Question

the most precise name for each quadrilateral.
16)
17)
18)
find the values of the variables. then find the side lengths.

1. square wxyz

w
x
\frac{3x}{5}
z
y
x - 8

1. rhombus abcd

a
m + 12
d
b
3m
c

1. rectangle qrst

q
r
x
3x - 10
t
s
4y
2y

1. square lmno

l
m
x + 7
o
n
3x + 1

Explanation:

Part 1: Name the Quadrilaterals

1. 15): Opposite sides are congruent, no right angles marked.
2. 16): All sides congruent, all angles right angles.
3. 17): Opposite sides congruent, all angles right angles.
4. 18): All sides congruent, no right angles marked.

19) Square WXYZ

Step1: Set sides equal

All square sides are congruent:
$$\frac{3x}{5} = x - 8$$

Step2: Solve for $x$

Multiply both sides by 5:
$$3x = 5(x - 8)$$
$$3x = 5x - 40$$
$$-2x = -40$$
$$x = 20$$

Step3: Calculate side length

Substitute $x=20$:
$$\frac{3(20)}{5} = 12$$

20) Rhombus ABCD

Step1: Set sides equal

All rhombus sides are congruent:
$$m + 12 = 3m$$

Step2: Solve for $m$

$$12 = 2m$$
$$m = 6$$

Step3: Calculate side length

Substitute $m=6$:
$$3(6) = 18$$

21) Rectangle QRST

Step1: Set opposite sides equal

Opposite rectangle sides are congruent:
$$x = 4y \quad \text{and} \quad 3x - 10 = 2y$$

Step2: Substitute $x=4y$

$$3(4y) - 10 = 2y$$
$$12y - 10 = 2y$$
$$10y = 10$$
$$y = 1$$

Step3: Solve for $x$

$$x = 4(1) = 4$$

Step4: Calculate side lengths

Length: $x=4$, Width: $2y=2(1)=2$

22) Square LMNO

Step1: Set sides equal

All square sides are congruent:
$$x + 7 = 3x + 1$$

Step2: Solve for $x$

$$6 = 2x$$
$$x = 3$$

Step3: Calculate side length

Substitute $x=3$:
$$3 + 7 = 10$$

Answer:

1. Parallelogram
2. Square
3. Rectangle
4. Rhombus

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Part 2: Find Variables and Side Lengths