Question

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

statement | reason
1 ∠i and ∠j are complementary | given
2 ∠k and ∠i are complementary | given
3 m∠i + m∠j = 90° | blank
4 m∠k + m∠i = 90° | blank
5 m∠i + m∠j = m∠k + m∠i | blank
6 m∠k = m∠j | properties of addition, subtraction, multiplication, and division
7 ∠k ≅ ∠j | definition of congruence

Explanation:

Step1: Reason for Statement 3

The definition of complementary angles states that if two angles are complementary, the sum of their measures is \( 90^\circ \). Since \( \angle I \) and \( \angle J \) are complementary (given in Statement 1), we can conclude \( m\angle I + m\angle J = 90^\circ \) by the definition of complementary angles.

Step2: Reason for Statement 4

Similarly, for \( \angle K \) and \( \angle I \) being complementary (given in Statement 2), by the definition of complementary angles, the sum of their measures is \( 90^\circ \). So \( m\angle K + m\angle I = 90^\circ \) is derived from the definition of complementary angles.

Step3: Reason for Statement 5

We know from Statement 3 that \( m\angle I + m\angle J = 90^\circ \) and from Statement 4 that \( m\angle K + m\angle I = 90^\circ \). Since both expressions equal \( 90^\circ \), we can set them equal to each other by the Transitive Property of Equality (if \( a = c \) and \( b = c \), then \( a = b \)). So \( m\angle I + m\angle J = m\angle K + m\angle I \) is by the Transitive Property of Equality.

Answer:

• For Statement 3: Definition of complementary angles
• For Statement 4: Definition of complementary angles
• For Statement 5: Transitive Property of Equality