Question

match the theorems below.
if pq + rs = ps and rs = xy, then pq + xy = p
jm = jm
if rs = tu and wv = tu, then rs = wv
if 2xy = yz, then xy = 1/2 yz
if q is between p and r, then pq + qr = pr
if segment ma = segment rg, then ma = rg
if hi = po, then hi + cd = cd + po
if 2kl = kl + mn, then kl = mn
if ab = xz, then xz = ab

Explanation:

• If \(PQ + RS=PS\) and \(RS = XY\), then \(PQ+XY = PS\): Since \(RS\) is replaced by \(XY\) (because \(RS = XY\)), this is the Substitution Property.
• \(JM = JM\): A quantity is equal to itself, which is the Reflexive Property.
• If \(RS = TU\) and \(WV=TU\), then \(RS = WV\): If \(a = b\) and \(c = b\), then \(a = c\), this is the Transitive Property.
• If \(2XY=YZ\), then \(XY=\frac{1}{2}YZ\): Dividing both sides of the equation \(2XY = YZ\) by \(2\) (using the formula \(a=b\Rightarrow\frac{a}{c}=\frac{b}{c},c

eq0\)), this is the Division Property of Equality.

• If \(Q\) is between \(P\) and \(R\), then \(PQ + QR=PR\): This is the definition of how segments on a line - segment add up, the Segment Addition Postulate.
• If Segment \(MA\cong\) Segment \(RG\), then \(MA = RG\): By the definition, if two segments are congruent, their lengths are equal, so this is the Definition of Congruence.
• If \(HI = PO\), then \(HI + CD=CD + PO\): Adding the same quantity (\(CD\)) to both sides of the equation \(HI = PO\) (using the formula \(a = b\Rightarrow a + c=b + c\)), this is the Addition Property of Equality.
• If \(2KL=KL + MN\), then \(KL = MN\): Subtracting \(KL\) from both sides of the equation \(2KL=KL + MN\) (using the formula \(a=b\Rightarrow a - c=b - c\)), this is the Subtraction Property of Equality.
• If \(AB = XZ\), then \(XZ = AB\): If \(a = b\), then \(b = a\), this is the Symmetric Property.

Answer:

1. Substitution Property
2. Reflexive Property
3. Transitive Property
4. Division Property of Equality
5. Segment Addition Postulate
6. Definition of Congruence
7. Addition Property of Equality
8. Subtraction Property of Equality
9. Symmetric Property