Question

transitive property of angle congruence m∠1 + m∠2 m∠1 = m∠3 definition of supplementary angles complete the two - column proof. 1. ∠1 and ∠2 are complementary. ∠1 and ∠3 are complementary. 1. given 2. 2. definition of complementa 3. m∠1 + m∠2 = m∠1 + m∠3 3. 4. 4. subtraction property of ec 5. ∠2 ≅ ∠3 5. m∠2 = m∠3 m∠1 + m∠2 = 90° m∠1 + m∠3 = 90° m∠1 + m∠2 = m∠2 + m∠3 m∠1 = m∠3 definition of congruent angles definition of complementary angles

Explanation:

Step1: Use the definition of complementary angles

Since \(\angle1\) and \(\angle2\) are complementary, \(m\angle1 + m\angle2=90^{\circ}\). Since \(\angle1\) and \(\angle3\) are complementary, \(m\angle1 + m\angle3 = 90^{\circ}\). So statement 2 is \(m\angle1 + m\angle2=90^{\circ}\) and \(m\angle1 + m\angle3=90^{\circ}\).

Step2: Use substitution property

We know that if \(a = b\) and \(a=c\), then \(b = c\). Here, since \(m\angle1 + m\angle2=90^{\circ}\) and \(m\angle1 + m\angle3=90^{\circ}\), by substitution property (if two expressions are equal to the same value, they are equal to each other), the reason for \(m\angle1 + m\angle2=m\angle1 + m\angle3\) (statement 3) is substitution property.

Step3: Use subtraction property of equality

Subtract \(m\angle1\) from both sides of the equation \(m\angle1 + m\angle2=m\angle1 + m\angle3\). We get \(m\angle2=m\angle3\) (statement 4).

Step4: Use the definition of congruent angles

If \(m\angle2=m\angle3\), then by the definition of congruent angles (two angles are congruent if their measures are equal), \(\angle2\cong\angle3\) (statement 5).

Answer:

1. \(m\angle1 + m\angle2 = 90^{\circ}\); \(m\angle1 + m\angle3=90^{\circ}\)
2. Substitution Property
3. \(m\angle2=m\angle3\)
4. Definition of congruent angles