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Grade 8 / Middle 2 (age 13-14)

grade 8 Parallel Lines & Proportional Segments

Parallel Lines & Proportional Segments

When parallels cross two lines, the ratio on one side matches the other. Inside a triangle, a parallel to one side splits the other two equally, and a segment joining two midpoints is parallel to the base and half as long. This midpoint theorem shows up constantly on tests. Change the ratio and the split point and the matching ratios and the half-length at the midpoints update.

Intuition — Proportions From Parallel Lines

2
👀 Parallel Line Magic
①When three parallel lines cut two transversals
②The segment ratio on one side equals
③The segment ratio on the other side — that is the parallel-line property!

Parallel Lines Inside a Triangle

0.5
Triangle Parallel Theorem
DE ∥ BC → AD:AB = AE:AC = DE:BC
A segment parallel to one side divides the other two in the same ratio

Midpoint Theorem

Midpoint Theorem
D, E midpoints → DE ∥ BC, DE = 12BC
A segment joining midpoints is parallel to the third side and half its length
💡 Key Idea of the Midpoint Theorem
①Join midpoints D, E of AB and AC
②DE is parallel to BC
③DE = BC / 2
④One of the most frequent test theorems!

Applications

📝 Common Patterns
①Parallel lines in a trapezoid → set up a proportion
②Parallel lines in a triangle → use the similarity ratio to find a side
③Midpoint theorem → find unknown side via half/double

Work It Out

Example 1
In △ABC, DE ∥ BC with AD = 3, DB = 6, AE = 4. Find EC.
1
A line parallel to a side splits the other two sides equally: AD : DB = AE : EC.
3 : 6 = 4 : EC
2
Solve the proportion.
3 × EC = 24 ⇒ EC = 8
▸ EC = 8
A line parallel to one side of a triangle divides the other two in the same ratio.
Example 2
Three parallel lines cut one transversal in ratio 1 : 2. On another the parts are 3 and x. Find x.
1
Parallel lines cut two transversals in the same ratio.
1 : 2 = 3 : x
2
Solve the proportion.
1 × x = 2 × 3 ⇒ x = 6
▸ x = 6
The segment ratio made by several parallels is the same on any transversal.

Exam Wrap-up

Core Idea
ℓ₁ ∥ ℓ₂ ∥ ℓ₃ → a:b = c:d (corresponding segment ratios are equal)
Parallel lines create proportions — the heart of this unit
Grade-8 school exam type
In △ABC, M and N are the midpoints of AB and AC. If BC = 10, what is MN?
①4
②5
③6
④8
⑤10
▸ ② 5
1
Midsegment theorem: MN is parallel to BC and half its length.
MN = (1/2) × BC
2
Substitute.
= (1/2) × 10 = 5
🎯 Exam Points
①3 parallel + 2 transversal lines → equal corresponding ratios
②Inside triangle: DE ∥ BC → AD:DB = AE:EC
③Midpoint theorem: top frequency — midpoints give parallel + half
④Solving proportions: a:b = c:d → ad = bc
⑤The converse holds inside a triangle when two sides are cut in the same ratio. Equal cuts on two transversals do not make three lines parallel
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