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.

two pairs of alternate angles, one is painted red and the other blue.

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Test yourself on angles in parallel lines!

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

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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:

A2 - Parallel lines

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

C -   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".

angles opposite 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° 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".

Example of 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.

Alter_angles_-_Exercise_1.1.original

Alterior_angles_-_Exercise_1.2.original

Alter_angles_-_Exercise_1.3.original

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.


Check your understanding

Exercise 2

Alter_angles_-_Exercise_2.original

Given the triangle ABC ABC as illustrated in the diagram.

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

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

Find the other angles of the triangle ABC ABC .

Solution:

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

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

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

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

Answer:

The angle A A measures 85° 85° degrees.

Angle C C measures 60° 60° degrees.


Exercise 3

In the following diagram given:

Alter_angles_-_Exercise_3.original

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

The angle K 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 C ) that cuts the two parallel lines (A A and B B ) and these angles are not on the same side of their respective parallel lines.

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

Answer:

Angle K measures 75° 75° degrees.


Do you think you will be able to solve it?

Exercise 4

a and b are parallel Find the marked angles

a and b are parallel

Find the marked angles

Solution:

a=104° ∡a=∡104° Alternate angles

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


Exercise 5:

a,b a,b and c c are parallel

Find the value of X X .

Exercise 5 a, b and c are parallel Find the value of X.

Solution:

Ilustración

a1b3=38° ∡a_1∡b_3=38° Supplementary angles, therefore equal.

Complementary therefore equal to: c1=b1=25° ∡c_1=∡b_1=25°

b1+b2+b3=180° ∡b_1+∡b_2+b_3=180° (Adjacent angles)

25°+3X+38°=180° 25°+3X+38°=180°

3X=117° 3X=117° /:3

X=39° X=39°

Answer:

X=39° X=39°


Exercise 6

Given the rectangle, find the value of X X .

Exercise 6 Given the rectangle, find the value of X

Solution:

The quadrilateral is a rectangle, therefore:

AB AB is parallel to CD CD

AD AD is parallel to BC BC

4,3 ∡4,∡3

Supplementary angles are equal, therefore:

3=4=84° ∡3=∡4=84°

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

2=5=26° ∡2=∡5=26°

6=45=84°26°=58° ∡6=∡4-∡5=84°-26°=58°

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

B=90° ∡B=90° is a right angle.

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

90°+X+58°=180° 90°+X+58°=180°

X=1905890=32 X=190-58-90=32

Answer:

X=32° X=32°


Test your knowledge

Exercise 7

Given the parallelogram

Exercise 7 Given the parallelogram

The marked angles are acute angles.

For what values of X 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<4X²+3X<90°

0<5X42°<90° 0<5X-42°<90°

5X42°=4X2+3X 5X-42°=4X²+3X

0=4X22X+42 0=4X²-2X+42

0=2X2X+21 0=2X²-X+21

0=2X2X+21 0=2X²-X+21

X1,2=b±b24ac2a X_{1,2}=\frac{-b±\sqrt{b²-4\cdot a\cdot c}}{2\cdot a}

a=2 a=2

b=1 b=-1

c=21 c=21

X1,2=1±1422122=1±1674=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.


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:

What are external alternate angles

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

b=g \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:

What is an internal alternating angle

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

d=e \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 A and B B be parallel. Calculate the value of the angle 5 \sphericalangle5 in the following drawing:

How to get the measure of internal alternate angles

We know that the 3=120o \sphericalangle3=120^o , therefore the 6 \sphericalangle6 also measures the same because they are internal alternate angles, then.

6=120o \sphericalangle6=120^o

And we can see, 5 \sphericalangle5 and 6 \sphericalangle6 are supplementary angles, therefore added together they should give us 180o 180^o

Therefore 5=180o6 \sphericalangle5=180^o-\sphericalangle6

5=180o120o \sphericalangle5=180^o-120^o

5=60o \sphericalangle5=60^o

Answer:

5=60o \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 AB A\parallel B Calculate the value of the angle 8 \sphericalangle8 , given that 1=75o \sphericalangle1=75^o

How to take the measure of an external alternating angle

Since the angles 1 \sphericalangle1 and 8 \sphericalangle8 , are external alternate angles, then by definition we know that they are equal, therefore:

1=8 \sphericalangle1=\sphericalangle8

8=75o \sphericalangle8=75^o

Answer:

8=75o \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.

Answer

Alternate

Exercise #2

a a is parallel to

b b

Determine which of the statements is correct.

αααβββγγγδδδaaabbb

Video Solution

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.

Answer

β,γ \beta,\gamma Colateralesγ,δ \gamma,\delta Adjacent

Exercise #3

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

Video Solution

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.

Answer

No

Exercise #4

In which of the diagrams are the angles α,β  \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.

Answer

αααβββ

Exercise #5

The lines a and b are parallel.

What are the corresponding angles?

αααβββγγγδδδaaabbb

Video Solution

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 α=β \alpha=\beta and therefore the corresponding angles

Answer

α,β \alpha,\beta

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