Question

guided practice
congruent complements theorem
given: ∠1 and ∠2 are complementary.
∠3 and ∠2 are complementary.
prove: ∠1 ≅ ∠3
you will complete a paragraph proof of this theorem by answering a series of questions.
click or tap proof at any time to view your progress on the proof.
proving angles congruent
which statement is equivalent to ∠1 ≅ ∠3?
a. ( mangle 3 + mangle 2 = 90 )
b. ( mangle 1 = mangle 2 )
c. ( mangle 1 + mangle 2 = 90 )
d. ( mangle 1 = mangle 3 )

Explanation:

To determine which statement is equivalent to \( \angle 1 \cong \angle 3 \), we recall the definition of congruent angles: two angles are congruent if their measures are equal. So \( \angle 1 \cong \angle 3 \) means \( m\angle 1 = m\angle 3 \). Let's analyze the options:

• Option A: \( m\angle 3 + m\angle 2 = 90 \) relates \( \angle 3 \) and \( \angle 2 \), not \( \angle 1 \) and \( \angle 3 \).
• Option B: \( m\angle 1 = m\angle 2 \) relates \( \angle 1 \) and \( \angle 2 \), not \( \angle 1 \) and \( \angle 3 \).
• Option C: \( m\angle 1 + m\angle 2 = 90 \) is about \( \angle 1 \) and \( \angle 2 \) being complementary, not about \( \angle 1 \) and \( \angle 3 \) congruence.
• Option D: \( m\angle 1 = m\angle 3 \) matches the definition of \( \angle 1 \cong \angle 3 \).

Answer:

D. \( m\angle 1 = m\angle 3 \)