Question
Which would show that \( F G H J \) is a parallelogram? Select all that apply. A. Show that slope of \( \overline{F G}= \) slope of \( \overline{H J} \) and \( F G=H J \). B. Show that slope of \( \overline{F G}= \) slope of \( \overline{H J} \) and \( F J=G H \). \( \square \) C. Show that slope of \( \overline{F G}= \) slope of \( \overline{H J} \) and slope of \( \overline{F J}= \) slope of \( \overline{G H} \) D. Show that \( F G=H J \) and \( F J=G H \).
Ask by Montgomery Cruz. in the United States
Feb 20,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
All options A, B, C, and D show that \( F G H J \) is a parallelogram.
Solution
To determine which options would show that \( F G H J \) is a parallelogram, we need to recall the properties of parallelograms. A quadrilateral is a parallelogram if:
1. Both pairs of opposite sides are equal in length.
2. Both pairs of opposite sides are parallel (which can be shown by having equal slopes).
3. One pair of opposite sides is both equal in length and parallel.
Now, let's analyze each option:
**A.** Show that slope of \( \overline{F G}= \) slope of \( \overline{H J} \) and \( F G=H J \).
- This option shows that one pair of opposite sides (\( F G \) and \( H J \)) are both equal in length and parallel. This is sufficient to prove that \( F G H J \) is a parallelogram.
**B.** Show that slope of \( \overline{F G}= \) slope of \( \overline{H J} \) and \( F J=G H \).
- This option shows that one pair of opposite sides (\( F G \) and \( H J \)) are parallel, and the other pair (\( F J \) and \( G H \)) are equal in length. This is also sufficient to prove that \( F G H J \) is a parallelogram.
**C.** Show that slope of \( \overline{F G}= \) slope of \( \overline{H J} \) and slope of \( \overline{F J}= \) slope of \( \overline{G H} \).
- This option shows that both pairs of opposite sides are parallel. This is sufficient to prove that \( F G H J \) is a parallelogram.
**D.** Show that \( F G=H J \) and \( F J=G H \).
- This option shows that both pairs of opposite sides are equal in length. This is also sufficient to prove that \( F G H J \) is a parallelogram.
Based on this analysis, the options that would show that \( F G H J \) is a parallelogram are:
- **A**
- **B**
- **C**
- **D**
Thus, all options (A, B, C, and D) apply.
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution The Deep Dive
To show that \( F G H J \) is a parallelogram, you can use several geometric properties. Firstly, if you can demonstrate that the opposite sides are both equal and parallel, you have successfully proven that you have a parallelogram on your hands! In this case, options A, B, and C each include evidence of parallel sides because they involve showing that the slopes of opposite sides are equal. Option D focuses on the lengths of opposite sides, which also fulfills the criteria for being a parallelogram. All options combined cover the necessary criteria to verify \( F G H J \) as a parallelogram!
