quadrilateral A B C D on a graph. Point A is at ( -5, -1), Point B is at ( 6, 1), Point C is at (4, -3), Point D is at (-7, -5).
Prove the quadrilateral is a parallelogram by using Theorem 5-8; if one pair of opposite sides of a quadrilateral is both congruent and parallel, then it is a parallelogram.
figure the length and slope of AB and CD. If they match, it's a ||-ogram.
To prove that the quadrilateral ABCD is a parallelogram using Theorem 5-8, we need to show that one pair of opposite sides of the quadrilateral is both congruent and parallel.
Step 1: Calculate the slopes of the lines containing opposite sides AB and CD.
The slope of a line can be calculated using the formula: slope = (change in y / change in x).
For AB:
- The coordinates of A are (-5, -1).
- The coordinates of B are (6, 1).
- Therefore, the slope of line AB = (1 - (-1)) / (6 - (-5)) = 2/11.
For CD:
- The coordinates of C are (4, -3).
- The coordinates of D are (-7, -5).
- Therefore, the slope of line CD = (-5 - (-3)) / (-7 - 4) = -2/11.
Step 2: Determine if the slopes of opposite sides are equal.
Since the slopes of lines AB and CD are both found to be 2/11, they are congruent.
Step 3: Calculate the slopes of the lines containing opposite sides BC and AD.
For BC:
- The coordinates of B are (6, 1).
- The coordinates of C are (4, -3).
- Therefore, the slope of line BC = (-3 - 1) / (4 - 6) = 2/-2 = -1.
For AD:
- The coordinates of A are (-5, -1).
- The coordinates of D are (-7, -5).
- Therefore, the slope of line AD = (-5 - (-1)) / (-7 - (-5)) = -4/-2 = 2.
Step 4: Determine if the slopes of opposite sides are equal.
Since the slopes of lines BC and AD are both found to be -1 and 2, respectively, they are not congruent.
Conclusion:
From the calculations, we can see that the slopes of opposite sides AB and CD are equal (2/11), and therefore, they are congruent. However, the slopes of opposite sides BC and AD are not equal. According to Theorem 5-8, if one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram. Therefore, based on the given information, we can conclude that the quadrilateral ABCD is a parallelogram.