Sum of Angles in a Polygon - Examples, Exercises and Solutions

Remember that an acute angle is smaller than 90 degrees, an obtuse angle is larger than 90 degrees, and a straight angle equals 180 degrees.

Since the lines are perpendicular to each other, the marked angles are right angles each equal to 90 degrees.

To find the size of the missing angle, we will use the property that the sum of angles on a straight line is $180^\circ$. Given that one angle is $80^\circ$, we can calculate the missing angle using the following steps:

• Step 1: Recognize that the given angle $\alpha = 80^\circ$ and the missing angle $\beta$ form a straight line.
• Step 2: Use the angle sum property for a straight line: $\alpha + \beta = 180^\circ$
• Step 3: Substitute the known value: $80^\circ + \beta = 180^\circ$
• Step 4: Solve for the missing angle $\beta$: $\beta = 180^\circ - 80^\circ = 100^\circ$

Therefore, the size of the missing angle is $100^\circ$.

Note that in drawing B, the two lines form a right angle, which is an angle of 90 degrees:

While the angle in drawing A is greater than 90 degrees:

Therefore, the angle in drawing A is larger.

Answer B is correct because the more closed the angle is, the more acute it is (less than 90 degrees), meaning it's smaller.

The more open the angle is, the more obtuse it is (greater than 90 degrees), meaning it's larger.

The angle in diagram (a) is more acute, meaning it is smaller:

Conversely, the angle in diagram (b) is more obtuse, making it larger.

Note that in drawing A, the angle is a straight angle equal to 180 degrees:

While in drawing B, we are given a right angle, equal to 90 degrees:

Therefore, the angle in drawing A is larger.

In drawing A, we can see that the angle is more closed:

While in drawing B, the angle is more open:

In other words, in diagram (a) the angle is more acute, while in diagram (b) the angle is more obtuse.

Remember that the more obtuse an angle is, the larger it is.

Therefore, the larger of the two angles appears in diagram (b).

In drawing A, we can see that the angle is an obtuse angle, meaning it is larger than 90 degrees:

While in drawing B, the angle is a right angle, meaning it equals 90 degrees:

Therefore, the larger angle appears in drawing A.

We can see from the diagram that angle DBC equals 100 degrees.

We can also see that the size of angle ABC is shown and equals 40 degrees.

Therefore, the answer is 40.

In order to calculate the value of angle ABC, we must calculate the sum of all the given angles.

That is:

$ABC=60+50$

$ABC=110$

The unmarked angle is adjacent to an angle of 160 degrees.

Remember: the sum of adjacent angles is 180 degrees.

Therefore, the size of the unknown angle is:

$180-160=20$

Remember that the sum of angles in a triangle is equal to 180.

Therefore, we will use the formula:

$A+B+C=180$

Let's input the known data:

$100+\alpha+90=180$

$190+\alpha=180$

$\alpha=180-190$

We should note that it's not possible to get a negative result, and therefore there is no solution.

Note that the sum of the angles in a triangle is equal to 180 degrees.

Therefore, we can use the formula:

$A+B+C=180$

Then we will substitute in the known data:

$\alpha+27.7+41=180$

$\alpha+68.7=180$

Finally, we will move the variable to the other side while maintaining the appropriate sign:

$\alpha=180-68.7$

$\alpha=111.3$

Let's remember that the sum of angles in a triangle is equal to 180 degrees.

Therefore, we will use the following formula:

$A+B+C=180$

Now let's input the known data:

$\alpha+69+23=180$

$\alpha+92=180$

We'll move the term to the other side and keep the appropriate sign:

$\alpha=180-92$

$\alpha=88$

Remember that the sum of angles in a triangle is equal to 180 degrees.

Therefore, we will use the following formula:

$A+B+C=180$

Now let's input the known data:

$80+55+\alpha=180$

$135+\alpha=180$

We'll move the term to the other side and keep the appropriate sign:

$\alpha=180-135$

$\alpha=45$