The scale factor of a dilation tells us what we multiply corresponding sides by to get the new side lengths. In this case, we want these lengths to be the same to get congruent triangles. Thus, we must be looking for the multiplicative identity, which is 1. 

ASA (Angle Side Angle) is a theorem to prove triangle congruency.

In this case, we only need two angles to prove that two triangles are similar, so the last side in ASA is unnecessary for this question.

For this purpose, we use the theorem AA instead. 

Two triangles are similar if and only if their side lengths are proportional.

In this case, two of the sides are proportional, leading us to a scale factor of 2.

However, with the last side,  which is not our side length.

Thus, these pair of sides are not proportional and therefore our triangles cannot be similar. 

We must remember that there are three ways to prove triangles are similar.

1. At least two angles in one triangle are congruent to angles in another (AA)
2. All three pairs of corresponding sides are proportional (SSS)
3. Two pairs of corresponding sides are proportional and the angles between those sides are congruent (SAS)

Comparing triangles I and II, we only have one angle and two sides in trinagle II, so attempting to use either AA or SSS for similarity will not work, leaving SAS as the only option.  If we compare the two given sides in each triangle, we notice that the ratio of the longer side in triangle I to the longer side in triangle II is

The ratio of the shorter sides in each triangle are

Notice we have equal ratios and thus a proportion.  However, we still must confirm that the included angles are congruent.  The measure for this angle is not given in triangle I, but we can calculate since all three angles must add up to 180 degrees.  Calculation tells us that the measure is 98 degrees, which unfortunately does not equal the 110 from triangle II.  Therefore, we have no SAS and therefore no similarity between I and II.

Transitioning to I and III, we only have angles in triangle III, so we are unable to use either SSS or SAS.  However, we previously calculated the measure third angle in triangle I to be 98.  Therefore, two of our angles are congruent, meaning we have AA and thus similarity.

Regarding II and III, we can use some logic.  Since we know I and III are similar, then if II and III were also similar, then we could use the transitive property to conclude that I and II are also similar.  But we know this is false, so II and III cannot be similar.

Therefore, the only two similar triangles are I and III.

To determine if the triangles are similar, set up a proportion.

2/4 = 4/8 = 5/10

 When we do this, we cross multiply to get a true statement.

Or, we can find the scale factor.

Since the scale factor is 2 for all three lengths, it becomes clear that these triangles are similar. 

For both triangles, we are given the "legs." Based on their relative lenghts, we can see that 2 corresponds with 3, and 7 corresponds with 10.5. First we need to make sure that these two triangles are similar. We can do this by comparing the ratios of corresponding sides:

There are a couple of ways to go from here. One would be to cross-multiply:

the ratios are equal, so the triangles are similar, and the scale factor is .

Based on their positions relative to the congruent angles, and their relative lengths, we can see that 1.5 corresponds to 6, and 8 corresponds to 30. If the ratios of corresponding sides are equal, then the triangles are congruent: 

We can compare these in a couple different ways. One would be to cross-multiply:

These triangles are not similar.

Since the angles are supplementary, their sum is 180 degrees.  Because they are in a ratio of 1:4, the following expression could be written:

The angles are equal. When two parallel lines are intersected by a transversal, the corresponding angles have the same measure.  

Let x equal the measure of angle DPB. Because the measure of angle APC is eighty-one degrees larger than the measure of DPB, we can represent this angle's measure as x + 81. Also, because the measure of angle CPD is equal to the measure of angle DPB, we can represent the measure of CPD as x.

Since APB is a straight line, the sum of the measures of angles DPB, APC, and CPD must all equal 180; therefore, we can write the following equation to find x:

x + (x + 81) + x = 180

Simplify by collecting the x terms.

3x + 81 = 180

Subtract 81 from both sides.

3x = 99

Divide by 3.

x = 33.

This means that the measures of angles DPB and CPD are both equal to 33 degrees. The original question asks us to find the measure of angle CPB, which is equal to the sum of the measures of angles DPB and CPD.

measure of CPB = 33 + 33 = 66.

The answer is 66.