Question

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

1 ∠h and ∠g are complementary
2 ∠i and ∠g are complementary
3 m∠h + m∠g = 90°
4 m∠i + m∠g = 90°
5 m∠h + m∠g = m∠i + m∠g
6 m∠i = m∠h
7 ∠i ≅ ∠h
choose
definition of complementary angles
substitution/transitive
✓ given
subtraction property
definition of congruence
1
2 given
3 choose
4 choose
5 choose
6 choose
7 choose

Explanation:

Step1: Reason for statement 3

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

Step2: Reason for statement 4

Since \( \angle I \) and \( \angle G \) are complementary, by the definition of complementary angles (two angles are complementary if the sum of their measures is \(90^{\circ}\)), we have \(m\angle I + m\angle G = 90^{\circ}\).

Step3: Reason for statement 5

We know that \(m\angle H + m\angle G=90^{\circ}\) (from statement 3) and \(m\angle I + m\angle G = 90^{\circ}\) (from statement 4). By the substitution/transitive property (if \(a = c\) and \(b=c\), then \(a = b\)), we can substitute \(90^{\circ}\) in the two equations, so \(m\angle H + m\angle G=m\angle I + m\angle G\).

Step4: Reason for statement 6

Starting with \(m\angle H + m\angle G=m\angle I + m\angle G\). By the subtraction property of equality (if \(a + b=c + b\), then \(a=c\)), we subtract \(m\angle G\) from both sides of the equation. So \(m\angle I=m\angle H\).

Step5: Reason for statement 7

By the definition of congruence (two angles are congruent if their measures are equal). Since \(m\angle I=m\angle H\), we have \( \angle I\cong\angle H\).

Answer:

1. Definition of Complementary Angles; 4. Definition of Complementary Angles; 5. Substitution/Transitive; 6. Subtraction Property; 7. Definition of Congruence.