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.

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

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 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".

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".

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.

Answer:

Diagram No 1: alternate angles.

DiagramNo 2: not alternate angles.

Diagram No 3: not alternate angles.

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.

Answer:

The angle $A$ measures $85°$ degrees.

Angle $C$ measures $60°$ degrees.

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.

Answer:

Angle K measures $75°$ degrees.

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

$X=39°$

Answer:

$X=39°$

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.

$90°+X+58°=180°$

$X=190-58-90=32$

Answer:

$X=32°$

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:

• Angle notation
• Sides, vertices, and angles
• Right angles
• Acute angle
• Obtuse angle
• Plane angles
• Adjacent angles
• Angles opposite at the vertex
• Corresponding angles
• Sum of the angles of a triangle
• Sum of the angles of a polygon
• Addition and difference of angles

In Tutorela you will find a variety of articles about mathematics.

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 $\sphericalangle5=180^o-\sphericalangle6$

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

$\sphericalangle5=60^o$

Answer:

$\sphericalangle5=60^o$

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:

$\sphericalangle1=\sphericalangle8$

$\sphericalangle8=75^o$

Answer:

$\sphericalangle8=75^o$