Question

given: ∠2 and ∠3 form a right angle; m∠1 + m∠2 = 90° prove: ∠1 ≅ ∠3 statements reasons 1. ∠2 and ∠3 form a right angle; m∠1 + m∠2 = 90° 1. given 2. ∠2 and ∠3 are complementary 2. 3. definition of complementary ∠s 3. 4. ∠1 ≅ ∠3 4.

Explanation:

Step1: Recall complementary - angle definition

If the sum of two angles is 90°, they are complementary. Since ∠2 and ∠3 form a right - angle (90°), by the definition of complementary angles, ∠2 and ∠3 are complementary.

Step2: Write the angle - sum equations

We know that \(m\angle1 + m\angle2=90^{\circ}\) and \(m\angle2 + m\angle3 = 90^{\circ}\) (because ∠2 and ∠3 are complementary).

Step3: Use the substitution or transitive property

If \(a + b=c\) and \(d + b=c\), then \(a=d\). Here, let \(a = m\angle1\), \(b = m\angle2\), \(d = m\angle3\), and \(c = 90^{\circ}\). So, \(m\angle1=m\angle3\).

Step4: Recall the definition of congruent angles

If two angles have the same measure, they are congruent. Since \(m\angle1=m\angle3\), then ∠1≅∠3.

Answer:

Statements	Reasons
2. ∠2 and ∠3 are complementary	2. Definition of complementary angles (sum of angles is 90°)
3. \(m\angle1 + m\angle2=m\angle2 + m\angle3\)	3. Substitution (since \(m\angle2 + m\angle3 = 90^{\circ}\) and \(m\angle1 + m\angle2 = 90^{\circ}\))
4. ∠1≅∠3	4. Definition of congruent angles (angles with equal measures are congruent)