Statement
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Reasons
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MN = SP
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Given
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2t = (t + 5)
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Substitution property of equality
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2t - 1t = t + 5 - t
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Subtraction property of equality
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t = 5
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MN = 2t
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Given
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MN = 2(5)
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Substitution
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MN = 10
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Statement
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Reason
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m<M = m<P
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Opposite angles of parallelograms are (congruent) therefore, they are equal
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45 = 3x
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Substitution
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45/3 = 3x/3
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Division property of equality
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15 = x
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NP = x
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Given
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NP = 15
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Substitution
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Statement
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Reasons
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SMNP is a parallelogram
m<MSP = 5x degrees;
m<P = x degrees
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Given
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m<MSP + m<P = 180 degrees
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Consecutive angles of a parallelogram are supplementary; therefore their measures equal 180 degrees
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5x + x = 180
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Substitution
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6x = 180
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Combine like terms
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6/6x = 180/6
x = 30
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Multiplication (division) property of equalities
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m<MSP = 5(30 degrees)
m<MSP = 150 degrees
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Substitution
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m<MSP = m<MNP
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Opposite angles of a parallelogram are congruent; therefore, they are equal.
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m< MNP = 150 degrees
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Transitive or Substitution Property of equality
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Statement
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Reasons
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m<M = (x + 20) degrees and
m<MNP = (2x + 10) degrees
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Given
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m<M + m<MNP = 180 degrees
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Consecutive angles of a parallelogram are supplementary.
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(x + 20) + (2x + 10) = 180
3x + 30 = 180
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Substitution
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3x + 30 - 30 = 180 - 30
3x = 150
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Subtraction property of equality
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3/3 x = 150/3
x = 50
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Division property of equality
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m<M = x + 20 (from above)
m<M = 50 + 20
m<M = 70
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Substitution
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Statement
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Reason
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m<P = 55 degrees,
m<M = (x + 5) degrees
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Given
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m<M = m<P
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Opposite angles of a parallelogram are congruent; therefore they are equal.
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x + 5 = 55
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Substitution
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x + 5 - 5 = 55 - 5
x = 50
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Subtraction property of equality
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NS = x - 15
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Given
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NS = 50 - 15
NS = 35
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Substitution
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Statement
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Reason
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SMNP is a parallelogram
m<MNS = (5x + 10) degrees
m<NSP = (x + 30) degrees
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given
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m<NSP = m<MNS
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Alternate interior angles are congruent; therefore, their measures are equal.
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5x + 10 = x + 30
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Substitution
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4x = 20
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Addition/subtraction property of equalities
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x = 5
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Multiplication/division property of equalities
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m<MNS = 5(5) + 10 degrees
m<MNS = 35 degrees
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substitution property of equalities
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m<NSP = 35 degrees
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substitution property of equalities
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Statement
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Reasons
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SMNP is a parallelogram
MN = 3x,
SP = (40 - x)
MS = 2x
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Given
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MN = SP
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The opposite sides of a parallelogram are congruent; therefore, by the segment congruence postulate, their measures are equal
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3x = 40 -x
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substitution
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4x = 40
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Addition property of equalities
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x = 10
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multiplication property of equalities
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MS = 2(10)
MS = 20
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Substitution property of equalities
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MP = MS
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The opposite sides of a parallelogram are congruent; therefore, by the segment congruence postulate, their measures are equal
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MP = 20
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Substitution
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SMNP is a parallelogram
m<P = 80 degrees
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Given
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<P and <MNP are supplementary
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Consecutive angles of a parallelogram are supplementary
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m<P + m<MNP = 180 degrees
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Definition of supplementary angles
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80 + m<MNP = 180 degrees
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substitution
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m<MNP = 100 degrees
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addition property of equalities
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Statement
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Reasons
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SMNP is a parallelogram
m<MNP = 120 degrees, m<MSP = 6x degrees
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Given
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m<MNP = m<MSP
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Opposite angles of a parallelogram are congruent; therefore, by the angle congruence theorem, their measures are equal.
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120 = 6x
20 = x
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substitution property of qualities
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x = 20
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Reflexive property of equalities
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NS = (x - 15)
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Given
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NS = (20 - 15)
NS = 5
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substitution property of equalities
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Statement
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Reasons
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SMNP is a parallelogram
MS = 15, NP = (x - 5), and m<P = x degrees
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Given
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NP = MS
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Opposite sides of a parallelogram are congruent; therefore by the segment congruence theorem, their measures are equal
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x - 5 = 15
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substitution property of equalities
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x = 20
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Addition/subtraction property of equalities
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m<P = 20 degrees
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substitution
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m<M = m<P
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Opposite angles of a parallelogram are congruent; therefore by the angle congruence theorem, their measures are equal
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m<M = 20 degrees
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Substitution or transitive property of equalities
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Statement
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Proof
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