Side Splitter Theorem
Essential Question: What is the Side Splitter Theorem
and how can you use it to find sides of a triangle?
Assessment: Students will demonstrate in writing
through examples in their notebooks.
Side Splitter Theorem
A line parallel to one side of
a triangle divides the other
two sides proportionally.
Prove the Side Splitter Theorem
Given: 𝐷𝐸||𝐵𝐶
Prove:
𝐵𝐷
𝐷𝐴
=
𝐶𝐸
𝐸𝐴
Statement
Reason
1. 𝐷𝐸||𝐵𝐶
1. Given
2. ∠A ≅ ∠A
2. Reflexive Property
3. ∠ADE ≅ ∠ABC
3. Corresponding Angles are congruent
4. ∆ ADE ~ ∆ABC
4. AA triangle similarity postulate
5.
𝐴𝐵
𝐴𝐷
=
𝐴𝐶
𝐴𝐸
5. Definition of polygon similarity
6. 𝐴𝐵 = 𝐵𝐷 + 𝐷𝐴 𝑎𝑛𝑑 𝐴𝐶 = 𝐶𝐸 + 𝐸𝐴
7.
𝐵𝐷+𝐷𝐴
𝐴𝐷
8.
𝐵𝐷 𝐷𝐴 𝐶𝐸 𝐸𝐴
+
=
+
𝐷𝐴 𝐷𝐴 𝐴𝐸 𝐸𝐴
9.
10.
=
𝐶𝐸+𝐸𝐴
𝐴𝐸
𝐵𝐷
𝐶𝐸
+1=
+1
𝐷𝐴
𝐸𝐴
𝐵𝐷 𝐶𝐸
=
𝐷𝐴 𝐸𝐴
6. Segment Addition Postulate
7. Substitution
8. Addition of fractions
9. Reduction of fractions
10. Subtraction Property
Examples
Find the value of x.
A student examines the triangle
shown above. He notices that
lines BE and CD look parallel. The
student claims that they are
parallel by look alone.
Is the student correct?
6
3
A) Yes, because =
B) Yes, because
6
12
12
9
=
3
9
6
9
3
12
6
3
12
9
C) Yes, because =
D) No, because ≠
A triangular city park has a path that
passes through it. The path is
represented by NQ and is parallel to
one side of the park, which is
represented by MR. Ray and Kevin
are arguing about the length of QR.
Ray thinks that the length of QR will
be less than the length of NM, while
Kevin thinks that it will be greater.
Find the length of QR, rounded to
the nearest foot.
Final Example
Carl is attempting to find the value of x by using the
Side Splitter Theorem. Carl set up the proportion
6/3=12/x. If Carl is correct, what should his answer be?
If Carl is incorrect, what is the correct proportion and
answer?
A) Carl is correct. X = 24
B) Carl is correct. X = 6
C) Carl is incorrect.
6
12
=
3
𝑥
and x = 1.5
D) Carl is incorrect. He cannot
find x due to lines BE and CD not
being parallel.