Exploring Triangle Proportionality Theorems

The video tutorial covers solving problems involving proportions and triangles. It begins with explaining triangle similarity and solving for x using proportions. The side splitter theorem is introduced to find the value of y when lines are parallel. The tutorial then calculates the sum of x and y using parallel lines and proportions. Finally, the angle bisector theorem is applied to find x, demonstrating how a bisected angle divides the opposite side proportionally to the adjacent sides.

If triangle ABC is similar to triangle DBE, and AB = 20, BD = 8, BE = 6, and BC = 6 + x, what is the value of x?

• 10

• 7

• 9

• 8

What is the ratio of the sides AB to BD in similar triangles ABC and DBE if AB = 20 and BD = 8?

• 3

• 1.5

• 2.5

• 2

Using the side-splitter theorem, if DE is parallel to AC and BD = 15, AD = 24, BE = y, and EC = 30, what is the value of y?

• 22.5

• 20

• 18.75

• 15

In the side-splitter theorem, if a line is parallel to one side of a triangle and intersects the other two sides, what does it do to those sides?

• None of the above

• Makes them congruent

• Divides them proportionally

• Divides them equally

If DE is parallel to BC and AE = 24, EC = 32, AD = 6x - 3, and DB = x^2 - 4, what is the value of x?

• 9

• 8

• 7

• 6

Using the side-splitter theorem, if AE = 24, EC = 32, AD = 6x - 3, and DB = x^2 - 4, what is the sum of x and y?

• 10

• 12

• 13

• 15

If three parallel lines intersect two transversals, and AC = 12, CE = 30, BD = 18, and DF = x, what is the value of x?

• 55

• 50

• 45

• 40

In a figure with four parallel lines cut by two transversals, if BH = 65 and the segments on the left are in the ratio 6:8:12, what is the length of segment DF?

• 20

• 18

• 22

• 15

According to the angle bisector theorem, if a ray bisects an angle of a triangle, what does it do to the opposite side?

• Makes it congruent to the adjacent sides

• Divides it proportionally to the adjacent sides

• Divides it equally

• None of the above

Using the angle bisector theorem, if AD = x, DC = 16, AB = 12, and BC = 20, what is the value of x?

• 8.4

• 9.6

• 11.4

• 10.2