Question

given: line m intersects line l; ∠1 ≅ ∠2. prove: m∠1 = 90° and m∠2 = 90° statements 1. line m intersects line l; ∠1 ≅ ∠2 2. ∠1 and ∠2 are a linear pair of angles 3. 4. m∠1 = 90° and m∠2 = 90° reasons 1. given 2. 3. theorem 3.4a 4. def. of perpendicular lines a. l ⊥ m b. def. of linear pair of angles c. m∠1 = m∠2 d. def. of congruent angles

Explanation:

Step1: Determine the reason for statement 2

By the definition of a linear pair of angles (two adjacent angles whose non - common sides are opposite rays), since line \(m\) intersects line \(\ell\), \(\angle1\) and \(\angle2\) are a linear pair. So the reason for statement 2 is "Def. of linear pair of angles".

Step2: Determine the statement for step 3

Since \(\angle1\cong\angle2\) (given) and \(\angle1\) and \(\angle2\) are a linear pair (so \(\angle1+\angle2 = 180^{\circ}\) by the linear - pair postulate), and if \(\angle1\cong\angle2\), then \(m\angle1=m\angle2\) (def. of congruent angles). Also, using the linear - pair postulate \(m\angle1 + m\angle2=180^{\circ}\), substituting \(m\angle2=m\angle1\) gives \(2m\angle1 = 180^{\circ}\), \(m\angle1=90^{\circ}\) and \(m\angle2 = 90^{\circ}\). But Theorem 3.4A (if two angles are a linear pair and congruent, then the lines are perpendicular) is used after we know that \(m\angle1=m\angle2\). The statement for step 3 is \(m\angle1=m\angle2\) (because \(\angle1\cong\angle2\) and by the def. of congruent angles \(m\angle1 = m\angle2\)).

Answer:

1. b. Def. of linear pair of angles; 3. c. \(m\angle1=m\angle2\)