Angles - Examples, Exercises and Solutions

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 $100^\circ$.

Let's review the definition of adjacent angles:

Adjacent angles are angles formed where there are two straight lines that intersect. These angles are formed at the point where the intersection occurs, one next to the other, and hence their name.

Now let's review the definition of collateral angles:

Two angles formed when two or more parallel lines are intersected by a third line. The collateral angles are on the same side of the intersecting line and even are at different heights in relation to the parallel line to which they are adjacent.

Therefore, answer C is correct for this definition.

Let's remember the definition of consecutive angles:

Consecutive angles are, in fact, a pair of angles that can be found on the same side of a straight line when this line crosses a pair of parallel straight lines.

These angles are on opposite levels with respect to the parallel line to which they belong.

The sum of a pair of angles on one side is one hundred eighty degrees.

Therefore, since line a is parallel to line b and according to the previous definition: the angles$\beta+\gamma=180$

are consecutive.

Given that line a is parallel to line b, the angles$\alpha_2,\beta_1$ are equal according to the definition of corresponding angles.

Also, the angles$\alpha_1,\gamma_1$are equal according to the definition of corresponding angles.

Now let's remember the definition of collateral angles:

Collateral angles are actually a pair of angles that can be found on the same side of a line when it crosses a pair of parallel lines.

These angles are on opposite levels with respect to the parallel line they belong to.

The sum of a pair of angles on one side is one hundred eighty degrees.

Therefore, since line a is parallel to line b and according to the previous definition: the angles

γ1​+γ2​=180

are the collateral angles

Given that according to the definition, the vertex angles are equal to each other, it can be argued that:

$115=2$Now we can calculate the second pair of vertex angles in the same circle:

$1=3$

Since the sum of a plane angle is 180 degrees, angle 1 and angle 3 are complementary to 180 degrees and equal to 65 degrees.

We now notice that between the parallel lines there are corresponding and equal angles, and they are:

$115=4$

Since angle 4 is opposite to angle 6, it is equal to it and also equal to 65 degrees.

Another pair of alternate angles are angle 1 and angle 5.

We have proven that:$1=3=65$

Therefore, angle 5 is also equal to 65 degrees.

Since angle 7 is opposite to angle 5, it is equal to it and also equal to 115 degrees.

That is:

$115=2=4=6$

$65=1=3=5=7$

In the question, we can observe that there are two pairs of parallel lines, lines a and b.

When a line crosses two parallel lines, different angles are formed

Angles alpha and the given angle of 104 are on different sides of the transversal line, but both are in the interior region between the two parallel lines,

This means they are alternate angles, and alternate angles are equal.

Therefore, 

Angle beta and the second given angle of 81 degrees are both on the same side of the transversal line, but each is in a different position relative to the parallel lines, one in the exterior region and one in the interior. Therefore, we can conclude that these are corresponding angles, and corresponding angles are equal.

Therefore,

Remember that adjacent angles are angles that are formed when two lines intersect one another.

These angles are created at the point of intersection, one adjacent to the other, and that's where their name comes from.

Adjacent angles always complement one another to one hundred and eighty degrees, meaning their sum is 180 degrees. 

Let's remember that vertical angles are angles that are formed when two lines intersect. They are are created at the point of intersection and are opposite each other.

Alternate angles are a pair of angles that can be found on the opposite side of a line that cuts two parallel lines.

Furthermore, these angles are located on the opposite level of the corresponding line that they belong to.

First let's remember that corresponding angles can be defined as a pair of angles that can be found on the same side of a transversal line that intersects two parallel lines.

Additionally, these angles are positioned at the same level relative to the parallel line to which they belong.

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.

Given that line a is parallel to line b, let us remind ourselves of the definition of corresponding angles between parallel lines:

Corresponding angles are angles located on the same side of the line that intersects the two parallels and are also situated at the same level with respect to the parallel line to which they are adjacent.

Corresponding angles are equal in size.

According to this definition $\alpha=\beta$and as such they are the corresponding angles.

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

Since we are given two angles, we can calculate $a$

$94+92=186$

We should note that the sum of the two given angles is greater than 180 degrees.

Therefore, there is no solution possible.

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.

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.