallel lines and one transversal also apply to a set of parallel lines and two transversals.

For Example: If we extend the lines of the parallelogram in figure 1, we get figure 2. Our parallelogram can now be seen as two parallel lines crossed by two parallel transversals. This helps us see more easily how the three relations mentioned above apply to parallelograms.

Steps For Solving Complex Figure Problems

Step 1 | Determine if the angles in question (the one with a given value and the one whose value you are asked to find) are Corresponding Angles, Alternate Interior Angles, Alternate Exterior Angles, Complementary Angles, Supplementary Angles, or Vertical angles. (Remember that in solving Complex Figure Problems you may have to apply the rules from more than one Step to the problem to reach a solution.)
Step 2 | If the angles in question are Corresponding Angles, they are equal. The value of the given angle is the same as the value of the other angle. Go to Step 8.
Step 3 | If the angles in question are Alternate Interior Angles, they are equal. The value of the given angle is the same as the value of the other angle. Go to Step 8.
Step 4 | If the angles in question are Alternate Exterior Angles, they are equal. The value of the given angle is the same as the value of the other angle. Go to Step 8.
Step 5 | If the angle is complementary, subtract the value of the given angle from to find the value of the other angle. Go to Step 8.
Step 6 | If the angle is supplementary, subtract the value of the given angle from . Go to Step 8.
Step 7 | If the angles are vertical, they are equal. The value of the given angle is the same as the value of the other angle. Go to Step 8.
Step 8 | Write the final answer.

figure 2

Example 1: Given angle a = 75 degrees, what is the measurement of angle c in degrees?

Step 1 | Determine if the angles in question (the one with a given value and the one whose value you are asked to find) are Corresponding Angles, Alternate Interior Angles, Alternate Exterior Angles, Complementary Angles, Supplementary Angles, or Vertical angles. (Remember that in solving Complex Figure Problems you may have to apply the rules from more than one Step to the problem to reach a solution.)
| Bring line ab and line cd together, as shown in figure 2. Notice that angle a and angle c have switched positions (as have angles d and b) and are now clearly supplementary angles.
Step 6 | If the angle is supplementary, subtract the value of the given angle from . Go to Step 8.
Step 8 | Write the final answer.
| The measurement of angle c in degrees is 105.

figure 2

figure 1

Example 2: Given angle a is 70 degrees, what is the measurement of angle d in degrees?

Step 1 | Determine if the angles in question (the one with a given value and the one whose value you are asked to find) are Corresponding Angles, Alternate Interior Angles, Alternate Exterior Angles, Complementary Angles, Supplementary Angles, or Vertical angles. (Remember that in solving Complex Figure Problems you may have to apply the rules from more than one Step to the problem to reach a solution.)
| Bring line ac and line bd together, as shown in figure 2. Notice that angle a and angle d have switched positions (as have angles b and c) and are now clearly Alternate Interior Angles.
Step 3 | If the angles in question are Alternate Interior Angles, they are equal. The value of the given angle is the same as the value of the other angle. Go to Step 8.
Step 8 | Write the final answer.
| The measurement of angle d in degrees is 75.

Example 3: Given angle d = 60 degrees, what is the measurement of angle c in degrees?

Step 1 | Determine if the angles in question (the one with a given value and the one whose value you are asked to find) are Corresponding Angles, Alternate Interior Angles, Alternate Exterior Angles, Complementary Angles, Supplementary Angles, or Vertical angles. (Remember that in solving Complex Figure Problems you may have to apply the rules from more than one Step to the problem to reach a solution.)
| Bring line ad and line bc together, as shown in figure 2. Notice that angle d and angle c have switched positions (as have angles b and a) and are now clearly supplementary angles.
Step 6 | If the angle is supplementary, subtract the value of the given angle from . Go to Step 8.
Step 8 | Write the final answer.
| The measurement of angle c in degrees is 120.