Question

1. defg is a rectangle. df = 5x - 5 and eg = x + 11. find the value of x and the length of each diagonal.
2. find the values of a and b.
3. find m∠1 and m∠3 in the kite.

Explanation:

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Step1: Use rectangle property

In a rectangle, the diagonals are equal. So, $DF = EG$. We set up the equation $5x - 5=x + 11$.

Step2: Solve for x

Subtract $x$ from both sides: $5x-x-5=x - x+11$, which simplifies to $4x-5 = 11$. Then add 5 to both sides: $4x-5 + 5=11 + 5$, giving $4x=16$. Divide both sides by 4: $x=\frac{16}{4}=4$.

Step3: Find the length of the diagonal

Substitute $x = 4$ into either $DF$ or $EG$. Using $EG=x + 11$, we get $EG=4 + 11=15$. Since $DF = EG$, $DF = 15$ too.

Step1: Use the property of a trapezoid (assuming it's an isosceles trapezoid as no other info is given)

The sum of the interior angles of a quadrilateral is $360^{\circ}$. Also, in an isosceles trapezoid, base - angles are equal.

Step2: Calculate a

$a=113^{\circ}$ (because the non - base angles of an isosceles trapezoid are equal).

Step3: Calculate b

We know that $36^{\circ}+113^{\circ}+113^{\circ}+b^{\circ}=360^{\circ}$. Combine like terms: $262^{\circ}+b^{\circ}=360^{\circ}$. Subtract $262^{\circ}$ from both sides: $b^{\circ}=360^{\circ}-262^{\circ}=98^{\circ}$.

Step1: Use the property of a kite

In a kite, the diagonals are perpendicular, so $\angle 2 = 90^{\circ}$. Also, the diagonal of a kite bisects the angles at the vertices.

Step2: Calculate m∠1

In right - triangle $ABD$, $\angle 1=90^{\circ}-39^{\circ}=51^{\circ}$.

Step3: Calculate m∠3

Since the diagonal $AC$ bisects $\angle DAB$, $\angle 3 = 39^{\circ}$.

Answer:

$x = 4$, length of each diagonal is 15

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