1. Write a two-column proof for the following conjecture. You may not need to use all of the rows of the two-column table provid
ed below. You may also add additional rows if needed. Given: Parallelogram ABCD Prove: Angle A and B are supplementary. C and d are supplementary.
2 answers:
Answer:
Given: ABCD is a parallelogram.
Prove: Angle A and B are supplementary and C and D are supplementary.
That is, m∠A + m∠B = 180° and m∠C + m∠D = 180°
Statement Reason
1. ABCD is a parallelogram 1. Given
2.m∠A=m∠C and m∠B=m∠D 2. Definition of parallelogram.
3.m∠A+m∠B+m∠C+m∠D=360° 3. Definition of quadrilateral
4. m∠A+m∠B+m∠A+m∠B=360° 4. By substitution
⇒ 2( m∠A + m∠B ) = 360°
⇒ m∠A + m∠B = 180°
Similarly, m∠C + m∠D = 180°
5.∠A and ∠C are supplementary, 5. Def. of Supplementary
∠ B and ∠D are supplementary
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Through similar steps, you should find the slope of line BC is -1.
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You should find that line CD has a slope of 1, so BC and CD are perpendicular. Lastly, line AD has a slope of -1. So all the slopes mentioned are either +1 or -1. The slopes alternate as you move around the figure. This all shows that any two adjacent sides meet at a 90 degree angle.
Therefore, this figure is a rectangle. Any rectangle has all four angles of 90 degrees.
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