Question

1. \if ∠1 and ∠2, then ∠1 ≅ ∠2\ a. definition of congruence b. angle addition postulate c. linear pair theorem d. vertical angles theorem \if ∠a and ∠b are complementary, then m∠a + m∠b = 90°\ a. complement theorem b. definition of a right angle c. congruent complements theorem d. definition of complementary angles given: \if l ⊥ m, then ∠1 is a right angle\ a. definition of complementary angles b. definition of a right angle c. definition of perpendicular d. complement theorem \if cd = ef, then \overline{cd} ≅ \overline{ef}\ a. definition of congruence b. reflexive property c. symmetric property d. definition of midpoint

Explanation:

• For the first statement “If ∠A and ∠B are complementary, then \(m\angle A + m\angle B=90^{\circ}\)”: By the definition of complementary angles, two angles are complementary if the sum of their measures is \(90^{\circ}\).
• For the second statement “If \(l\perp m\), then \(\angle1\) is a right angle”: When two lines are perpendicular, the angles formed are right angles. This is the definition of perpendicular lines.
• For the third statement “If \(CD = EF\), then \(\overline{CD}\cong\overline{EF}\)”: Congruent segments are defined as segments that have the same length. If \(CD = EF\) (i.e., they have the same length), then by the definition of congruence for segments, \(\overline{CD}\cong\overline{EF}\).
• For the fourth statement “If \(\angle1\cong\angle2\)”: Vertical angles are congruent. When two lines intersect, the opposite angles (vertical angles) are equal in measure, so by the Vertical Angles Theorem, \(\angle1\cong\angle2\).

Answer:

1. D. Definition of Complementary Angles
2. C. Definition of Perpendicular
3. A. Definition of Congruence
4. D. Vertical Angles Theorem