Sum and Difference of Angles: Using angles in a triangle

Given that it is an isosceles triangle:$AB=BC$

It is possible to argue that:$BAC=ACB=45$

Since the sum of the angles of a triangle is 180, the angle ABC will be equal to:

$180-45-45=90$

Since the angle ABC measures 90 degrees, it is a right triangle.

As we know that triangle ABC is isosceles.

$B=C=2X$

It is known that in a triangle the sum of the angles is 180.

Therefore, we can calculate in the following way:

$2X+2X+4X=180$

$4X+4X=180$

$8X=180$

We divide the two sections by 8:

$\frac{8X}{8}=\frac{180}{8}$

$X=22.5$

Let's look at triangle ABD and note that we are given two angles: 90 and 17.

Since the sum of angles in a triangle equals 180, we can calculate angle BAD as follows:

$180-90-17=73$

Since we are given angle BAC, we can calculate angle CAD as follows:

$73-61=12$

Note from the diagram that we can know the value of angle BAC, since its corresponding angle is equal to 105.

Now in the triangle we are given two angles:

$BAC=105$

$BCA=45$

Since the sum of angles in a triangle is equal to 180, we can calculate angle ABC:

$180-105-45=30$

Note that angle BDA equals 90 degrees, therefore angle BDC also equals 90 degrees.

Let's look at triangle ABC and calculate angle C, since angles A and B are given to us:

$180-90-30=60$

Now let's focus on triangle BDC and calculate angle alpha, since we have calculated the other two angles.

$180-90-60=30$

Since we are given an angle of 147, let's calculate the angle ACB which complements it to 180 degrees:

$180-147=33$

Now we have two angles in triangle ABC, and we can calculate angle BAC:

$180-90-33=57$

Let's remember that an angle bisector divides the angle into 2 equal parts, therefore:

$BAD=DAC=20$

We should also note that we are given:

$ADB=ADC=90$

Since the sum of angles in a triangle is 180, we can determine the size of angle ACB as follows:

Let's focus on triangle ACD, where we know 2 angles and calculate:

$ACB=180-90-20$

$ACB=70$

First, let's find the value of angle B.

Since the sum of angles in a triangle is equal to 180, the formula is:

$180-86-28=66$

Now let's look at angle ADB, we can calculate its value since we are given angle ADC.

$180-122=58$

Now we can calculate angle BAD:

$180-66-58=56$

Remember that a straight angle is equal to 180 degrees.

Let's note that angle 111 and angle ACB together form a straight angle.

Let's calculate angle ACB as follows:

$ACB=180-111$

$ACB=69$

Now we can calculate X.

Remember that the sum of angles in a triangle is equal to 180 degrees, therefore:

$X=180-69-60$

$X=51$

First, let's note that in triangle ACB we are given two angles.

Angle ABC equals 47 degrees, angle ACB equals 90 degrees.

Since the sum of angles in a triangle equals 180, we can calculate angle BAC and find the value of Y as follows:

$180-90-47=43$

Now let's look at triangle ACD, where we are also given two angles.

Angle CAD equals 43 degrees, angle ACD equals 90 degrees.

Since the sum of angles in a triangle equals 180, we can calculate angle ADC and find the value of X as follows:

$180-90-43=47$

Angle Y complements 180 and we can calculate it since we know the adjacent angle.

Let's calculate it as follows:

$180-105=75$

Now that we found angle Y, we can calculate angle X since we have the other two angles in the same triangle: 72 and 75.

$180-75-72=33$

We can calculate angle Z since we have two angles in the triangle: 25 and 105

The sum of angles in a triangle is 180, so we'll calculate Z as follows:

$180-105-25=50$

Since we are given that AM bisects BC, we can claim that AM is a median, therefore:

$AM=BM=MC$

As a result, we have created an isosceles triangle BMA, where $BM=MA$

Since we are given that angle B is equal to 50, and in an isosceles triangle the base angles are equal to each other, we can claim:

$B=MAB=50$

Let's look at triangle ABC, where we are given two angles.

Let's remember that the sum of angles in a triangle equals 180, and we'll calculate angle C in the following way:

$180-90-50=40$

Now let's look at triangle CMD, where we are given two angles.

Let's remember that the sum of angles in a triangle equals 180, and we'll calculate angle CDM in the following way:

$180-90-40=50$

Due to the fact that the lines are parallel, we will begin by drawing a further imaginary parallel line that crosses the 110 angle.

The angle adjacent to the angle 105 is equal to 75 (a straight angle is equal to 180 degrees) This angle is alternate with the angle that was divided using the imaginary line, therefore it is also equal to 75.

In the picture we are shown that the whole angle is equal to 110. Considering that we found only a part of it, we will indicate the second part of the angle as X since it alternates and is equal to the existing X angle.

Therefore we can say that:

$75+x=100$

$x=110-75=35$

First, let's draw another line parallel to the existing lines that will divide the given angle of 120 degrees in the following way:

Note that the line we drew creates two adjacent and straight angles, each equal to 90 degrees.

Now we can calculate the missing part of the angle known to us using the formula:

$120-90=30$

Let's write down the known data as follows:

Note that from the drawing we can see that angle alpha and the angle equal to 30 degrees are alternate angles, therefore they are equal to each other.

$\alpha=30$