Question

part 6: kite
use kite abcd and the given information to complete the following.

1. if ( ab = x + 3 ), ( bc = x + 4 ), ( cd = 2x - 1 ), and ( ad = 3x - y ), solve for ( x ) and ( y ).
2. if ( angle aeb = 5x - 10 ) and ( de = 6x ), find ( db ).

part 7: rhombus
use rhombus abcd and the given information to complete the following.

1. if ( angle adb = 27 ), find ( angle adc ).
2. if ( angle dec = 5x ), find the value of ( x ).
3. if ( de = 13 ), find ( db ).

part 8: squares
use square abcd and the given information to complete the following.

1. if ( bc = 3x + 14 ) and ( dc = 5x - 8 ), find the value of ( x ).
2. if ( angle aeb = 3x ), find the value of ( x ).
3. if ( angle bac = 5x ), find the value of ( x ).

Explanation:

Step1: Solve for x (kite congruent sides)

A kite has two pairs of congruent adjacent sides, so $AB = AD$ and $BC = CD$.
First use $BC = CD$:
$x + 4 = 2x - 1$
$4 + 1 = 2x - x$
$x = 5$

Step2: Solve for y (kite congruent sides)

Substitute $x=5$ into $AB = AD$:
$AB = 5 + 3 = 8$
$AD = 3(5) - y = 15 - y$
$8 = 15 - y$
$y = 15 - 8 = 7$

Step3: Find x (kite right angle)

Kite diagonals are perpendicular, so $\angle AEB = 90^\circ$:
$5x - 10 = 90$
$5x = 100$
$x = 20$

Step4: Find DB (kite diagonal segments)

Kite diagonals bisect each other at right angles, so $DB = 2 \cdot DE$. Substitute $x=20$:
$DE = 6(20) = 120$
$DB = 2 \cdot 120 = 240$

Step5: Find $\angle ADC$ (rhombus diagonal)

Rhombus diagonals bisect interior angles, so $\angle ADC = 2 \cdot \angle ADB$:
$\angle ADC = 2 \cdot 27^\circ = 54^\circ$

Step6: Find x (rhombus perpendicular diagonals)

Rhombus diagonals are perpendicular, so $\angle DEC = 90^\circ$:
$5x = 90$
$x = 18$

Step7: Find DB (rhombus diagonal segments)

Rhombus diagonals bisect each other, so $DB = 2 \cdot DE$:
$DB = 2 \cdot 13 = 26$

Step8: Find x (square congruent sides)

Square sides are congruent, so $BC = DC$:
$3x + 14 = 5x - 8$
$14 + 8 = 5x - 3x$
$22 = 2x$
$x = 11$

Step9: Find x (square perpendicular diagonals)

Square diagonals are perpendicular, so $\angle AEB = 90^\circ$:
$3x = 90$
$x = 30$

Step10: Find x (square angle bisector)

Square diagonals bisect right angles, so $\angle BAC = 45^\circ$:
$5x = 45$
$x = 9$

Answer:

1. $x=5$, $y=7$
2. $DB=240$
3. $\angle ADC=54^\circ$
4. $x=18$
5. $DB=26$
6. $x=11$
7. $x=30$
8. $x=9$