Question

4th. find the measure of the unlabeled angles in each item

a

1. find the value of x in the rhombus below

Explanation:

Problem 29: Find the measures of the numbered angles in each item.

Part a

Step1: Identify triangle angle sum

The sum of angles in a triangle is $180^\circ$. For the triangle with $101^\circ$ and $45^\circ$:
$180^\circ - 101^\circ - 45^\circ = 34^\circ$
This equals $\angle 1$.

Step2: Use vertical angle property

$\angle 2$ is vertical to the $45^\circ$ angle, so $\angle 2 = 45^\circ$.

Step3: Use right angle property

The triangle has a right angle ($90^\circ$). For $\angle 3$:
$180^\circ - 90^\circ - 45^\circ = 45^\circ$

Step4: Use linear pair property

$\angle 4$ forms a linear pair with $\angle 3$:
$180^\circ - 45^\circ = 135^\circ$

Step1: Use rhombus angle properties

In a rhombus, opposite angles are equal, so the angle opposite $116^\circ$ is also $116^\circ$. The adjacent angles sum to $180^\circ$:
$180^\circ - 116^\circ = 64^\circ$
This is the measure of the adjacent pair of angles.

Step2: Bisect angles for $\angle 1, \angle 2$

Diagonals in a rhombus bisect the angles. For the $64^\circ$ angle:
$\angle 1 = \angle 2 = \frac{64^\circ}{2} = 32^\circ$

Step3: Bisect angles for $\angle 3, \angle 4$

For the $116^\circ$ angle:
$\angle 3 = \angle 4 = \frac{116^\circ}{2} = 58^\circ$

Step4: Confirm right angle $\angle 5$

Diagonals in a rhombus are perpendicular, so $\angle 5 = 90^\circ$

Step1: Rhombus side equality

All sides of a rhombus are equal, so set $4x+1 = 17$.

Step2: Solve for $x$

Subtract 1 from both sides:
$4x = 17 - 1 = 16$
Divide by 4:
$x = \frac{16}{4} = 4$

Step3: Verify with other side

Check $6x-3$ with $x=4$:
$6(4)-3 = 24-3 = 21$, which matches the given side length.

Answer:

$\angle 1 = 34^\circ$, $\angle 2 = 45^\circ$, $\angle 3 = 45^\circ$, $\angle 4 = 135^\circ$

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Part b