Triangle Proportionality Theorem
1. Name the two triangles that are in figure 1.
_______________________________________________
2. Explain why
(illustrated in figure 2) is a
transversal.___________________________________
3. Explain why the two triangles are similar.
4. Solve for the unknown quantities in each of the following figures. Assume that lines which look parallel in
each figure are parallel.
a.
c.
b.
Triangle Proportionality Theorem
If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides
proportionally.
Proof #1:
Given:
Prove:
Complete the proof:
Show that ΔAEF ~ ΔABC
Since
you can conclude that
and 3
4 by __________
So ΔAEF ~ ΔABC by ____________
Use the fact that corresponding sides of similar triangles are proportional to complete the proof
____________ Corresponding sides are proportional
____________ Segment Addition Postulate
_______________ Use the property that
_______________ Subtract 1 from both sides.
________________ Take the reciprocal of both sides.
Converse of the Triangle Proportionality Theorem
If a line divides two sides of a triangle proportionally, then it is parallel to the third side
Proof #2:
Given:
Prove:
Complete the proof. Show that ΔAEF ~ ΔABC
It is given that
and taking the reciprocal of both sides show that __________________.
Now add 1 to both sides by adding
to the left side and
to the right side.
_____________________________
Adding and using the Segment Addition Postulate gives _______________________.
Since
by ____________________.
As corresponding angles of similar triangles, AEF
So,
by _________________.
________________.
Let’s practice finding the length of a segment since you know how to prove the Triangle Proportionality
Theorem and its converse.
1. What is the length of NR?
3. Given the diagram, determine whether
2) What is the length of DF?
is parallel to
.