Question

module 2 mid topic 2 assessment review
1.) identify the property that justifies each statement below.
(a) if ( mangle1 = 134^circ ) and ( mangle2 = 134^circ ), then ( mangle1 = mangle2 )
(b) if ( gh = jk ), then ( gh - rs = jk - rs )
(c) if ( angle efg cong angle lmn ) and ( angle lmn cong angle spt ), then ( angle efg cong angle spt )
(d) ( mangle pqr = mangle pqr )

Explanation:

Part (a)

Step 1: Analyze the statement

We have two angles, \( \angle 1 \) and \( \angle 2 \), both with a measure of \( 134^\circ \). The conclusion is that their measures are equal.

Step 2: Identify the property

This is the Transitive Property of Equality? No, wait. Wait, if two quantities are equal to the same quantity, then they are equal to each other. Wait, actually, the property that if \( a = c \) and \( b = c \), then \( a = b \) is the Transitive Property? No, wait, the Reflexive Property is \( a = a \), Symmetric is if \( a = b \) then \( b = a \), and Transitive is if \( a = b \) and \( b = c \), then \( a = c \). Wait, in this case, \( m\angle1 = 134^\circ \) and \( m\angle2 = 134^\circ \), so \( m\angle1 = m\angle2 \) because they are both equal to \( 134^\circ \). This is the Transitive Property of Equality? Wait, no, actually, it's the Substitution Property or the Transitive Property? Wait, the Transitive Property of Equality states that if \( a = b \) and \( b = c \), then \( a = c \). Here, \( a = m\angle1 \), \( b = 134^\circ \), \( c = m\angle2 \). So since \( m\angle1 = 134^\circ \) ( \( a = b \)) and \( m\angle2 = 134^\circ \) ( \( b = c \)), then \( m\angle1 = m\angle2 \) ( \( a = c \)). So that's the Transitive Property of Equality. Wait, but sometimes it's also called the "Substitution Property" when you substitute the value. But more accurately, the Transitive Property of Equality: If \( a = b \) and \( b = c \), then \( a = c \).

Wait, maybe I made a mistake. Let's recall:

• Reflexive Property: \( a = a \) (a quantity is equal to itself)
• Symmetric Property: If \( a = b \), then \( b = a \)
• Transitive Property: If \( a = b \) and \( b = c \), then \( a = c \)
• Substitution Property: If \( a = b \), then \( a \) can be substituted for \( b \) in any equation or expression.

In this case, since \( m\angle1 = 134^\circ \) and \( m\angle2 = 134^\circ \), we can say \( m\angle1 = m\angle2 \) because they are both equal to the same angle measure. So this is the Transitive Property of Equality? Wait, no, the Transitive Property is about three quantities. Wait, maybe it's the Substitution Property? Let's see: Since \( m\angle2 = 134^\circ \), and \( m\angle1 = 134^\circ \), so substituting \( 134^\circ \) with \( m\angle2 \) in the first equation, we get \( m\angle1 = m\angle2 \). So that's the Substitution Property. But actually, the correct property here is the Transitive Property of Equality? Wait, no, the Transitive Property requires three terms. Let me check a reference. The Transitive Property of Equality: If \( a = b \) and \( b = c \), then \( a = c \). In this problem, \( a = m\angle1 \), \( b = 134^\circ \), \( c = m\angle2 \). So \( a = b \) ( \( m\angle1 = 134^\circ \)) and \( b = c \) ( \( 134^\circ = m\angle2 \)), so by Transitive Property, \( a = c \) ( \( m\angle1 = m\angle2 \)). So yes, Transitive Property of Equality.

Wait, but another way: The property that if two quantities are equal to the same quantity, then they are equal to each other is the Transitive Property. So the answer for (a) is the Transitive Property of Equality? Wait, no, maybe the Substitution Property? Let me confirm. The Substitution Property states that if \( a = b \), then \( a \) can be replaced with \( b \) in any equation or expression. So since \( m\angle2 = 134^\circ \), we can replace \( 134^\circ \) with \( m\angle2 \) in the equation \( m\angle1 = 134^\circ \), resulting in \( m\angle1 = m\angle2 \). So that's the Substitution Property. Hmm, maybe I confused Transitiv…

Step 1: Analyze the statement

We have \( GH = JK \), and then we subtract \( RS \) from both sides to get \( GH - RS = JK - RS \).

Step 2: Identify the property

This is the Subtraction Property of Equality, which states that if \( a = b \), then \( a - c = b - c \) for any real number \( c \). Here, \( a = GH \), \( b = JK \), and \( c = RS \). So by the Subtraction Property of Equality, if \( GH = JK \), then \( GH - RS = JK - RS \).

Step 1: Analyze the statement

We have \( \angle EFG \cong \angle LMN \) and \( \angle LMN \cong \angle SPT \), and we conclude that \( \angle EFG \cong \angle SPT \).

Step 2: Identify the property

This is the Transitive Property of Congruence, which states that if \( \angle A \cong \angle B \) and \( \angle B \cong \angle C \), then \( \angle A \cong \angle C \). Here, \( \angle A = \angle EFG \), \( \angle B = \angle LMN \), \( \angle C = \angle SPT \). So by the Transitive Property of Congruence, \( \angle EFG \cong \angle SPT \).

Answer:

Transitive Property of Equality

Part (b)