Angles and Transversals: Alternate, Corresponding, and Interior Angles

Interior Angles	Exterior Angles
$∠ 2,$ $∠ 4,$ $∠ 6,$ $∠ 8$	$∠ 1,$ $∠ 3,$ $∠ 5,$ $∠ 7$

Interior Angles

Exterior Angles

∠ 2,

∠ 4,

∠ 6,

∠ 8

∠ 1,

∠ 3,

∠ 5,

∠ 7

If	Then
$ℓ_{1} ∥ ℓ_{2}$	$∠ 3 ≅ ∠ 5$ and $∠ 4 ≅ ∠ 6$
$∠ 3 ≅ ∠ 5$ or $∠ 4 ≅ ∠ 6$	$ℓ_{1} ∥ ℓ_{2}$

Then

ℓ_{1} ∥ ℓ_{2}

∠ 3 ≅ ∠ 5

and

∠ 4 ≅ ∠ 6

∠ 3 ≅ ∠ 5

∠ 4 ≅ ∠ 6

ℓ_{1} ∥ ℓ_{2}

If	Then
$ℓ_{1} ∥ ℓ_{2}$	$∠ 1 ≅ ∠ 7$ and $∠ 2 ≅ ∠ 8$
$∠ 1 ≅ ∠ 7$ or $∠ 2 ≅ ∠ 8$	$ℓ_{1} ∥ ℓ_{2}$

Then

ℓ_{1} ∥ ℓ_{2}

∠ 1 ≅ ∠ 7

and

∠ 2 ≅ ∠ 8

∠ 1 ≅ ∠ 7

∠ 2 ≅ ∠ 8

ℓ_{1} ∥ ℓ_{2}

If	Then
$ℓ_{1} ∥ ℓ_{2}$	$m ∠ 3 + m ∠ 6 = 18 0^{\circ}$ and $m ∠ 4 + m ∠ 5 = 18 0^{\circ}$
$m ∠ 3 + m ∠ 6 = 18 0^{\circ}$ or $m ∠ 4 + m ∠ 5 = 18 0^{\circ}$	$ℓ_{1} ∥ ℓ_{2}$

Then

ℓ_{1} ∥ ℓ_{2}

m ∠ 3 + m ∠ 6 = 18 0^{\circ}

and

m ∠ 4 + m ∠ 5 = 18 0^{\circ}

m ∠ 3 + m ∠ 6 = 18 0^{\circ}

m ∠ 4 + m ∠ 5 = 18 0^{\circ}

ℓ_{1} ∥ ℓ_{2}

If	Then
$ℓ_{1} ∥ ℓ_{2}$	$m ∠ 1 + m ∠ 8 = 18 0^{\circ}$ and $m ∠ 2 + m ∠ 7 = 18 0^{\circ}$
$m ∠ 1 + m ∠ 8 = 18 0^{\circ}$ or $m ∠ 2 + m ∠ 7 = 18 0^{\circ}$	$ℓ_{1} ∥ ℓ_{2}$

Then

ℓ_{1} ∥ ℓ_{2}

m ∠ 1 + m ∠ 8 = 18 0^{\circ}

and

m ∠ 2 + m ∠ 7 = 18 0^{\circ}

m ∠ 1 + m ∠ 8 = 18 0^{\circ}

m ∠ 2 + m ∠ 7 = 18 0^{\circ}

ℓ_{1} ∥ ℓ_{2}

Consider what I drew on Zimbabwe's flag. The number of countries in all of Africa equals three times $x .$ How many countries are there in Africa?

Pair	Position Relative to the Vertex	Position Relative to the Transversal
$∠ 1$ and $∠ 5$	Northeast	Right
$∠ 2$ and $∠ 6$	Northwest	Left
$∠ 3$ and $∠ 7$	Southwest	Left
$∠ 4$ and $∠ 8$	Southeast	Right

Pair

Position Relative to the Vertex

Position Relative to the Transversal

∠ 1

and

∠ 5

Northeast

Right

∠ 2

and

∠ 6

Northwest

Left

∠ 3

and

∠ 7

Southwest

Left

∠ 4

and

∠ 8

Southeast

Right

If	Then
$ℓ_{1} ∥ ℓ_{2}$	$∠ 1 ≅ ∠ 5,$ $∠ 2 ≅ ∠ 6,$ $∠ 3 ≅ ∠ 7,$ and $∠ 4 ≅ ∠ 8$
$∠ 1 ≅ ∠ 5,$ $∠ 2 ≅ ∠ 6,$ $∠ 3 ≅ ∠ 7,$ or $∠ 4 ≅ ∠ 8$	$ℓ_{1} ∥ ℓ_{2}$

Then

ℓ_{1} ∥ ℓ_{2}

∠ 1 ≅ ∠ 5,

∠ 2 ≅ ∠ 6,

∠ 3 ≅ ∠ 7,

and

∠ 4 ≅ ∠ 8

∠ 1 ≅ ∠ 5,

∠ 2 ≅ ∠ 6,

∠ 3 ≅ ∠ 7,

∠ 4 ≅ ∠ 8

ℓ_{1} ∥ ℓ_{2}

Angles	Relation
$∠ 7$ and $∠ 4$	Alternate Interior Angles
$∠ 7$ and $∠ 3$	Consecutive Interior Angles
$∠ 7$ and $∠ 2$	Corresponding Angles
$∠ 7$ and $∠ 5$	Vertical Angles

Angles

Relation

∠ 7

and

∠ 4

Alternate Interior Angles

∠ 7

and

∠ 3

Consecutive Interior Angles

∠ 7

and

∠ 2

Corresponding Angles

∠ 7

and

∠ 5

Vertical Angles

Angles	Relation	Relationship Between Measures
$∠ 7$ and $∠ 4$	Alternate Interior Angles	$Congruent$
$∠ 7$ and $∠ 3$	Consecutive Interior Angles	$Supplementary$
$∠ 7$ and $∠ 2$	Corresponding Angles	$Congruent$
$∠ 7$ and $∠ 5$	Vertical Angles	$Congruent$

Angles

Relation

Relationship Between Measures

∠ 7

and

∠ 4

Alternate Interior Angles

Congruent

∠ 7

and

∠ 3

Consecutive Interior Angles

Supplementary

∠ 7

and

∠ 2

Corresponding Angles

Congruent

∠ 7

and

∠ 5

Vertical Angles

Congruent

Angles	Relation	Relationship Between Measures
$∠ 1$ and $∠ 8$	Alternate Exterior Angles	$Congruent$
$∠ 1$ and $∠ 5$	Same-Side Exterior Angles	$Supplementary$
$∠ 1$ and $∠ 6$	Corresponding Angles	$Congruent$
$∠ 1$ and $∠ 3$	Vertical Angles	$Congruent$

Angles

Relation

Relationship Between Measures

∠ 1

and

∠ 8

Alternate Exterior Angles

Congruent

∠ 1

and

∠ 5

Same-Side Exterior Angles

Supplementary

∠ 1

and

∠ 6

Corresponding Angles

Congruent

∠ 1

and

∠ 3

Vertical Angles

Congruent

In geography class, Heichi saw a picture of the pyramids of Giza. He was inspired to design the following scene.

The right-hand sides of the pyramids are parallel to each other. Find the value of

x .

Notice that the line passing through the three bases is a transversal that cuts the three right-hand sides. Let's make a diagram to illustrate it.

The angles with measures 5x^(∘) and 2y^(∘) are on the same position relative to their vertices — both are located northwest. Then, these angles are corresponding angles. The corresponding angles are congruent because the lines l_2 and l_3 are parallel. As a result, the angles have the same measure. 5x^(∘) &= 2y^(∘) &⇓ x &= 2y/5 As the equation shows, we can determine the value of x if we know the value of y.

Finding the Value of y

The angles with measures 4y^(∘) and 2y^(∘) are between lines l_1 and l_2 and on the same side of the transversal. With these characteristics, these two angles are consecutive interior angles. Even more, these angles are supplementary because the lines l_1 and l_2 are parallel. Their measures add up to 180^(∘). 4y^(∘) + 2y^(∘) = 180^(∘) Let's solve the equation for y.

Finding the Value of x

Let's substitute y=30 into the equation we wrote at the beginning. This way we will find the value of x.

Our calculations tell us that the value of x is 12. Lastly, let's update Heichi's diagram.

Consider two parallel lines

ℓ_{1}

and

ℓ_{2}

and a transversal

t .

t

is perpendicular to

ℓ_{1},

which of the following statements are true?

Let's begin by drawing a pair of parallel lines l_1 and l_2. Also, let's draw a transversal t that is perpendicular to l_1.

Corresponding angles are congruent when the lines being cut are parallel. This means that the upper-right exterior angle is also a right angle.

The transversal is also perpendicular to l_2, by definition of perpendicular lines. Statement I is true. I.The transversal is also perpendicular tol_2. ✓ Recall that when two lines are perpendicular, they form four right angles. For this reason, the eight angles in the diagram are right angles.

All right angles have the same measure. This implies that the eight angles are congruent. We conclude that statements III and IV are true. III.& The eight angles formed are congruent. ✓ IV.& The eight angles are right angles. ✓ Finally, notice that each angle has a measure of 90^(∘). As a result, the sum of the measures of any two angles is equal to 180^(∘). This characteristic suggests that any pair of angles are supplementary. Statement II is also true. II.& Of the angles formed, any pair is & supplementary. ✓ All four given statements are true.

Find the value of $x$ in each of the following diagrams.

Let's try to relate ∠ ABC to ∠ JHE somehow. These angles have nothing in common. Let's use the angles formed at angle E to connect them. Notice that AD is parallel to FI and CG is a transversal.

We can see that ∠ BEH and ∠ ABC are corresponding angles. Also, they are congruent because FI and AD are parallel. This implies that m∠ BEH = 60^(∘).

We also see that ∠ BEH and ∠ JHE are same-side interior angles.

Now focus on ∠ BEH and ∠ JHE. They are same-side interior angles formed by CG, JK, and FI. Also, these angles are supplementary because CG ∥ JK. In other words, their measures add up to 180^(∘). We can find the value of x using this information.

Let's relate ∠ JHK and ∠ ABC to determine the value of x. These two angles have nothing in common. Then, let's try to relate them using the angles formed at E. Notice that CD is parallel to FJ and AG is a transversal.

We see that ∠ ABC and ∠ GEF are alternate exterior angles. Since CD and FJ are parallel, these angles are congruent. We can then say that m∠ GEF = 40^(∘).

The diagram shows that ∠ GEF and ∠ JHK are alternate exterior angles. Even more, the angles are congruent because AG and IK are parallel. ∠ JHK &≅ ∠ GEF &⇓ m∠ JHK &= m∠ GEF &⇓ x &= 40 The value of x is 40.

Consider the following diagram.

Two pairs of parallel lines cut by one transversal.

Assume

m ∠ 1 = 7 1^{\circ}

and

m ∠ 2 = 4 1^{\circ} .

What is the measure of

∠ 10 ?

Assume

m ∠ 4 = 7 7^{\circ}

and

m ∠ 5 = 14 0^{\circ} .

Find the measure of

∠ 6 .

Notice that ∠ 10 is formed by the transversal v and the line r. Also line r is parallel to line s. We can determine the measure of ∠ 10 if we know the measure of one of the angles formed by s and v. With this in mind, let's try to find the measure of one of these angles.

The diagram shows that ∠ 10 and ∠ 17 are supplementary. Also, ∠ 17 and the angle formed by ∠ 3 and ∠ 11 are corresponding angles and so, they are congruent.

We can determine the measure of ∠ 17 if we find the measures of ∠ 3 and ∠ 11. Once done, we can find the measure of ∠ 10. Let's write two equations using this information. m∠ 10 + m∠ 17 = 180^(∘) & (I) m∠ 17 = m∠ 3 + m∠ 11 & (II) Our plan became to find the measures of ∠ 3 and ∠ 11. Funny, huh? Notice that ∠ 1 and ∠ 3 are corresponding angles. Also, these angles are congruent because r and s are parallel. m∠ 3=71^(∘) Similarly, ∠ 2 and ∠ 11 are corresponding angles. Even more, these angles are congruent because t∥ u and v is a transversal. m∠ 11=41^(∘) Let's now substitute the measures of ∠ 3 and ∠ 11 into Equation II.

Finally, let's substitute the measure of ∠ 17 into Equation I.

Let's take a look at the angles formed by the transversal s and the parallel lines t and u.

We can see that ∠ 4 can be related to some of the angles formed below, but not directly to ∠ 6. This is because ∠ 6 is formed by the other transversal. We also have that ∠ 3 and ∠ 4 are congruent because t∥ u and the angles are alternate exterior angles. ∠ 3 &≅ ∠ 4 &⇓ m∠ 3 &= 77^(∘) Let's study the angles formed by the transversal v and the parallel lines t and u.

The diagram shows that ∠ 5 and the angle formed by ∠ 3 and ∠ 6 are corresponding angles. Therefore, they are congruent. This means that the measure of ∠ 5 is equal to the sum of the measures of ∠ 3 and ∠ 6. m∠ 5 = m∠ 3 + m∠ 6 We can determine the measure of ∠ 6 by solving this equation. Remember that we were given the measure of ∠ 5 and we already found the measure of ∠ 3.

Consider the following diagram.

Use the diagram's characteristics to complete the following sentences.

$∠ 5$ and are corresponding angles.

Select all the options that apply!

$∠ 3$ and $∠ 13$ are $.$

$∠ 10$ and are alternate interior angles.

Select all the options that apply!

Our task is to identify those angles that are corresponding to ∠ 5. With this in mind, let's recall the definition of corresponding angles.

Corresponding Angles |- Consider a pair of lines cut by a transversal. The pairs of angles with different vertices that lie on the same position relative to the vertex are called corresponding angles.

Let's consider only one pair of parallel lines and a transversal at a time. We can start with lines r and s and the transversal m.

Notice that ∠ 4 and ∠ 5 both lie northeast of their vertices. This means they are corresponding angles. Let's now focus on the parallel lines m and n and the transversal s.

We see that ∠ 15 is northeast of the vertex, as is ∠ 5. Then, ∠ 5 and ∠ 15 are also corresponding angles. & ∠ 5and ∠ 4are corresponding angles. & ∠ 5and ∠ 15 are corresponding angles. Be aware that none of the angles formed by lines n and r can be corresponding to ∠ 5. This is because they do not have a line that is common to ∠ 5.

The first thing we note is that ∠ 3 and ∠ 13 are not formed by the same transversal. As a result, this pair of angles cannot be corresponding, nor alternate interior or exterior, nor same-side interior or exterior angles.

However, we can still relate them. Notice that ∠ 3 and ∠ 9 are corresponding angles — both are southeast of their vertices. Then, these angles are congruent. Also, the diagrams shows that ∠ 9 and ∠ 13 are same-side interior angles.

Both ∠ 9 and ∠ 13 are formed by two parallel lines and a transversal. This implies that these angles are supplementary and so, their measures add up to 180^(∘). m∠ 9 + m∠ 13 = 180^(∘) Remember that ∠ 3 ≅ ∠ 9 and so, they have the same measure. For this reason, we can substitute m∠ 3 for m∠ 9 into the equation. m∠ 3 + m∠ 13 = 180^(∘) The last equation implies that ∠ 3 and ∠ 13 are supplementary angles. This is the relation we were looking for!

This time we are asked to identify the angles that are alternate interior with ∠ 10. Then, let's recall that definition first.

Alternate Interior Angles |- Consider a pair of lines cut by a transversal. The pairs of interior angles with different vertices that lie on opposite sides of the transversal are called alternate interior angles.

With the definition in mind, let's consider the parallel lines m and n and the transversal r.

Notice that ∠ 2 is an interior angle whose vertex is different from the vertex of ∠ 10. Additionally, ∠ 2 and ∠ 10 are on opposite sides of the transversal. This implies that ∠ 10 and ∠ 2 are alternate interior angles. Let's move on and focus on the parallel lines r and s and the transversal n.

As before, we see that ∠ 10 and ∠ 13 are on opposite sides of the transversal, have different vertices, and are interior angles. All these characteristics make them alternate interior angles. & ∠ 10and ∠ 2are alternate interior angles. & ∠ 10and ∠ 13 are alternate interior angles. None of the angles formed by lines m and s can be alternate interior with ∠ 10. This is because they do not have a line in common to the lines forming ∠ 10.

	16 Theory slides
	13 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Angles and Transversals

Catch-Up and Review

Hint

Solution

Extra

Hint

Solution

Hint

Solution

Extra

Hint

Solution

Hint

Solution

Extra

Finding the Value of y

Finding the Value of x

Angles and Transversals

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