Math Problem Statement

Complete the proof that \( \angle J \cong \angle K \) given that \( \angle K \) and \( \angle I \) are complementary and \( \angle J \) and \( \angle I \) are complementary.

Solution

The image shows a proof table where we need to complete the reasoning to show that ∠J≅∠K\angle J \cong \angle K. Let's complete the proof.

Given:

  • ∠K\angle K and ∠I\angle I are complementary.
  • ∠J\angle J and ∠I\angle I are complementary.

Proof Steps:

StatementReason
1. ∠K\angle K and ∠I\angle I are complementaryGiven
2. ∠J\angle J and ∠I\angle I are complementaryGiven
3. m∠K+m∠I=90∘m\angle K + m\angle I = 90^\circDefinition of complementary angles
4. m∠J+m∠I=90∘m\angle J + m\angle I = 90^\circDefinition of complementary angles
5. m∠K+m∠I=m∠J+m∠Im\angle K + m\angle I = m\angle J + m\angle ITransitive Property of Equality (both sums equal 90°)
6. m∠J=m∠Km\angle J = m\angle KProperties of addition, subtraction (subtract m∠Im\angle I from both sides)
7. ∠J≅∠K\angle J \cong \angle KDefinition of congruent angles (equal measures)

This completes the proof that ∠J≅∠K\angle J \cong \angle K.

Would you like further details or explanations on any part of this proof? Here are a few related questions that could deepen understanding:

  1. Why are angles considered complementary when their sum is 90 degrees?
  2. Can the transitive property of equality be applied in cases other than equality of angles?
  3. How is the definition of congruent angles connected to their measures?
  4. Could a similar approach be used to prove angles are supplementary instead of complementary?
  5. What happens if two angles are neither complementary nor supplementary?

Tip: When proving congruence or equality of angles, always ensure that the definitions of complementary, supplementary, and congruent angles are clearly understood.

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Math Problem Analysis

Mathematical Concepts

Geometry
Angle Relationships

Formulas

m∠K + m∠I = 90°
m∠J + m∠I = 90°

Theorems

Definition of Complementary Angles
Transitive Property of Equality
Definition of Congruent Angles

Suitable Grade Level

Grades 7-10

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