1. **Problem Statement:** Given angles in a diagram with four intersecting lines forming 14 angles, find the measures of angles 2, 4, 5, 6, 7, 8, 9, 11, 13, and 14 using the given angles 1=35°, 3=125°, 10=90°, and 12=40°. 2. **Key Concepts:** - Vertical angles are equal. - Supplementary angles sum to 180°. - Complementary angles sum to 90°. - Adjacent angles share a common side. 3. **Find missing angles:** - Since ∠1=35°, vertical angle ∠5=35°. - Since ∠3=125°, vertical angle ∠4=125°. - ∠2 and ∠6 are vertical angles, and since ∠1 and ∠2 are supplementary (on a straight line), ∠2=180°−35°=145°, so ∠6=145°. - ∠7 and ∠8 are vertical angles; since ∠3 and ∠8 are supplementary, ∠8=180°−125°=55°, so ∠7=55°. - ∠9 and ∠12 are vertical angles, so ∠9=40°. - ∠11 and ∠14 are vertical angles; since ∠10=90° and ∠11 are supplementary, ∠11=90°, so ∠14=90°. - ∠13 and ∠14 are vertical angles, so ∠13=90°. 4. **Summary of angle measures:** - ∠2=145° - ∠4=125° - ∠5=35° - ∠6=145° - ∠7=55° - ∠8=55° - ∠9=40° - ∠11=90° - ∠13=90° - ∠14=90° 5. **Classify angle pairs:** - Supplementary pairs (sum 180°): ∠2 & ∠6, ∠3 & ∠4, ∠9 & ∠10, ∠11 & ∠12 - Complementary pairs (sum 90°): ∠7 & ∠8 - Adjacent angles (share a side): ∠1 & ∠5, ∠4 & ∠7, ∠9 & ∠12, ∠11 & ∠14 - Vertical angles (equal): ∠2 & ∠6, ∠3 & ∠4, ∠7 & ∠8, ∠1 & ∠5, ∠9 & ∠12, ∠11 & ∠14, ∠13 & ∠14 Final answers: 1. m∠2 = 145° 2. m∠4 = 125° 3. m∠5 = 35° 4. m∠6 = 145° 5. m∠7 = 55° 6. m∠8 = 55° 7. m∠9 = 40° 8. m∠11 = 90° 9. m∠13 = 90° 10. m∠14 = 90° Supplementary: ∠2 & ∠6, ∠3 & ∠4, ∠9 & ∠10, ∠11 & ∠12 Complementary: ∠7 & ∠8 Adjacent: ∠1 & ∠5, ∠4 & ∠7, ∠9 & ∠12, ∠11 & ∠14 Vertical: ∠2 & ∠6, ∠3 & ∠4, ∠7 & ∠8, ∠1 & ∠5, ∠9 & ∠12, ∠11 & ∠14, ∠13 & ∠14