Proof Number \(1\)
Given:
\(\overrightarrow{MN} \perp \overrightarrow{MY},\)
\(\overrightarrow{MN}\) bisects \(\angle AMD\)
Prove:
\(\angle 1 \) and \(\angle 3 \) are complementary
\(\,\,\,\,\,\,\,\)
| \(\underline{\text{Statement}}\) | \(\underline{\text{Reason}}\) |
|---|---|
\(\overrightarrow{MN} \perp \overrightarrow{MY},\) \(\overrightarrow{MN}\) bisects \(\angle AMD\) |
Given </ |
\(m\angle NMY=90^{\circ}\) |
Definition of perpendicular \((\perp)\) |
\(m\angle 2+ m\angle 3= m\angle NMY\) |
Angle Addition Postulate |
\(m\angle 2+ m\angle 3= 90^{\circ}\) |
Transitive Property or Substitution Property |
\(m\angle 1= m\angle 2\) |
Definition of bisects |
\(m\angle 1+ m\angle 3= 90^{\circ}\) |
Substitution Property |
\(\angle 1 \) and \(\angle 3 \) are complementary |
Definition of Complementary |
Proof Number \(2\)
Given:
\(\angle A\) and \(\angle N\) are complementary
\(\angle D\) and \(\angle N\) are complementary
Prove:
\(\angle A \cong \angle D\)
| \(\underline{\text{Statement}}\) | \(\underline{\text{Reason}}\) |
|---|---|
\(\angle A\) & \(\angle N\) are complementary \(\angle D\) & \(\angle N\) are complementary |
Given |
\(m\angle A + m\angle N=90^{\circ}\) \(m\angle D + m\angle N=90^{\circ}\) |
Definition of Complementary Angles |
\(m\angle A + m\angle N=m\angle D + m\angle N\) |
Transitive Property |
\(m\angle A=m\angle D\) |
Subtraction Property of Equality |
\(\angle A \cong \angle D\) |
Definition of Congruent Angles |
Proof Number \(3\)
Given:
\(AS=8,\, SH=8,\, \overline{AS} \cong \overline{AH} \)
Prove:
\(\overline{SH} \cong \overline{AH}\)
\(\,\,\,\,\,\,\,\)
| \(\underline{\text{Statement}}\) | \(\underline{\text{Reason}}\) |
|---|---|
\(AS=8\) \(SH=8\) \(\overline{AS} \cong \overline{AH} \) |
Given |
AS=SH |
Transitive Property |
\(\overline{AS} \cong \overline{SH}\) |
Definition of congruent segments |
\(\overline{SH} \cong \overline{AH}\) |
Substitution Property |
Proof Number \(4\)
Given:
\(\overrightarrow{TD}\) bisects \(\angle ATM\)
\(\angle 1 \cong \angle 4\)
Prove:
\(\angle 2 \cong \angle 3\)
\(\,\,\,\,\,\,\,\)
| \(\underline{\text{Statement}}\) | \(\underline{\text{Reason}}\) |
|---|---|
\(\overrightarrow{TD}\) bisects \(\angle ATM\) \(\angle 1 \cong \angle 4\) |
Given |
\(m\angle ATD = m\angle DTM\) |
Definition of Bisects |
\(m\angle 1 = m\angle 4\) |
Definition of Congruent Angles |
\(m\angle 1 + m\angle 2=m \angle ATD\) \(m\angle 3 + m\angle 4=m \angle DTM\) |
Angle Addition Postulate |
\(m\angle 1 + m\angle 2=m \angle 3 + m \angle 4\) |
Substitution Property |
\(m\angle 1 + m\angle 2=m \angle 3 + m \angle 1\) |
Substitution Property |
\(m\angle 2=m \angle 3\) |
Subtraction Property of Equality |
\(\angle 2 \cong \angle 3\) |
Definition of Congruent Angles |
Proof Number \(5\)
Given:
\(\angle A\) and \(\angle N\) are supplementary
\(\angle D\) and \(\angle N\) are supplementary
Prove:
\(\angle A \cong \angle D\)
| \(\underline{\text{Statement}}\) | \(\underline{\text{Reason}}\) |
|---|---|
\(\angle A\) & \(\angle N\) are supplementary \(\angle D\) & \(\angle N\) are supplementary |
Given |
\(m\angle A + m\angle N=180^{\circ}\) \(m\angle D + m\angle N=180^{\circ}\) |
Definition of Supplementary Angles |
\(m\angle A + m\angle N=m\angle D + m\angle N\) |
Transitive Property |
\(m\angle A=m\angle D\) |
Subtraction Property of Equality |
\(\angle A \cong \angle D\) |
Definition of Congruent Angles |
Proof Number \(6\)
Given:
\(\overrightarrow{MD} \perp \overline{AT}\)
\(\angle 1 \cong \angle 4\)
Prove:
\(\angle 2 \cong \angle 3\)
| \(\underline{\text{Statement}}\) | \(\underline{\text{Reason}}\) |
|---|---|
\(\overrightarrow{MD} \perp \overline{AT}\) \(\angle 1 \cong \angle 4\) |
Given |
\(m \angle 1 = m \angle 4\) |
Definition of congruent angles |
\(m \angle AMD = 90^{\circ}\) \(m \angle DMT = 90^{\circ}\) |
Definition of Perpendicular \((\perp)\) |
\( m \angle AMD= m \angle DMT\) |
Transitive Property |
\(m \angle 1 + m \angle 2= m \angle AMD\) \(m \angle 3 + m \angle 4= m \angle DMT\) |
Angle Addition Postulate |
\(m \angle 1 + m \angle 2= m \angle 3 + m \angle 4\) |
Substitution Property |
\(m \angle 1 + m \angle 2= m \angle 3 + m \angle 1\) |
Substitution Property |
\(m \angle 2= m \angle 3\) |
Subtraction Property of Equality |
\(\angle 2 \cong \angle 3\) |
Definition of Congruent Angles |
Proof Number \(7\)
Given:
\(\angle AMN \cong \angle DMY\)
Prove:
\(\angle AMD \cong \angle NMY\)
\(\,\,\,\,\,\,\,\)
| \(\underline{\text{Statement}}\) | \(\underline{\text{Reason}}\) |
|---|---|
\(\angle AMN \cong \angle DMY\) |
Given |
\(m \angle AMN = m \angle DMY\) |
Definition of Congruent Angles |
\( m \angle NMD = m \angle NMD\) |
Reflexive Property |
\(m \angle AMN + m \angle NMD = m \angle DMY + m \angle NMD\) |
Addition Property of Equality |
\(m \angle AMN + m \angle NMD =m \angle AMD\) \(m \angle DMY + m \angle NMD= m \angle NMY\) |
Angle Addition Postulate |
\(m \angle AMD = m \angle NMY\) |
Substitution Property |
\(\angle AMD \cong \angle NMY\) |
Definition of Congruent Angles |
Proof Number \(8\)
Given:
\(\overrightarrow{MA} \perp \overrightarrow{MD}\)
\(\overrightarrow{MN} \perp \overrightarrow{MY}\)
Prove:
\(\angle 1 \cong \angle 3\)
\(\,\,\,\,\,\,\,\)
| \(\underline{\text{Statement}}\) | \(\underline{\text{Reason}}\) |
|---|---|
\(\overrightarrow{MA} \perp \overrightarrow{MD}\) \(\overrightarrow{MN} \perp \overrightarrow{MY}\) |
Given |
\(m\angle AMD=90^{\circ}\) \(m\angle NMY=90^{\circ}\) |
Definition of perpendicular \((\perp)\) |
\(m\angle AMD= m \angle NMY\) |
Transitive Property |
\(m\angle 1 + m \angle 2= m \angle AMD\) \(m\angle 2 + m \angle 3= m \angle NMY\) |
Angle Addition Postulate |
\(m\angle 1 + m \angle 2= m \angle 2 + m \angle 3\) |
Substitution Property |
\(m\angle 1 = m \angle 3\) |
Subtraction Property of Equality |
\(\angle 1 \cong \angle 3\) |
Definition of Congruent Angles |
Proof Number \(9\)
Given:
\(\angle 1 \cong \angle 2\)
Prove:
\(\angle 2 \cong \angle 3\)
\(\,\,\,\,\,\,\,\)
| \(\underline{\text{Statement}}\) | \(\underline{\text{Reason}}\) |
|---|---|
\(\angle 1 \cong \angle 2\) |
Given |
\(\angle 1 \cong \angle 3\) |
Vertical Angles are Congruent |
\(\angle 2 \cong \angle 3\) |
Substitution Property |
Proof Number \(10\)
Given:
\(U\) is the midpoint of \(\overline{FN}\) \(\)
Prove:
\(x=4\)
\(\,\,\,\,\,\,\,\)
| \(\underline{\text{Statement}}\) | \(\underline{\text{Reason}}\) |
|---|---|
\(U\) is the midpoint of \(\overline{FN}\) |
Given |
\(FU=UN\) |
Definition of Midpoint |
\(5x+8=7x\) |
Substitution Property |
\(8=2x\) |
Subtraction Property of Equality |
\(4=x\) |
Division Property of Equality |
\(x=4\) |
Symmetric Property |
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