Question

given: ( mangle o + mangle r = mangle t ) and ( mangle s = mangle o + mangle r ) prove: ( mangle s = mangle t ) a. reflexive property b. betweenness of rays c. addition property d. transitive property e. symmetric property

Explanation:

To prove \( m\angle S = m\angle T \), we start with the given information: \( m\angle O + m\angle R = m\angle T \) and \( m\angle S = m\angle O + m\angle R \). The Transitive Property of Equality states that if \( a = b \) and \( b = c \), then \( a = c \). Here, let \( a = m\angle S \), \( b = m\angle O + m\angle R \), and \( c = m\angle T \). Since \( m\angle S = m\angle O + m\angle R \) (so \( a = b \)) and \( m\angle O + m\angle R = m\angle T \) (so \( b = c \)), by the Transitive Property, we can conclude \( m\angle S = m\angle T \). The other properties do not apply here: the Reflexive Property is \( a = a \), Betweenness of Rays is about ray positions, the Addition Property involves adding equal quantities to both sides, and the Symmetric Property is \( \text{if } a = b \text{, then } b = a \), none of which fit this situation.

Answer:

D. Transitive Property