# Alternate angles

🏆Practice angles in parallel lines

Alternate angles are on opposite sides of the transversal that intersects two parallel lines and are not on the same side of the parallel lines to which they belong. Alternate angles are equal.

The following sketch illustrates two pairs of alternate angles, one is painted red and the other blue.

## Test yourself on angles in parallel lines!

If one of two corresponding angles is a right angle, then the other angle will also be a right angle.

## What are alternate angles?

Before seeing a formal definition of alternate angles, it is helpful to understand the circumstances under which these angles can be formed. The simplest way to describe it is with a drawing of two parallel lines with a transversal that intersects them (for more details, see the article that deals with the topic of "Parallel lines "), as can be seen in this illustration:

As mentioned, here we see two parallel lines $A$ and $B$ with a transversal $C$ intersecting them.

## Other angles in brief

There are other types of angles that are formed under the circumstances just described. We will briefly review them:

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### Corresponding angles

Corresponding angles are on the same side of the transversal that cuts two parallel lines and are on the same side of their respective parallel lines. The corresponding angles are equal. For more details refer to the article that deals with the subject of "corresponding angles".

### Vertically opposite angles

Vertically opposite angles are formed by two straight lines that intersect, have a vertex in common and are opposite each other. Opposite angles are equal. For more details see the specific article that deals with the subject of "angles opposite at the vertex".

Do you know what the answer is?

### Collateral angles

Collateral angles are on the same side of the transversal that intersects two parallel straight lines and are not at the same side of their respective parallel lines. Together they equal $180°$, that is, the sum of two collateral angles is equal to one hundred and eighty degrees. For more details go to the specific article that deals with "collateral angles".

## Examples and exercises

### Exercise 1

In each of the following illustrations, indicate if the angles are alternate angles or not. In both cases explain why.

Solution:

Diagram No 1: In this case we are dealing with alternate angles since they meet the criteria, that is, they are two angles that are on opposite sides of the transversal that cuts the two parallel lines and the angles are not on the same side of their respective parallel lines.

Diagram No 2: In this case we are not dealing with alternate angles since they do not meet the criteria, i.e., we are dealing with two angles that are on the same side of the transversal that cuts the two parallel lines and are also on the same side of their respective parallel lines.

Diagram No 3: In this case we are not dealing with alternate angles since they do not meet the criteria, i.e., we are dealing with two angles that lie on the same side of the transversal that cuts the two parallel lines and the two angles are not on the same side of their respective parallel lines.

Diagram No 1: alternate angles.

DiagramNo 2: not alternate angles.

Diagram No 3: not alternate angles.

### Exercise 2

Given the triangle $ABC$ as illustrated in the diagram.

The angle $B$ of the triangle equals $35°$ degrees.

Also, we know that the line $AK$ and the edge (the side) $BC$ are parallel.

Find the other angles of the triangle $ABC$.

Solution:

We will look at the illustration and see that, in fact, we have two parallel lines ($AK$ and $BC$) that are cut by a transversal (the edge $AC$). The angle $C$ of the triangle is equal to the angle $CAK$ since they are alternate angles, that is, they are two angles located on opposite sides of the transversal ($AC$) that cuts the two parallel lines ($AK$ and $BC$) and these angles are not on the same side of their respective parallel lines.

It follows that the angle $C$ of the triangle is equal to $60°$ degrees.

The sum of the three angles of any triangle equals $180°$ degrees.

Therefore, the angle $A$ equals $180°-35°-60°=85°$degrees.

The angle $A$ measures $85°$ degrees.

Angle $C$ measures $60°$ degrees.

### Exercise 3

In the following diagram given:

In this illustration two parallel lines and a transversal line cutting them are given.

The angle $K$ is to be found based on the data given in the sketch.

Solution:

According to the given information we can see that the two angles indicated in the diagram are alternate angles. That is, they are two angles located on opposite sides of the transversal ($C$) that cuts the two parallel lines ($A$ and $B$) and these angles are not on the same side of their respective parallel lines.

The alternate angles are equal, therefore, the angle $K$ also measures $75°$ degrees.

Angle K measures $75°$ degrees.

Do you think you will be able to solve it?

### Exercise 4

a and b are parallel

Find the marked angles

Solution:

$∡a=∡104°$ Alternate angles

$∡β=∡81°\text{ }$ Corresponding angles

Exercise 5:

$a,b$ and $c$ are parallel

Find the value of $X$.

Solution:

Ilustración

$∡a_1∡b_3=38°$ Supplementary angles, therefore equal.

Complementary therefore equal to: $∡c_1=∡b_1=25°$

$∡b_1+∡b_2+b_3=180°$ (Adjacent angles)

$25°+3X+38°=180°$

$3X=117°$ /:3

$X=39°$

$X=39°$

### Exercise 6

Given the rectangle, find the value of $X$.

Solution:

The quadrilateral is a rectangle, therefore:

$AB$ is parallel to $CD$

$AD$ is parallel to $BC$

$∡4,∡3$

Supplementary angles are equal, therefore:

$∡3=∡4=84°$

$∡2,∡5$ are opposite angles , therefore equal:

$∡2=∡5=26°$

$∡6=∡4-∡5=84°-26°=58°$

$∡7=∡1=X$ are opposite angles, therefore equal $∡7, ∡1$

$∡B=90°$ is a right angle.

$∡B=90°+∡7+∡6=180°$ Sum of the angles in the triangle.

$90°+X+58°=180°$

$X=190-58-90=32$

$X=32°$

### Exercise 7

Given the parallelogram

The marked angles are acute angles.

For what values of $X$ is there a solution?

Solution:

The given polygon is a parallelogram, so all opposite sides are parallel.

The angles in the figure are supplementary and therefore equal.

• Acute angles

$0<4X²+3X<90°$

$0<5X-42°<90°$

$5X-42°=4X²+3X$

$0=4X²-2X+42$

$0=2X²-X+21$

$0=2X²-X+21$

$X_{1,2}=\frac{-b±\sqrt{b²-4\cdot a\cdot c}}{2\cdot a}$

$a=2$

$b=-1$

$c=21$

$X_{1,2}=\frac{1±\sqrt{1-4\cdot2\cdot21}}{2\cdot2}=\frac{1±\sqrt{-167}}{4}=$

The discriminant is negative, so there is no solution.

If you are interested in learning more about angles, you can access one of the following articles:

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## Review questions

What are external alternate angles?

They are angles that, as their name implies, are alternate on the outside of the two parallel lines and are equal, as illustrated in the following:

$\sphericalangle a=\sphericalangle h$ (alternate external angles)

$\sphericalangle b=\sphericalangle g$ (alternate external angles)

What are internal alternate angles?

They are the angles that are in the internal part of two parallel lines cut by a transversal, but in an alternate way. These angles are equal. Let's see in the following:

$\sphericalangle c=\sphericalangle f$ (internal alternate angles)

$\sphericalangle d=\sphericalangle e$ (internal alternate angles)

How to calculate internal alternate angles?

The internal alternate angles are equal, let's see an example of how to calculate these angles:

Let the lines $A$ and $B$ be parallel. Calculate the value of the angle $\sphericalangle5$ in the following drawing:

We know that the $\sphericalangle3=120^o$, therefore the $\sphericalangle6$ also measures the same because they are internal alternate angles, then.

$\sphericalangle6=120^o$

And we can see, $\sphericalangle5$ and $\sphericalangle6$ are supplementary angles, therefore added together they should give us $180^o$

Therefore $\sphericalangle5=180^o-\sphericalangle6$

$\sphericalangle5=180^o-120^o$

$\sphericalangle5=60^o$

$\sphericalangle5=60^o$

How do we calculate external alternate angles?

Remember that the external alternate angles are equal. Let's see an example:

We are given that the straight lines $A\parallel B$ Calculate the value of the angle $\sphericalangle8$, given that $\sphericalangle1=75^o$

Since the angles $\sphericalangle1$ and $\sphericalangle8$, are external alternate angles, then by definition we know that they are equal, therefore:

$\sphericalangle1=\sphericalangle8$

$\sphericalangle8=75^o$

$\sphericalangle8=75^o$

Do you know what the answer is?

## examples with solutions for alternate angles

### Exercise #1

Which type of angles are shown in the figure below?

### Step-by-Step Solution

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.

Alternate

### Exercise #2

$a$ is parallel to

$b$

Determine which of the statements is correct.

### Step-by-Step Solution

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.

$\beta,\gamma$ Colaterales$\gamma,\delta$ Adjacent

### Exercise #3

Is it possible to have two adjacent angles, one of which is obtuse and the other right?

### Step-by-Step Solution

Remember the definition of adjacent angles:

Adjacent angles always complement each other up to one hundred eighty degrees, that is, their sum is 180 degrees.

This situation is impossible since a right angle equals 90 degrees, an obtuse angle is greater than 90 degrees.

Therefore, together their sum will be greater than 180 degrees.

No

### Exercise #4

In which of the diagrams are the angles $\alpha,\beta\text{ }$ vertically opposite?

### Step-by-Step Solution

Remember the definition of angles opposite by the vertex:

Angles opposite by the vertex are angles whose formation is possible when two lines cross, and they are formed at the point of intersection, one facing the other. The acute angles are equal in size.

The drawing in answer A corresponds to this definition.

### Exercise #5

The lines a and b are parallel.

What are the corresponding angles?

### Step-by-Step Solution

Given that line a is parallel to line b, let's remember 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 therefore the corresponding angles

$\alpha,\beta$