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
\\( m \angle abc = 90 ^ { \circ }, \\)
\\( m \angle abc = m \angle def \\)
\\( \angle abc \cong \angle def \\)
right angles
\\( m \angle def = 90 ^ { \circ } \\)
given
a
b
c
a
b
c
definition of congruent angles
definition of equal angles
substitution property
symmetric property

Explanation:

Step1: Definition of a right angle

A right angle is defined as an angle whose measure is \(90^{\circ}\). So, if \(\angle ABC\) and \(\angle DEF\) are right angles, by the definition of a right angle, \(m\angle ABC = 90^{\circ}\) and \(m\angle DEF=90^{\circ}\).

Step2: Substitution property

Since \(m\angle ABC = 90^{\circ}\) and \(m\angle DEF = 90^{\circ}\), we can substitute \(90^{\circ}\) for \(m\angle DEF\) (or vice - versa) in the equation \(m\angle ABC=m\angle DEF\). The substitution property states that if \(a = b\) and \(b = c\), then \(a = c\). Here, \(a=m\angle ABC\), \(b = 90^{\circ}\), and \(c=m\angle DEF\).

Step3: Definition of congruent angles

Two angles are congruent if and only if their measures are equal. 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; C. definition of congruent angles