Sum and Difference of Angles: Calculate Angles in Quadrilaterals

$$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

Let's observe triangle CAD, the sum of angles in a triangle is 180 degrees, hence we can determine angle DAC:

$CAD+90+30=180$

$CAD+120=180$

$CAD=180-120$

$CAD=60$

Given that ABCD is a rectangle, all angles are equal to 90 degrees.

Therefore angle CAB equals:

$90-CAD=90-60=30$

Furthermore we can deduce that CAD equals 30 degrees, since ABCD is a rectangle all angles are equal to 90 degrees.

CAB equals 60 degrees.

Therefore:

$CAD=BCA=30,ACD=CAB=60$

According to the properties of a quadrilateral, each pair of opposite angles are equal to each other.

Therefore:

$B=C=80$

$A=D$

Additionally, we know that the sum of the angles in a quadrilateral equals 360 degrees.

Therefore, we can calculate angles A and D as follows:

$360-80-80=200$

$200:2=100$

Angle A is equal to 100.

We know that the sum of the angles of a quadrilateral is 360°, that is:

$A+B+C+D=360$

We replace the known data within the following formula:

$80+B+95+45=360$

$B+220=360$

We move the integers to one side, making sure to keep the appropriate sign:

$B=360-220$

$B=140$

Note that angle BAG is part of angle BAD.

Therefore, we can write the following equation:

$BAG+GAD=BAD$

From the data provided in the question, we know that angle A is equal to 90 degrees and angle GAD is equal to 50 degrees.

Let's now substitute the known values into the formula:

$BAG+50=90$

We'll then move like terms to one side, maintaining the appropriate sign:

$BAG=90-50$

$BAG=40$

Let's break down angle FCD for an angle addition exercise:

$FCD=FCE+ECD$

Let's write down the known information from the question:

$FCD=30+ECD$

Since angle ECD is not given to us, we will calculate it in the following way:

Let's look at triangle EDC, where we have 2 angles.

Since we know that the sum of angles in a triangle equals 180 degrees, let's write down the data in the formula:

$ECD+CED+EDC=180$

$ECD+70+90=180$

Let's move terms and keep the appropriate sign:

$ECD=180-90-70$

$ECD=20$

Now we can substitute ECD in the formula we wrote earlier:

$FCD=30+ECD$

$FCD=30+20$

$FCD=50$

Note that the GDC angle is part of the EDC angle.

Therefore, we can write the following expression:

$GDC+EDG=EDC$

Since we know that angle D equals 90 degrees, we will substitute the values in the formula:

$GDC+40=90$

We will simplify the expression and keep the appropriate sign:

$GDC=90-40$

$GDC=50$

Let's pay attention to the data in the question.

We know that angle DEB is equal to 95 degrees.

Let's break it down into an addition exercise:

$CEB+CED=DEB$

Now let's substitute the known data into the formula:

$CEB+70=95$

We'll move a term and keep the appropriate sign:

$CEB=95-70$

$CEB=25$

Let's look at angle BCD and break it down into the angles that compose it:

$BCD=BCF+FCE+ECD$

Note that the angle values we wrote in the formula are given to us in the diagram, and now we'll substitute them:

$BCD=25+30+20$

$BCD=75$

Let's break down angle BCE into an angle addition exercise:

$BCE=BCF+FCE$

Now let's input the known data from the diagram:

$BCE=25+30$

$BCE=55$

The sum of angles in a trapezoid is 360 degrees.

Therefore:

$360=A+B+C+D$

Let's substitute the known data into the above formula:

$360=110+130+70+D$

$360=310+D$

We'll move terms and maintain the appropriate sign:

$360-310=D$

$50=D$

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 $\boxed{74}$.

As we know, the sum of the angles in a square is equal to 360 degrees, therefore:

$360=A+B+C+D$

We replace the data we have in the previous formula:

$360=140+B+80+90$

$360=310+B$

Rearrange the sides and use the appropriate sign:

$360-310=B$

$50=B$

Since we know that ABCD is a rectangle, we know that AC is parallel to BD.

Therefore, angles ACB and CBD are equal (30 degrees).

In a rectangle, we know that all angles are equal to 90 degrees, meaning angle ABD is equal to 90.

Now we can calculate angle ABC as follows:

$90-30=60$

Since we are dealing with a parallelogram, there are 2 pairs of parallel lines.

As a result, we know that angle ADB and angle DBC are alternate angles between parallel lines and therefore both are equal to each other (44 degrees):

$ADB=DBC=44$

Now we can calculate angle ABC as follows:

$ABC=ABD+DBC$

Finally, let's substitute in our values:

$ABC=40+44=84$

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 $103^\circ$.