Question

∠a ≅ ∠b, ∠c ≅ ∠b given
∠b ≅ ∠c symmetric prop.
∠a ≅ ∠c ?
m∠a = m∠c def. of ≅ ∠s
what is the missing justification?
○ transitive property
● reflexive property
○ symmetric property
○ substitution property

Explanation:

Step1: Recall Properties of Congruence

The transitive property of congruence states that if \( \angle X \cong \angle Y \) and \( \angle Y \cong \angle Z \), then \( \angle X \cong \angle Z \). The reflexive property is \( \angle X \cong \angle X \), symmetric is if \( \angle X \cong \angle Y \) then \( \angle Y \cong \angle X \), and substitution is replacing a quantity with an equal one.

Step2: Analyze Given Information

We have \( \angle A \cong \angle B \) (given) and \( \angle B \cong \angle C \) (from symmetric property on \( \angle C \cong \angle B \)). To get \( \angle A \cong \angle C \), we use the transitive property: if \( \angle A \cong \angle B \) and \( \angle B \cong \angle C \), then \( \angle A \cong \angle C \). Wait, but the original selected was reflexive, which is wrong. Wait, let's re - check.

Wait, the steps: First, \( \angle A\cong\angle B \), \( \angle C\cong\angle B \) (given). Then \( \angle B\cong\angle C \) (symmetric property, since symmetric property of congruence says if \( \angle X\cong\angle Y \), then \( \angle Y\cong\angle X \), so here \( X = C \), \( Y = B \), so \( \angle C\cong\angle B\implies\angle B\cong\angle C \)). Then we have \( \angle A\cong\angle B \) and \( \angle B\cong\angle C \). The transitive property of congruence (for angles) states that if \( \angle A\cong\angle B \) and \( \angle B\cong\angle C \), then \( \angle A\cong\angle C \). So the missing justification is the transitive property.

Wait, the original option selected was reflexive, which is incorrect. Let's correct:

The transitive property of congruence (for angles) is: If \( \angle X\cong\angle Y \) and \( \angle Y\cong\angle Z \), then \( \angle X\cong\angle Z \). Here, \( X = A \), \( Y = B \), \( Z = C \). We have \( \angle A\cong\angle B \) (given) and \( \angle B\cong\angle C \) (from symmetric property on \( \angle C\cong\angle B \)). So by transitive property, \( \angle A\cong\angle C \).

Answer:

The correct missing justification is the transitive property. So the correct option is: transitive property