1. **Problem statement:** We have 6 parallel lines cut by two transversals, creating segments with some missing lengths $x$, $y$, $z$, and $w$. We need to find these missing lengths using properties of parallel lines and similar triangles. 2. **Key formula and rule:** When parallel lines are cut by transversals, corresponding segments on the transversals are proportional. This means for segments on two transversals, the ratios of corresponding segments are equal: $$\frac{a}{b} = \frac{c}{d}$$ where $a,b$ are segments on one transversal and $c,d$ on the other. --- ### a) Given segments: - On first transversal: 4, 4, 6 - On second transversal: 3, $x$, $y$ Using proportionality: $$\frac{4}{3} = \frac{4}{x} = \frac{6}{y}$$ Step 1: Find $x$: $$\frac{4}{3} = \frac{4}{x} \implies 4x = 12 \implies x = 3$$ Step 2: Find $y$: $$\frac{4}{3} = \frac{6}{y} \implies 4y = 18 \implies y = 4.5$$ --- ### b) Given segments: - Lower transversal: 3.6, $S$, 2.4 - Upper transversal: 1.8, $x$, 1.8 Since $S$ is between 3.6 and 2.4, total lower length is $3.6 + 2.4 = 6$. Total upper length is $1.8 + 1.8 = 3.6 + x$ (but $x$ is missing, so we use proportionality between segments). Using proportionality for segments around $S$: $$\frac{x}{1.8} = \frac{3.6}{2.4} = 1.5$$ Step 1: Find $x$: $$x = 1.8 \times 1.5 = 2.7$$ Step 2: Find $y$ (given as 1.8, so no calculation needed). --- ### c) Given segments: - Upper transversal: 10.5, $S$, 8.4 - Lower transversal: 3.2, 7.4, 6.2 - Missing lengths: $x, y, z, w$ Using proportionality: $$\frac{10.5}{3.2} = \frac{y}{7.4} = \frac{z}{6.2} = \frac{w}{?}$$ Step 1: Calculate ratio: $$r = \frac{10.5}{3.2} \approx 3.28125$$ Step 2: Find $y$: $$y = 7.4 \times r = 7.4 \times 3.28125 = 24.28$$ Step 3: Find $z$: $$z = 6.2 \times r = 6.2 \times 3.28125 = 20.34$$ Step 4: Find $w$ (assuming $w$ corresponds to a segment of length 7.4 on lower transversal): $$w = 7.4 \times r = 24.28$$ --- **Final answers:** - a) $x=3$, $y=4.5$ - b) $x=2.7$, $y=1.8$ - c) $y \approx 24.28$, $z \approx 20.34$, $w \approx 24.28$ (assuming $w$ corresponds to 7.4 segment)