Question

name the property of equality that the statement illustrates. if ( mangle a = 29^circ ) and ( mangle b = 29^circ ), then ( mangle a = mangle b ). the statement illustrates the property of equality.

Explanation:

Step1: Recall equality properties

The Transitive Property of Equality states that if \( a = b \) and \( b = c \), then \( a = c \). But here, we have two quantities equal to the same value (\( m\angle A = 29^\circ \) and \( m\angle B = 29^\circ \)), so \( m\angle A = m\angle B \). Wait, actually, the Substitution Property or the Transitive? Wait, no, the **Transitive Property** (or more precisely, when two things are equal to the same thing, they are equal to each other, which is a form of transitive: if \( a = c \) and \( b = c \), then \( a = b \)). Alternatively, the **Substitution Property** (substituting \( 29^\circ \) for \( m\angle A \) and \( m\angle B \)). But the key here is that if two quantities are equal to the same quantity, they are equal to each other, which is the **Transitive Property of Equality** (or sometimes called the "Substitution" or "Equality of the Same Quantity" property, but the formal name here is the **Transitive Property**? Wait, no, the Transitive Property is \( a = b, b = c \implies a = c \). Here, \( m\angle A = 29^\circ \), \( m\angle B = 29^\circ \), so let \( c = 29^\circ \), then \( a = c \), \( b = c \), so \( a = b \). So this is the Transitive Property? Wait, actually, it's also called the **Substitution Property** or the **Property of Equality for Equal Quantities** (if two quantities are equal to the same quantity, they are equal to each other). But the standard name here is the **Transitive Property of Equality**? Wait, no, the Transitive Property is for three terms. Wait, maybe it's the **Reflexive**? No, Reflexive is \( a = a \). Symmetric is \( a = b \implies b = a \). Transitive is \( a = b, b = c \implies a = c \). So in this case, since \( m\angle A = 29^\circ \) and \( m\angle B = 29^\circ \), we can say \( m\angle A = m\angle B \) by the Transitive Property (treating \( 29^\circ \) as the middle term). Alternatively, the **Substitution Property**, where we substitute \( 29^\circ \) with \( m\angle A \) and \( m\angle B \). But the correct property here is the **Transitive Property of Equality** (or sometimes called the "Equality of the Same Quantity" property, but the formal term is Transitive? Wait, no, let's check: If \( a = c \) and \( b = c \), then \( a = b \) is a special case of Transitive (let \( b = c \), so \( a = c \) and \( c = b \) (by Symmetric), then \( a = b \) by Transitive). So the property is the **Transitive Property of Equality**? Wait, no, actually, the correct property here is the **Substitution Property** or the **Property of Equality for Equal Quantities**. Wait, maybe I'm overcomplicating. The statement is: If two angles have the same measure (both \( 29^\circ \)), then their measures are equal. So this is the **Transitive Property of Equality** (since \( m\angle A = 29^\circ \) and \( m\angle B = 29^\circ \), so \( m\angle A = m\angle B \) because they're both equal to \( 29^\circ \)). Alternatively, the **Substitution Property**, where we substitute \( 29^\circ \) with \( m\angle A \) and \( m\angle B \). But the standard answer here is the **Transitive Property**? Wait, no, let's recall: The Transitive Property of Equality: If \( a = b \) and \( b = c \), then \( a = c \). In this case, \( a = m\angle A \), \( b = 29^\circ \), \( c = m\angle B \). So \( m\angle A = 29^\circ \) ( \( a = b \) ) and \( 29^\circ = m\angle B \) ( \( b = c \) ), so by Transitive, \( m\angle A = m\angle B \) ( \( a = c \) ). So yes, Transitive Property of Equality.

Wait, but another way: The **Substitution Property** allows replacing a quantity with its equal. So since \( m\angle A = 29^\circ \) and \( m\angle B = 29^\circ \), we can substitute \( 29^\circ \) with \( m\angle A \) in the second equation, getting \( m\angle A = m\angle B \). So either Transitive or Substitution. But in most textbooks, this is the **Transitive Property of Equality** (or sometimes called the "Equality of the Same Quantity" property). Wait, maybe the correct term is the **Transitive Property**? Let me check a reference. The Transitive Property of Equality: If \( a = b \) and \( b = c \), then \( a = c \). So in this problem, \( a = m\angle A \), \( b = 29^\circ \), \( c = m\angle B \). So \( a = b \) ( \( m\angle A = 29^\circ \) ) and \( b = c \) ( \( 29^\circ = m\angle B \) ), so \( a = c \) ( \( m\angle A = m\angle B \) ). So that's Transitive.

Alternatively, the **Substitution Property** states that if \( a = b \), then \( a \) can be substituted for \( b \) in any equation or expression. So since \( m\angle A = 29^\circ \) and \( m\angle B = 29^\circ \), we can substitute \( m\angle A \) for \( 29^\circ \) in the second equation, giving \( m\angle A = m\angle B \). So both properties apply, but the more direct one here is the **Transitive Property** (since it's about two quantities equal to the same quantity being equal to each other).

Answer:

Transitive