Question

unit 9 scaffold review
① label \\(\overline{ab} \cong \overline{cd}\\) and \\(\overline{ac} \cong \overline{bd}\\)
② label \\(\angle dca \cong \angle abd\\)
③ if \\(\overline{ab} = 9x - 11\\) and \\(\overline{cd} = 4x + 14\\), what is \\(x\\)? \\(x = \\
④ if \\(\overline{ab} = 17\\), what is \\(cd\\)? \\
⑤ \\(\overline{ac} = 2(x + 6)\\). \\(\overline{bd} = 3(x - 2)\\). what is \\(x\\)? \\(x = \\
⑥ what are the names of your diagonals? \\(\underline{quad}\\) and \\
⑦ what are the possible quadrilaterals that \\(\square abcd\\) can be? why?

Explanation:

Step1: Set equal congruent segments

$9x - 11 = 4x + 14$

Step2: Isolate x terms

$9x - 4x = 14 + 11$

Step3: Simplify and solve for x

$5x = 25$
$x = \frac{25}{5} = 5$

Step4: Use segment congruence for CD

Since $\overline{AB} \cong \overline{CD}$, $CD = AB = 17$

Step5: Set equal congruent diagonals

$2(x + 6) = 3(x - 2)$

Step6: Expand both sides

$2x + 12 = 3x - 6$

Step7: Isolate x to solve

$12 + 6 = 3x - 2x$
$x = 18$

Answer:

1. $x=5$
2. $17$
3. $x=18$
4. $\overline{AC}$ and $\overline{BD}$
5. Possible quadrilaterals are a rectangle or a square. This is because the figure has congruent opposite sides (defining a parallelogram) and congruent diagonals; in parallelograms, congruent diagonals only occur in rectangles (and squares, which are a special type of rectangle).