HONORS GEOMETRY
7.4. Parallel Lines and Proportional Parts
Do Now
Homework
• Questions?
• Comments?
• Confusions?
• Concerns?
• ASK ASK ASK!
Triangle Proportionality Theorem
• If a line is parallel to one side of a triangle and intersects
the other two sides, then it divides the sides into
segments of proportional lengths.
Example One:
• In ∆PQR below, 𝑆𝑇 ∥ 𝑅𝑄. If 𝑃𝑇 = 7.5, 𝑇𝑄 = 3, and 𝑆𝑅 =
2.5, find 𝑃𝑆.
Example Two:
• Find x.
Example Three:
• Find a
Example Four:
• Why doesn’t the triangle proportionality theorem apply to
this problem?
You Try!
• Assuming that 𝐴𝐷 ∥ 𝐵𝐶, find x.
Converse of Triangle Proportionality Theorem
• If a line intersects two sides of a triangle and separates
the sides into proportional corresponding segments, then
the line is parallel to the third side of the triangle.
Example Five:
• Is 𝐷𝐸 ∥ 𝐶𝐵? Justify your answer
Example Six:
You Try!
• Is 𝐷𝐸 ∥ 𝐵𝐴? Justify your answer
Example Seven:
• What must x be in order for 𝐷𝐸 ∥ 𝐵𝐶?
Proportional Parts of Parallel Lines
• If three or more parallel lines intersect two transversals,
then they cut off the transversals proportionally.
Example Eight:
• Find x
Example Nine:
• Find x
Example Ten:
• Find x and y!
You Try!
• Find AG and GF and x
Do Now:
• Find the missing variables
Slightly Trickier:
• Ignore the numbers given to you. Instead assume that…
• 𝐴𝐵 = 12
• 𝐵𝐶 = 6
• 𝐶𝐷 = 4
• 𝐴𝐺 = 2𝑥 − 𝑦
• 𝐺𝐹 = 3𝑦 − 2𝑥
• 𝐹𝐸 = 8
• Find x and y
You Try!
• Ignore the numbers given to you. Instead assume that…
• 𝐴𝐵 = 9
• 𝐵𝐶 = 6
• 𝐶𝐷 = 1
• 𝐴𝐺 = 3𝑦 + 2𝑥
• 𝐺𝐹 = 4𝑦 − 6𝑥
• 𝐹𝐸 = 13
• Find x and y
Ponder…..
• What happens if the scale factor of two triangles is one?
Congruent Parts of Parallel Lines
• If three or more parallel lines cut off congruent segments
on one transversal, then they cut off congruent segments
on every transversal.
Example Eleven:
You Try!
• Find x and y in the picture to the left and x in the picture to
the right.
Practice Problems
• Try some on your own/in your table groups
• As always don’t hesitate to ask me and/or your
tablemates questions if you are confused!
Exit Ticket:
• Find x