Quadrilaterals · Ages 14–15
The intercept theorem
Equal pieces on one crossing line mean equal pieces on any other.
How to do it
Method
Equal pieces on one crossing line mean equal pieces on any other
AB = BC: DE = EF; 2x + 1 = x + 4, x = 3
Another way
Even lines cut equal parts
10 cm across 5 gaps: 10 ÷ 5 = 2 cm
In full
The intercept theorem: if three or more parallel lines cut off equal lengths on one line crossing them, they cut off equal lengths on any other line crossing them (intercept thm.). AB = BC, so DE = EF. The lengths on the two crossing lines needn't be the same: AB = BC = 3 cm and DE might be 4 cm, but then EF is 4 cm too. One way: use the theorem straight. Another: see it as the mid-point theorem in a trapezium, or picture the parallels as evenly spaced lines. A use: dividing a segment into equal parts — lay a 10 cm strip slantwise across 5 gaps of ruled paper and it is cut into 5 pieces of 2 cm. With an unknown, make an equation: DE = (2x + 1) cm, EF = (x + 4) cm, 2x + 1 = x + 4, x = 3.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
The four level lines in the picture are parallel. x =
Show the answer and working
Answer: 7 cm
AB = BC = CD = 6 cm, so EF = FG = GH (intercept thm.): x = 7.
Example 2
The three level lines in the picture are parallel. x =
Show the answer and working
Answer: 4 cm
AB = BC = 4 cm, so DE = EF (intercept thm.): x = 4.
Example 3
Three parallel lines cut two lines at A, B, C and at D, E, F. AB = 2 cm and BC = 5 cm, so DE = EF.
True or false?
Show the answer and working
Answer: False
The intercept theorem needs AB = BC to give DE = EF; 2 cm and 5 cm must be equal.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.The four level lines in the picture are parallel. x =
Answer:
2.Three parallel lines cut two lines at A, B, C and at D, E, F, with AB = BC, DE = (2x + 6) cm and EF = (3x − 1) cm. x =
Answer:
3.Three parallel lines cut two lines at A, B, C and at D, E, F. AB = 4 cm and BC = 4 cm, so DE = EF.
True or false?
Answer:
4.The four level lines in the picture are parallel, and EH = 15 cm. x =
Answer:
5.Four parallel lines cut two lines at A, B, C, D and at E, F, G, H, with AB = BC = CD, EF = (x + 4) cm and FG = (3x − 2) cm. GH =
Answer:
All the answers
- 1. 6 cm
- 2. 7
- 3. True
- 4. 5 cm
- 5. 7 cm
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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