Question

∠h and ∠j are complementary and ∠h and ∠i are complementary. complete the proof that ∠j ≅ ∠i.

statement
1 ∠h and ∠j are complementary
2 ∠h and ∠i are complementary
3 m∠h + m∠j = 90°
4 m∠h + m∠i = 90°
5 m∠h + m∠j = m∠h + m∠i
6 m∠j = m∠i
7 ∠j ≅ ∠i

reason
given
given

definition of complementary angles

properties of addition, subtraction, multiplication, and division

Explanation:

Step1: Definition of complementary angles

Since \(\angle H\) and \(\angle J\) are complementary, by the definition of complementary angles (two angles whose sum is \(90^{\circ}\)), we have \(m\angle H + m\angle J=90^{\circ}\).

Step2: Substitution property of equality

We know \(m\angle H + m\angle J = 90^{\circ}\) (from Step 1) and \(m\angle H + m\angle I=90^{\circ}\) (given as the definition of complementary angles for \(\angle H\) and \(\angle I\)). So, by the substitution property of equality (if \(a = b\) and \(a = c\), then \(b = c\)), \(m\angle H + m\angle J=m\angle H + m\angle I\).

Step3: Definition of congruent angles

Since \(m\angle J=m\angle I\) (from Step 2, using the subtraction property of equality: subtract \(m\angle H\) from both sides of \(m\angle H + m\angle J=m\angle H + m\angle I\)), and by the definition of congruent angles (two angles are congruent if their measures are equal), \(\angle J\cong\angle I\).

Answer:

1. Definition of complementary angles; 5. Substitution property of equality; 7. Definition of congruent angles.