Sum of the Interior Angles of a Polygon

$$The data in the drawing (which we will first write mathematically, using conventional notation):

$$$$$\sphericalangle BAD=101\degree\\ \sphericalangle ABC=87\degree\\ \sphericalangle CDA=68\degree$

Find:

$\sphericalangle BCD=\text{?}$Solution:

We'll use the fact that the sum of angles in a concave quadrilateral is $360\degree$ meaning that:

1. $\sphericalangle BAD+ \sphericalangle ABC+ \sphericalangle BCD+ \sphericalangle CDA=360\degree$

$$Let's substitute the above data in 1:

2. $101 \degree+ 87 \degree+ \sphericalangle BCD+ 68 \degree=360\degree$

Now let's solve the resulting equation for the requested angle, we'll do this by moving terms:

3. $\sphericalangle BCD=360\degree- 101 \degree- 87 \degree - 68 \degree$

4. $\sphericalangle BCD=104\degree$Therefore the correct answer is answer B

To find the measure of angle $∢\text{BAD}$ in quadrilateral $ABCD$, we apply the formula for the sum of interior angles of a quadrilateral:

• The sum of the interior angles in any quadrilateral is $360^\circ$.
• Therefore, we have the equation: $∢\text{BAD} + 71^\circ + 95^\circ + 120^\circ = 360^\circ$.

Solving for $∢\text{BAD}$:

• Add the given angles: $71^\circ + 95^\circ + 120^\circ = 286^\circ$.
• Subtract the sum from $360^\circ$: $360^\circ - 286^\circ = 74^\circ$.

Therefore, the measure of angle $∢\text{BAD}$ is $\boxed{74^\circ}$.

The correct answer to the problem is $\boxed{74}$.

To solve this problem, follow these steps:

• Step 1: Identify all given angles and understand the setup.

• Step 2: Apply the sum of angles in a quadrilateral formula.

• Step 3: Calculate the unknown angle.

Now, let's solve:
Step 1: The problem states:

• $\angle \text{DAB} = 48^\circ$

• $\angle \text{ADC} = 119^\circ$

• $\angle \text{ABC} = 90^\circ$ since it's marked as a right angle.

Step 2: Use the sum of angles in quadrilateral $ABCD$: $\angle \text{DAB} + \angle \text{ABC} + \angle \text{BCD} + \angle \text{ADC} = 360^\circ$ Substituting the known values: $48^\circ + 90^\circ + \angle \text{BCD} + 119^\circ = 360^\circ$ Step 3: Simplify and solve for $\angle \text{BCD}$: $157^\circ + \angle \text{BCD} = 360^\circ$$\angle \text{BCD} = 360^\circ - 157^\circ = 203^\circ$ Therefore, the measure of $\angle \text{BCD}$ is $103^\circ$.

Thus, the size of angle $\angle \text{BCD}$ is $103^\circ$.

To solve this problem, we'll follow these steps:

• Step 1: Set up the equation for the sum of the angles of a quadrilateral.
• Step 2: Solve for the variable $x$.
• Step 3: Calculate angle $\angle BAD$ using the determined value of $x$.

Now, let's work through each step:

Step 1: We know that the sum of the interior angles in a quadrilateral is $360^\circ$. Therefore, we have:

$(x + 3) + (2x - 2) + (5x - 2) + (2x + 11) = 360$

Step 2: Simplify the equation:

$x + 3 + 2x - 2 + 5x - 2 + 2x + 11 = 360$

$10x + 10 = 360$

Solve for $x$ by subtracting 10 from both sides:

$10x = 350$

Divide both sides by 10:

$x = 35$

Step 3: Calculate $\angle BAD$ using $x = 35$:

$\angle BAD = x + 3 = 35 + 3 = 38^\circ$

Therefore, the solution to the problem is $\mathbf{38^\circ}$.

To solve this problem, we'll follow these steps:

• Step 1: Set up the equation for the sum of the interior angles.
• Step 2: Solve for $x$.
• Step 3: Calculate $∠D$ using the value of $x$.

Now, let's work through each step:

Step 1: The angles of quadrilateral ABCD are given as follows:
$∠A = 2x + 8$, $∠B = 3x - 4$, $∠C = 6x + 10$, and $∠D = x - 2$.
According to the angle sum property of a quadrilateral, we have:

$(2x + 8) + (3x - 4) + (6x + 10) + (x - 2) = 360$

Step 2: Simplify and solve for $x$:

Combine like terms:
$2x + 3x + 6x + x + 8 - 4 + 10 - 2 = 360$

This simplifies to:
$12x + 12 = 360$

Subtract 12 from both sides:
$12x = 348$

Divide both sides by 12 to isolate $x$:
$x = 29$

Step 3: Substitute $x = 29$ into the expression for $∠D = x - 2$:
$∠D = 29 - 2 = 27$

Therefore, the measure of angle $∢\text{BDC}$ or $∠D$ is 27 degrees.