Question

find the measures of the numbered angles in each rhombus.
6)

m∠1= , m∠2=
m∠3= , m∠4=
7)

m∠1= , m∠2=
m∠3= , m∠4=
8)

m∠1= , m∠2=
m∠3= , m∠4=
9)

m∠1= , m∠2=
m∠3= , m∠4=
10)

m∠1= , m∠2=
m∠3= , m∠4=
11)

m∠1= , m∠2=
m∠3= , m∠4=
hijk is a rectangle. find the value of x and the length of each diagonal.

1. hj = 3x + 7 and ik = 6x - 11
2. hj = 19 + 2x and ik = 3x + 22

Explanation:

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Problem 7)

Step1: Find $m\angle1$

In a rhombus, opposite angles are equal, and adjacent angles sum to $180^\circ$. The given angle is $68^\circ$, so:
$m\angle1 = \frac{180^\circ - 68^\circ}{2} = 56^\circ$

Step2: Confirm $m\angle2$

$\angle2$ is equal to the given $68^\circ$ (opposite angles in rhombus, or alternate interior angles):
$m\angle2 = 68^\circ$

Step3: Find $m\angle3$

$\angle3$ is equal to $m\angle1$ (opposite angles of congruent triangles formed by diagonals):
$m\angle3 = 56^\circ$

Step4: Confirm $m\angle4$

Diagonals of a rhombus are perpendicular:
$m\angle4 = 90^\circ$

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Problem 8)

Step1: Find $m\angle1$

Adjacent angles in a rhombus sum to $180^\circ$. The given angle is $20^\circ$, so:
$m\angle1 = 180^\circ - 20^\circ = 160^\circ$

Step2: Find $m\angle2$

Diagonals bisect rhombus angles:
$m\angle2 = \frac{160^\circ}{2} = 80^\circ$

Step3: Find $m\angle3$

Diagonals bisect rhombus angles:
$m\angle3 = \frac{20^\circ}{2} = 10^\circ$

Step4: Find $m\angle4$

In a triangle, angles sum to $180^\circ$, and diagonals are perpendicular:
$m\angle4 = 180^\circ - 90^\circ - 10^\circ = 80^\circ$

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Problem 9)

Step1: Find $m\angle1$

$\angle1$ is equal to the given $52^\circ$ (alternate interior angles in rhombus):
$m\angle1 = 52^\circ$

Step2: Find $m\angle2$

Diagonals of a rhombus are perpendicular:
$m\angle2 = 90^\circ$

Step3: Find $m\angle3$

$\angle3$ is equal to $\angle2$ (vertical angles from intersecting diagonals):
$m\angle3 = 90^\circ$

Step4: Find $m\angle4$

In a triangle, angles sum to $180^\circ$:
$m\angle4 = 180^\circ - 90^\circ - 52^\circ = 38^\circ$

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Problem 10)

Step1: Find $m\angle1$

$\angle1$ is equal to the given $59^\circ$ (alternate interior angles in rhombus):
$m\angle1 = 59^\circ$

Step2: Find $m\angle2$

Diagonals of a rhombus are perpendicular:
$m\angle2 = 90^\circ$

Step3: Find $m\angle3$

$\angle3$ is equal to $\angle2$ (vertical angles from intersecting diagonals):
$m\angle3 = 90^\circ$

Step4: Find $m\angle4$

In a triangle, angles sum to $180^\circ$:
$m\angle4 = 180^\circ - 90^\circ - 59^\circ = 31^\circ$

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Problem 11)

Step1: Find $m\angle1$

In a rhombus, opposite angles are equal, so $\angle1$ matches the given $39^\circ$:
$m\angle1 = 39^\circ$

Step2: Find $m\angle2$

Adjacent angles in a rhombus sum to $180^\circ$:
$m\angle2 = 180^\circ - 39^\circ = 141^\circ$

Step3: Find $m\angle3$

Diagonals bisect rhombus angles:
$m\angle3 = \frac{39^\circ}{2} = 19.5^\circ$

Step4: Find $m\angle4$

Diagonals of a rhombus are perpendicular:
$m\angle4 = 90^\circ$

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Problem 13)

Step1: Set diagonals equal

In a rectangle, diagonals are congruent:
$3x + 7 = 6x - 11$

Step2: Solve for $x$

Rearrange and solve:
$7 + 11 = 6x - 3x \implies 18 = 3x \implies x = 6$

Step3: Calculate diagonal length

Substitute $x=6$ into $HJ$:
$HJ = 3(6) + 7 = 25$

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Problem 14)

Step1: Set diagonals equal

In a rectangle, diagonals are congruent:
$19 + 2x = 3x + 22$

Step2: Solve for $x$

Rearrange and solve:
$19 - 22 = 3x - 2x \implies x = -3$

Step3: Calculate diagonal length

Substitute $x=-3$ into $HJ$:
$HJ = 19 + 2(-3) = 13$

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Answer:

1. $m\angle1=56^\circ$, $m\angle2=68^\circ$, $m\angle3=56^\circ$, $m\angle4=90^\circ$
2. $m\angle1=160^\circ$, $m\angle2=80^\circ$, $m\angle3=10^\circ$, $m\angle4=80^\circ$
3. $m\angle1=52^\circ$, $m\angle2=90^\circ$, $m\angle3=90^\circ$, $m\angle4=38^\circ$
4. $m\angle1=59^\circ$, $m\angle2=90^\circ$, $m\angle3=90^\circ$, $m\angle4=31^\circ$
5. $m\angle1=39^\circ$, $m\angle2=141^\circ$, $m\angle3=19.5^\circ$, $m\angle4=90^\circ$
6. $x=6$, length of each diagonal $=25$
7. $x=-3$, length of each diagonal $=13$