Question

question 22 (1 point) given: ∠1 and ∠2 are complementary, and ∠2 and ∠3 are complementary. prove: ∠1 ≅ ∠3

Explanation:

Step1: Recall the definition of complementary angles

If two angles are complementary, the sum of their measures is \( 90^\circ \). So, since \( \angle 1 \) and \( \angle 2 \) are complementary, we have \( m\angle 1 + m\angle 2 = 90^\circ \). Also, since \( \angle 2 \) and \( \angle 3 \) are complementary, we have \( m\angle 2 + m\angle 3 = 90^\circ \).

Step2: Set the two equations equal to each other

From Step 1, we have \( m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3 \) (because both equal \( 90^\circ \)).

Step3: Subtract \( m\angle 2 \) from both sides

Subtracting \( m\angle 2 \) from both sides of the equation \( m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3 \), we get \( m\angle 1 = m\angle 3 \).

Step4: Recall the definition of congruent angles

If two angles have equal measures, then they are congruent. Since \( m\angle 1 = m\angle 3 \), we can conclude that \( \angle 1 \cong \angle 3 \).

Answer:

By the definition of complementary angles, \( m\angle 1 + m\angle 2 = 90^\circ \) and \( m\angle 2 + m\angle 3 = 90^\circ \). Setting these equal gives \( m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3 \). Subtracting \( m\angle 2 \) from both sides yields \( m\angle 1 = m\angle 3 \), so \( \angle 1 \cong \angle 3 \) (by the definition of congruent angles).