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Proof Number \(1\)

Given:
\(\overrightarrow{MN} \perp \overrightarrow{MY},\)
\(\overrightarrow{MN}\) bisects \(\angle AMD\)

Prove:
\(\angle 1 \) and \(\angle 3 \) are complementary

\(\,\,\,\,\,\,\,\)

\(\underline{\text{Statement}}\)	\(\underline{\text{Reason}}\)
CLICK ME \(\overrightarrow{MN} \perp \overrightarrow{MY},\) \(\overrightarrow{MN}\) bisects \(\angle AMD\)	CLICK ME Given
CLICK ME \(m\angle NMY=90^{\circ}\)	CLICK ME Definition of perpendicular \((\perp)\)
CLICK ME \(m\angle 2+ m\angle 3= m\angle NMY\)	CLICK ME Angle Addition Postulate
CLICK ME \(m\angle 2+ m\angle 3= 90^{\circ}\)	CLICK ME Transitive Property or Substitution Property
CLICK ME \(m\angle 1= m\angle 2\)	CLICK ME Definition of bisects
CLICK ME \(m\angle 1+ m\angle 3= 90^{\circ}\)	CLICK ME Substitution Property
CLICK ME \(\angle 1 \) and \(\angle 3 \) are complementary	CLICK ME Definition of Complementary

Proof Number \(2\)

Given:
\(\angle A\) and \(\angle N\) are complementary
\(\angle D\) and \(\angle N\) are complementary

Prove:
\(\angle A \cong \angle D\)

\(\underline{\text{Statement}}\)	\(\underline{\text{Reason}}\)
CLICK ME \(\angle A\) & \(\angle N\) are complementary \(\angle D\) & \(\angle N\) are complementary	CLICK ME Given
CLICK ME \(m\angle A + m\angle N=90^{\circ}\) \(m\angle D + m\angle N=90^{\circ}\)	CLICK ME Definition of Complementary Angles
CLICK ME \(m\angle A + m\angle N=m\angle D + m\angle N\)	CLICK ME Transitive Property
CLICK ME \(m\angle A=m\angle D\)	CLICK ME Subtraction Property of Equality
CLICK ME \(\angle A \cong \angle D\)	CLICK ME Definition of Congruent Angles

Proof Number \(3\)

Given:
\(AS=8,\, SH=8,\, \overline{AS} \cong \overline{AH} \)

Prove:
\(\overline{SH} \cong \overline{AH}\)

\(\,\,\,\,\,\,\,\)

\(\underline{\text{Statement}}\)	\(\underline{\text{Reason}}\)
CLICK ME \(AS=8\) \(SH=8\) \(\overline{AS} \cong \overline{AH} \)	CLICK ME Given
CLICK ME AS=SH	CLICK ME Transitive Property
CLICK ME \(\overline{AS} \cong \overline{SH}\)	CLICK ME Definition of congruent segments
CLICK ME \(\overline{SH} \cong \overline{AH}\)	CLICK ME Substitution Property

Proof Number \(4\)

Given:
\(\overrightarrow{TD}\) bisects \(\angle ATM\)
\(\angle 1 \cong \angle 4\)

Prove:
\(\angle 2 \cong \angle 3\)

\(\,\,\,\,\,\,\,\)

\(\underline{\text{Statement}}\)	\(\underline{\text{Reason}}\)
CLICK ME \(\overrightarrow{TD}\) bisects \(\angle ATM\) \(\angle 1 \cong \angle 4\)	CLICK ME Given
CLICK ME \(m\angle ATD = m\angle DTM\)	CLICK ME Definition of Bisects
CLICK ME \(m\angle 1 = m\angle 4\)	CLICK ME Definition of Congruent Angles
CLICK ME \(m\angle 1 + m\angle 2=m \angle ATD\) \(m\angle 3 + m\angle 4=m \angle DTM\)	CLICK ME Angle Addition Postulate
CLICK ME \(m\angle 1 + m\angle 2=m \angle 3 + m \angle 4\)	CLICK ME Substitution Property
CLICK ME \(m\angle 1 + m\angle 2=m \angle 3 + m \angle 1\)	CLICK ME Substitution Property
CLICK ME \(m\angle 2=m \angle 3\)	CLICK ME Subtraction Property of Equality
CLICK ME \(\angle 2 \cong \angle 3\)	CLICK ME Definition of Congruent Angles

Proof Number \(5\)

Given:
\(\angle A\) and \(\angle N\) are supplementary
\(\angle D\) and \(\angle N\) are supplementary

Prove:
\(\angle A \cong \angle D\)

\(\underline{\text{Statement}}\)	\(\underline{\text{Reason}}\)
CLICK ME \(\angle A\) & \(\angle N\) are supplementary \(\angle D\) & \(\angle N\) are supplementary	CLICK ME Given
CLICK ME \(m\angle A + m\angle N=180^{\circ}\) \(m\angle D + m\angle N=180^{\circ}\)	CLICK ME Definition of Supplementary Angles
CLICK ME \(m\angle A + m\angle N=m\angle D + m\angle N\)	CLICK ME Transitive Property
CLICK ME \(m\angle A=m\angle D\)	CLICK ME Subtraction Property of Equality
CLICK ME \(\angle A \cong \angle D\)	CLICK ME Definition of Congruent Angles

Proof Number \(6\)

Given:
\(\overrightarrow{MD} \perp \overline{AT}\)
\(\angle 1 \cong \angle 4\)

Prove:
\(\angle 2 \cong \angle 3\)

\(\underline{\text{Statement}}\)	\(\underline{\text{Reason}}\)
CLICK ME \(\overrightarrow{MD} \perp \overline{AT}\) \(\angle 1 \cong \angle 4\)	CLICK ME Given
CLICK ME \(m \angle 1 = m \angle 4\)	CLICK ME Definition of congruent angles
CLICK ME \(m \angle AMD = 90^{\circ}\) \(m \angle DMT = 90^{\circ}\)	CLICK ME Definition of Perpendicular \((\perp)\)
CLICK ME \( m \angle AMD= m \angle DMT\)	CLICK ME Transitive Property
CLICK ME \(m \angle 1 + m \angle 2= m \angle AMD\) \(m \angle 3 + m \angle 4= m \angle DMT\)	CLICK ME Angle Addition Postulate
CLICK ME \(m \angle 1 + m \angle 2= m \angle 3 + m \angle 4\)	CLICK ME Substitution Property
CLICK ME \(m \angle 1 + m \angle 2= m \angle 3 + m \angle 1\)	CLICK ME Substitution Property
CLICK ME \(m \angle 2= m \angle 3\)	CLICK ME Subtraction Property of Equality
CLICK ME \(\angle 2 \cong \angle 3\)	CLICK ME Definition of Congruent Angles

Proof Number \(7\)

Given:
\(\angle AMN \cong \angle DMY\)

Prove:
\(\angle AMD \cong \angle NMY\)

\(\,\,\,\,\,\,\,\)

\(\underline{\text{Statement}}\)	\(\underline{\text{Reason}}\)
CLICK ME \(\angle AMN \cong \angle DMY\)	CLICK ME Given
CLICK ME \(m \angle AMN = m \angle DMY\)	CLICK ME Definition of Congruent Angles
CLICK ME \( m \angle NMD = m \angle NMD\)	CLICK ME Reflexive Property
CLICK ME \(m \angle AMN + m \angle NMD = m \angle DMY + m \angle NMD\)	CLICK ME Addition Property of Equality
CLICK ME \(m \angle AMN + m \angle NMD =m \angle AMD\) \(m \angle DMY + m \angle NMD= m \angle NMY\)	CLICK ME Angle Addition Postulate
CLICK ME \(m \angle AMD = m \angle NMY\)	CLICK ME Substitution Property
CLICK ME \(\angle AMD \cong \angle NMY\)	CLICK ME Definition of Congruent Angles

Proof Number \(8\)

Given:
\(\overrightarrow{MA} \perp \overrightarrow{MD}\)
\(\overrightarrow{MN} \perp \overrightarrow{MY}\)

Prove:
\(\angle 1 \cong \angle 3\)

\(\,\,\,\,\,\,\,\)

\(\underline{\text{Statement}}\)	\(\underline{\text{Reason}}\)
CLICK ME \(\overrightarrow{MA} \perp \overrightarrow{MD}\) \(\overrightarrow{MN} \perp \overrightarrow{MY}\)	CLICK ME Given
CLICK ME \(m\angle AMD=90^{\circ}\) \(m\angle NMY=90^{\circ}\)	CLICK ME Definition of perpendicular \((\perp)\)
CLICK ME \(m\angle AMD= m \angle NMY\)	CLICK ME Transitive Property
CLICK ME \(m\angle 1 + m \angle 2= m \angle AMD\) \(m\angle 2 + m \angle 3= m \angle NMY\)	CLICK ME Angle Addition Postulate
CLICK ME \(m\angle 1 + m \angle 2= m \angle 2 + m \angle 3\)	CLICK ME Substitution Property
CLICK ME \(m\angle 1 = m \angle 3\)	CLICK ME Subtraction Property of Equality
CLICK ME \(\angle 1 \cong \angle 3\)	CLICK ME Definition of Congruent Angles

Proof Number \(9\)

Given:
\(\angle 1 \cong \angle 2\)

Prove:
\(\angle 2 \cong \angle 3\)

\(\,\,\,\,\,\,\,\)

\(\underline{\text{Statement}}\)	\(\underline{\text{Reason}}\)
CLICK ME \(\angle 1 \cong \angle 2\)	CLICK ME Given
CLICK ME \(\angle 1 \cong \angle 3\)	CLICK ME Vertical Angles are Congruent
CLICK ME \(\angle 2 \cong \angle 3\)	CLICK ME Substitution Property

Proof Number \(10\)

Given:
\(U\) is the midpoint of \(\overline{FN}\) \(\)

Prove:
\(x=4\)

\(\,\,\,\,\,\,\,\)

\(\underline{\text{Statement}}\)	\(\underline{\text{Reason}}\)
CLICK ME \(U\) is the midpoint of \(\overline{FN}\)	CLICK ME Given
CLICK ME \(FU=UN\)	CLICK ME Definition of Midpoint
CLICK ME \(5x+8=7x\)	CLICK ME Substitution Property
CLICK ME \(8=2x\)	CLICK ME Subtraction Property of Equality
CLICK ME \(4=x\)	CLICK ME Division Property of Equality
CLICK ME \(x=4\)	CLICK ME Symmetric Property

See Related Pages\(\)

\(\bullet\text{ Geometry Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)

\(\bullet\text{ Proving Triangle Congruence SAS}\)
\(\,\,\,\,\,\,\,\,\)\(…\)

\(\bullet\text{ Proving Triangle Congruence SSS}\)
\(\,\,\,\,\,\,\,\,\) \(…\)

\(\bullet\text{ Proving Triangle Congruence ASA}\)
\(\,\,\,\,\,\,\,\,\) \(…\)

\(\bullet\text{ Proving Triangle Congruence AAS}\)
\(\,\,\,\,\,\,\,\,\) \(…\)

\(\bullet\text{ Proving Triangle Congruence CPCTC}\)
\(\,\,\,\,\,\,\,\,\text{Corresponding Parts of Congruent Triangles are}\)\(…\)

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