Question

date: this is a 2 - page document! directions: if each quadrilateral below is a rectangle, find the missing measures. 1.
vw =
hx =
yw =
zx =
vx =
2.
gf =
ge =
df =
hf =
dg =
*gh = 14 3.
m∠1 =
m∠2 =
m∠3 =
m∠4 =
m∠5 =
m∠6 =
m∠7 =
m∠8 =
m∠9 =
m∠10 =
m∠11 =

Explanation:

Problem 1

Step1: Identify rectangle properties

In a rectangle, opposite sides are equal, diagonals are equal and bisect each other.

Step2: Find VW

$VW = YX = 31$

Step3: Find WX

$WX = VY = 19$

Step4: Find YW

Diagonals of rectangle are equal: $YW = ZX = VX$. Use Pythagorean theorem: $YW = \sqrt{VY^2 + YX^2} = \sqrt{19^2 + 31^2} = \sqrt{361 + 961} = \sqrt{1322} \approx 36.36$

Step5: Find ZX

$ZX = YW = \sqrt{1322} \approx 36.36$

Step6: Find VX

$VX = YW = \sqrt{1322} \approx 36.36$

Step1: Identify rectangle properties

Opposite sides equal, diagonals equal and bisect each other.

Step2: Find GF

$GF = DE = 11$

Step3: Find GE

$GE = DF$. Use Pythagorean theorem: $GE = \sqrt{GH^2 + GF^2} = \sqrt{14^2 + 11^2} = \sqrt{196 + 121} = \sqrt{317} \approx 17.80$

Step4: Find DF

$DF = GE = \sqrt{317} \approx 17.80$

Step5: Find HF

Diagonals bisect each other: $HF = \frac{1}{2}DF = \frac{\sqrt{317}}{2} \approx 8.90$

Step6: Find DG

$DG = GH = 14$

Step1: Use rectangle angle properties

All angles are $90^\circ$, alternate interior angles equal, vertical angles equal, linear pairs sum to $180^\circ$.

Step2: Find $m\angle1$

$m\angle1 = 90^\circ - 59^\circ = 31^\circ$

Step3: Find $m\angle2$

$m\angle2 = 59^\circ$ (alternate interior angle)

Step4: Find $m\angle3$

$m\angle3 = m\angle1 = 31^\circ$ (vertical angle)

Step5: Find $m\angle4$

$m\angle4 = m\angle2 = 59^\circ$ (vertical angle)

Step6: Find $m\angle5$

$m\angle5 = m\angle1 = 31^\circ$ (alternate interior angle)

Step7: Find $m\angle6$

$m\angle6 = 59^\circ$

Step8: Find $m\angle7$

$m\angle7 = 180^\circ - 2\times31^\circ = 118^\circ$

Step9: Find $m\angle8$

$m\angle8 = 180^\circ - m\angle7 = 62^\circ$

Step10: Find $m\angle9$

$m\angle9 = m\angle7 = 118^\circ$ (vertical angle)

Step11: Find $m\angle10$

$m\angle10 = m\angle8 = 62^\circ$ (vertical angle)

Step12: Find $m\angle11$

$m\angle11 = 180^\circ - 59^\circ = 121^\circ$

Answer:

$VW = 31$
$WX = 19$
$YW = \sqrt{1322} \approx 36.36$
$ZX = \sqrt{1322} \approx 36.36$
$VX = \sqrt{1322} \approx 36.36$

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Problem 2