1. **Identify all pairs of parallel segments.** Given the parallelogram and segments: - a) BD || AE - b) FD || AC - c) BF || CE Since in a parallelogram opposite sides are parallel, the pairs BD || AE and BF || CE are parallel segments. 2. **Name the segment parallel to the given segment.** - a) YZ || RT - b) RS || XZ - c) XY || ST In the parallelogram, opposite sides are parallel, so: - YZ is parallel to RT - RS is parallel to XZ - XY is parallel to ST 3. **Find each measure in ΔJKL with midpoints F, G, H:** Given: FG = 37, KL = 48, GH = 30 - a) FH = ? - b) JL = ? - c) KJ = ? - d) FJ = ? Since F, G, H are midpoints, segment connecting midpoints is half the length of the third side. - FH is parallel and half of KL, so $$FH = \frac{1}{2} \times 48 = 24$$ - JL is the same as KL, so $$JL = 48$$ - KJ is the same as JK, but no value given, so assume $$KJ = JL = 48$$ - FJ is half of KJ, so $$FJ = \frac{1}{2} \times 48 = 24$$ 4. **Find each measure in ΔAEN with midpoints C, P, T:** Given: PT = 13, EN = 43, CP = 29 - a) AE = ? - b) AN = ? - c) CT = ? - d) Perimeter of ΔAEN = ? Using midpoint theorem: - AE is twice PT, so $$AE = 2 \times 13 = 26$$ - AN is twice CP, so $$AN = 2 \times 29 = 58$$ - CT is half of AN, so $$CT = \frac{1}{2} \times 58 = 29$$ - Perimeter = AE + EN + AN = $$26 + 43 + 58 = 127$$ 5. **Solve for x in triangle with sides 10x + 44 and 8x - 23:** Assuming these sides are equal: $$10x + 44 = 8x - 23$$ Subtract 8x: $$10x - 8x + 44 = -23$$ $$2x + 44 = -23$$ Subtract 44: $$2x = -23 - 44$$ $$2x = -67$$ Divide by 2: $$x = \frac{\cancel{2}x}{\cancel{2}} = \frac{-67}{2} = -33.5$$ 6. **Solve for x in triangle with sides 19x - 28 and 6x + 7:** Set equal: $$19x - 28 = 6x + 7$$ Subtract 6x: $$19x - 6x - 28 = 7$$ $$13x - 28 = 7$$ Add 28: $$13x = 35$$ Divide by 13: $$x = \frac{35}{13} \approx 2.69$$ 7. **Find JL in triangle with sides 5x - 16 and 4x + 34:** Set equal: $$5x - 16 = 4x + 34$$ Subtract 4x: $$5x - 4x - 16 = 34$$ $$x - 16 = 34$$ Add 16: $$x = 50$$ Find JL: $$JL = 5x - 16 = 5(50) - 16 = 250 - 16 = 234$$ 8. **Find GH in polygon with sides 3x - 4 and 9x - 59:** Set equal: $$3x - 4 = 9x - 59$$ Subtract 3x: $$-4 = 6x - 59$$ Add 59: $$55 = 6x$$ Divide by 6: $$x = \frac{55}{6} \approx 9.17$$ Find GH: $$GH = 3x - 4 = 3 \times 9.17 - 4 = 27.5 - 4 = 23.5$$