Geometry Test Questions on Proportional Segments Cut by Parallel Lines

📐 Quick Study Guide

• 📏 Parallel Lines: Lines that never intersect.
• ✂️ Transversal: A line that intersects two or more other lines.
• ✨ Theorem: If parallel lines cut transversals, the segments are proportional.
• ✍️ Basic Proportionality Theorem (Thales' Theorem): If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio. Mathematically, if line DE is parallel to BC in triangle ABC, then $\frac{AD}{DB} = \frac{AE}{EC}$.
• ➕ Corollary: If three or more parallel lines are intersected by two transversals, then they divide the transversals proportionally. If lines l, m, and n are parallel, and transversals t and u intersect them, then $\frac{AB}{BC} = \frac{DE}{EF}$.
• 🧮 Solving for Unknowns: Set up proportions and cross-multiply to solve for unknown segment lengths. For example, if $\frac{x}{5} = \frac{8}{10}$, then $10x = 40$, and $x = 4$.

🧪 Practice Quiz

1. Question 1: In triangle ABC, DE is parallel to BC. AD = 4, DB = 6, AE = 5. Find EC.
   1. A) 7.5
   2. B) 6.5
   3. C) 8
   4. D) 5
2. Question 2: Three parallel lines intersect two transversals. On the first transversal, the segments are 3 and 9. On the second transversal, the first segment is 5. Find the second segment on the second transversal.
   1. A) 10
   2. B) 15
   3. C) 20
   4. D) 25
3. Question 3: If $\frac{AB}{BC} = \frac{DE}{EF}$, AB = 6, BC = 8, and EF = 12, find DE.
   1. A) 7
   2. B) 8
   3. C) 9
   4. D) 10
4. Question 4: Two parallel lines cut a transversal. One segment is x+2 and the other is 3. On another transversal the corresponding segments are 5 and 1. Solve for x.
   1. A) 10
   2. B) 13
   3. C) 16
   4. D) 19
5. Question 5: In triangle PQR, ST is parallel to QR. PS = 2, SQ = 6, PT = 3. What is TR?
   1. A) 6
   2. B) 9
   3. C) 12
   4. D) 15
6. Question 6: Parallel lines l, m, and n are intersected by transversals p and q. On p, the segments are 4 and x. On q, the corresponding segments are 6 and 9. What is x?
   1. A) 4
   2. B) 5
   3. C) 6
   4. D) 7
7. Question 7: In triangle XYZ, AW is parallel to YZ, XA = 5, AY = 7, XW = 6. Find WZ.
   1. A) 7.4
   2. B) 8.4
   3. C) 9.4
   4. D) 10.4