Question

given: ∠abc is a right angle and ∠def is a right angle. prove: all right angles are congruent by showing that ∠abc ≅ ∠def. what are the missing reasons in the steps of the proof? chart: ∠abc, ∠def are right angles (given) → m∠abc = 90°, m∠def = 90° (a) → m∠abc = m∠def (b) → ∠abc ≅ ∠def (c) a: definition of right angle (dropdown) b: options: identity property, reflexive property, substitution property, symmetric property c: dropdown

Explanation:

Step1: Analyze Step A

A right angle is defined as an angle with measure \( 90^\circ \). Since \( \angle ABC \) and \( \angle DEF \) are right angles, by the definition of a right angle, their measures are \( 90^\circ \). So the reason for Step A is "definition of right angle" (which is already selected, and this checks out).

Step2: Analyze Step B

We know \( m\angle ABC = 90^\circ \) and \( m\angle DEF = 90^\circ \). To conclude \( m\angle ABC = m\angle DEF \), we substitute the value \( 90^\circ \) for both angles. The substitution property allows us to replace one expression with an equal expression. So the missing reason for Step B is "substitution property".

Step3: Analyze Step C

If two angles have equal measures, then the angles are congruent (this is the definition of congruent angles: \( \angle A \cong \angle B \) if and only if \( m\angle A = m\angle B \)). So the reason for Step C is "definition of congruent angles" (though the options for C aren't fully shown, but focusing on B here).

Answer:

For Step B, the missing reason is "substitution property".