find the measure of each numbered angle. 1. 2. ans: angle 1 : 90o – 60o = 30o ans : angle 1 : 180o – ( 66o+ 58 o ) = 56o angle 2 : 90o – 30o = 60o angle 2 = angle 1 = 56o angle 3 : 180o – ( 56o+ 50 o ) = 74o 3. 4. ans: angle 1 : 180o – ( 35o+ 36 o ) = 109o ans : angle 1 = angle 2 = (180o - 146 o) / 2 = 17o angle rvs = angle 1 = 109o angle 2 : angle rvs – 80o = 109o – 80o = 29o angle 3 : 180o – angle 1 = 180o – 109o = 71o example 2 find each measure. 5. m∠abc 6. m∠f ans : 2xo + 95 o = 145o ans : xo + x o = 58o 2xo = 145o - 95 o = 50 o 2xo = 58 o xo = 50 o / 2 = 25o xo = 58 o / 2 = 29o 7. towers a lookout tower sits on a network of struts and posts. leslie measured three angles on the tower. if m∠1 = (7x − 7)°, m∠2 = (4x + 2)°, and m∠3 = (2x + 6)°, what is m∠1? ans : from the triangle with external angle m∠1 and internal angles m∠2 and m∠3 : m∠1 = m∠2 + m∠3 7x – 7 = (4x + 2) + (2x + 6) = 6x + 8 7x - 6x = 8 + 7 = 15 thus , m∠1 = 15° 8. gardening a gardener uses a grow light to grow vegetables indoors. if m∠1 = 8x° and m∠2 = (7x − 4)°, what is m∠1? ans : m∠1 + m∠2 = 116° 8x° + (7x − 4)° = 116° 15x° − 4° = 116° 15x° = 116°+ 4° = 120° x ° = 120°/ 15 = 8° m∠1 = 8x° = 8 * 8° = 64° example 3 find each measure. 9. m∠1 = 90° − 28° = 62° 10. m∠2 = 90° − 51° = 39° 11. m∠3 = 51° − 25° = 26° 12. m∠4 = 180° − ( 35°+ 90°) = 55° 13. m∠5 = 90° − 35° = 55° 14. m∠6 = 90° − m∠4 = 90° − 55° = 35° mixed exercises find the value of x. then find the measure of each angle. 15. 16. 17. ans : 2x°+3x°+4x° = 180° ans: x° + 2x° = 90° ans : (5x + 62)° = 9x° = 180° 3x° = 90° 37°+(3x+47)° x° = 180°/9 = 20° x° = 90°/3 = 30° 5x° + 62° = 3x° + 84° 5x° - 3x° = 84° - 62° 2x° = 22° x° = 22°/2 = 11° 18. proof determine whether the following statement is true or false. if false, give a counterexample. if true, write a narrative proof to support your conclusion. if the sum of two acute angles of a triangle is greater than 90°, then the triangle is acute. the statement is false. a counterexample to this statement would be an obtuse triangle. in an obtuse triangle, one of the angles is greater than 90°. let's say one angle is 100°, and the other two angles are acute and sum up to less than 90°. in this case, the sum of two acute angles (each less than 90°) is greater than 90°, but the triangle itself is not acute as it contains an angle greater than 90°. therefore, the statement that if the sum of two acute angles of a triangle is greater than 90°, then the triangle is acute is false. proof write the specified type of proof for each corollary. 19. flow proof of corollary 5.1 given: ∠r is a right angle.