It appears that you've provided a series of statements and properties related to the angles and sides of a parallelogram, specifically regarding triangle \( ABC \) within parallelogram \( ABCD \). Below, I’ll clarify the steps involved in proving certain properties about angles and sides in this parallelogram context.
Proof Steps on Properties of a Parallelogram
- Given: \( ABCD \) is a parallelogram.
- Property: \( DB \parallel AC \) (Opposite sides of a parallelogram are parallel).
- Property: \( DA \parallel CB \) (Opposite sides of a parallelogram are parallel).
- Property: \( AB = BA \) (Reflexive Property of Congruence).
- Property: \( AB = DC \) and \( AD = BC \) (Opposite sides of a parallelogram are congruent).
- Property: \( \angle BAD = \angle ABC \) (Consecutive angles of a parallelogram are congruent).
- Relationship: \( m\angle BAD = m\angle ABC \) (Definition of congruence).
- Supplementary Angles: \( m\angle BAD + m\angle ABC = 180^\circ \) (Consecutive angles of a parallelogram are supplementary).
- Combine Like Terms: \( 2m\angle BAD = 180^\circ \) (From previous statements).
- Dividing Both Sides: \( m\angle BAD = 90^\circ \) (Using Division Property of Equality).
- Conclusion: \( m\angle ABC = 90^\circ \) (Since \( m\angle BAD = m\angle ABC \)).
- Properties of Angles: Consequently, \( m\angle ADC = m\angle DAB = 90^\circ \) (Opposite angles are congruent due to the properties of a parallelogram).
- Final Conclusion: Hence, angles \( \angle BAD, \angle DCB, \angle ABC, \) and \( \angle ADC \) are all right angles, confirming that \( ABCD \) is a rectangle.
Summary of Properties Used:
- Reflexive Property of Congruence: An object is congruent to itself.
- SSS Congruence Theorem: A triangle is congruent if its sides are congruent.
- Definition of Congruence: If two angles are congruent, they have the same measure.
- Consecutive Angles of a Parallelogram are Supplementary: The angles that are adjacent (next to each other) in a parallelogram sum up to 180°.
- Transitive Property of Equality: If \( a = b \) and \( b = c \), then \( a = c \).
- Division Property of Equality: If \( a = b \), then \( a/c = b/c \) (for non-zero \( c \)).
- Definition of a Right Angle: An angle that measures 90°.
This structured approach helps in establishing relationships and proving properties essential for understanding the nature of parallelograms, specifically when that parallelogram forms right angles, leading to the conclusion that it is a rectangle. If you have further questions or another topic in mind, feel free to ask!
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