Question

given: \\( \angle abc \\) is a right angle and \\( \angle def \\) is a right angle.
prove: all right angles are congruent by showing that \\( \angle abc \cong \angle def \\).
what are the missing reasons in the steps of the proof?
\\( \angle abc, \angle def \\) are right angles \\( \
ightarrow \\) \\( m \angle abc = 90 ^ { \circ }, m \angle def = 90 ^ { \circ } \\) \\( \
ightarrow \\) \\( m \angle abc = m \angle def \\) \\( \
ightarrow \\) \\( \angle abc \cong \angle def \\)
given a b c
a:

b:

c:

Explanation:

• A: By the definition of a right angle, if an angle is a right angle, its measure is \(90^{\circ}\). So, for \(\angle ABC\) and \(\angle DEF\) (which are given as right angles), \(m\angle ABC = 90^{\circ}\) and \(m\angle DEF=90^{\circ}\) because of the definition of a right angle.
• B: Since \(m\angle ABC = 90^{\circ}\) and \(m\angle DEF = 90^{\circ}\), we can use the substitution property of equality (if \(a = c\) and \(b = c\), then \(a = b\)). Here, \(a=m\angle ABC\), \(b = m\angle DEF\), and \(c = 90^{\circ}\), so \(m\angle ABC=m\angle DEF\) by substitution.
• C: Congruent angles are defined as angles that have equal measures. Since \(m\angle ABC=m\angle DEF\), by the definition of congruent angles, \(\angle ABC\cong\angle DEF\).

Answer:

A: Definition of a right angle
B: Substitution property of equality
C: Definition of congruent angles