Understanding Alternate Interior Angles Theorem and Theorems About Lines and Angles

In this lesson, some theorems about lines and angles will be explored and proven. The theorems will be applied using real-life examples.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Parallel lines
Perpendicular lines
Vertical angles
Supplementary angles
Rigid motions
Corresponding angles
Alternate interior angles
Perpendicular bisector

Explore

Investigating Properties of Vertical Angles

Before any theorems will be introduced, try to discover some properties of angles using the interactive applet. While exploring, think about how those properties could be proven. The applet allows for translations and rotations of the angles. Consider a pair of vertical angles.

Is it possible to map one angle onto another by performing a transformation?
What conjecture can be made about any two vertical angles?

Extra

How to Use the Applet

To translate an angle, click on the sides of the angle or in the region between the sides.
To rotate an angle about move the corresponding slider.

Discussion

Congruence of Vertical Angles

Vertical angles can be mapped onto each other by using a rotation. Since rotations are rigid motions, the angle measures are preserved. This leads to the following theorem.

Rule

Vertical Angles Theorem

Vertical angles are always congruent.

Based on the characteristics of the diagram, the following relations hold true.

Proof

Geometric Approach

Analyzing the diagram, it can be seen that and form a straight angle, so these are supplementary angles. Similarly, and are also supplementary angles.

Therefore, by the Angle Addition Postulate, the sum of and is and the sum of and is also These facts can be used to express in terms of and in terms of

Angle Addition Postulate	Isolate

By the Transitive Property of Equality, the expressions representing

can be set equal to each other.

Then the equation can be simplified.

SubEqn

MultEqn

By the definition of congruent angles, this means that the vertical angles

and

are congruent angles. The same process can be used to prove

and

congruent.

Two-Column Proof

The previous proof can be summarized in the following two-column table.

Statements	Reasons
and lines	Given
and supplementary	Definition of straight angle
	Definition of supplementary angles
	Subtraction Property of Equality
and supplementary	Definition of straight angle
	Definition of supplementary angles
	Subtraction Property of Equality
	Transitive Property of Equality
	Subtraction and Multiplication Properties of Equality

Proof

Using Transformations

Consider the points and on each ray that starts at the point of intersection of the two lines.

Suppose that points

and

are rotated

about point

The points

and

are mapped onto the points

and

after the rotation. This means that

is mapped onto

Since rotations are a rigid motion,

and

are congruent angles.

Since the point

lies on

and point

lies on

is congruent to

By applying the Transitive Property of Congruence, it can be confirmed that

is congruent to

Example

Solving Problems Using Vertical Angles

In Flowerland Village, there is a crossroad between Tulip Street and Rose Street. There is a plan to continue the construction of Tulip Street toward the southwest. At the moment, the crossroad forms two angles, whose measures are expressed by

and

respectively.

A crossroad before and after the construction

Find the measures of all four angles the crossroad will form after the construction of Tulip Street is finished.

Hint

Use the given expressions to form an equation for Identify the relationship between and as well as and by analyzing their positions.

Solution

The angles formed by the crossroad before the construction form a linear pair. Therefore, they are supplementary angles. Using this information, the following equation can be formed.

By solving it, the value of

can be determined.

▼

Solve for

AddSubTerms

Add and subtract terms

SubEqn

DivEqn

Now that the value of

is known, the measures of

and

can be calculated.

Next, by analyzing the position of

and

as well as

and

it can be noted that these are vertical angles. Therefore, by the Vertical Angles Theorem, they are two pairs of congruent angles.

In this way, it was obtained that

and

are each

and

are each

Explore

Investigating Parallel Lines Cut by a Transversal

Consider two parallel lines cut by a transversal. The applet shows a pair of corresponding angles, and Is it possible to translate one line so that one of these angles maps onto another?

What conjecture can be made about any two corresponding angles?

Discussion

Corresponding Angles Theorem and Its Converse

The observed relation between corresponding angles is presented and proven in the following theorem.

Rule

Corresponding Angles Theorem

If parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

Two parallel lines intersected by a transversal forming four pairs of corresponding angles

Based on the characteristics of the diagram, the following relations hold true.

If then and

Proof

It is given that

and

are parallel lines. Therefore, by the definition of parallel lines, there is a translation that maps one line onto another.

It should be noted that when translating one line onto another, the pairs of corresponding angles overlap and seem to have the same measures. In fact, because a translation is a rigid motion, and a rigid motion preserves length and angle measures, the pairs of corresponding angles are congruent angles.

Note that the converse statement is also true.

Rule

Converse Corresponding Angles Theorem

If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel.

Based on the characteristics of the diagram, the following relation holds true.

If or then

Proof

This theorem can be proven by an indirect proof. Let and be two lines intersected by a transversal line forming corresponding congruent angles and

Since the goal is to prove that

is parallel to

, it will be temporarily assumed that

and

are not parallel.

By the Parallel Postulate, there exists a line

parallel to

that passes through the point of intersection between

and

This line forms

and

By the Angle Addition Postulate,

is equal to the sum of

and

Since

and

are parallel lines that are cut by a transversal, by the Corresponding Angles Theorem,

and

are congruent. By the definition of congruence, these angles have the same measure.

By the Substitution Property of Equality,

can be substituted for

into the equation for

From the above equation and since

is a positive number, it can be concluded that

is greater than

This contradicts the given fact that

and

are congruent. The contradiction came from assuming that

and

are not parallel lines. Therefore,

and

must be parallel lines.

Example

Solving Problems Using Corresponding Angles

In a Flowerland Village house, there are stairs with hand railings like shown in the diagram. The measures of and are expressed as and respectively.

What are the measures of

and

Hint

How do measures of and relate to each other? Use the given expressions to form an equation for

Solution

Analyzing the diagram, it can be noted that

and

are corresponding angles formed by two parallel lines and a transversal. Therefore, by the Corresponding Angles Theorem, these angles are congruent. Hence, the measures of these angles are the same.

By substituting

with

and

with

the equation for

can be formed.

SubstituteII

▼

Solve for

SubEqn

AddEqn

Now that the value of

is known, the measure of each of the angles can be calculated.

▼

Substitute

for

and evaluate

Substitute

Multiply

SubTerm

Subtract term

Since the angles are congruent, it can be concluded that they both measure to be

Discussion

Alternate Interior Angles Theorem and Its Converse

Like corresponding angles, alternate interior angles are also formed by two parallel lines cut by a transversal.

Rule

Alternate Interior Angles Theorem

If parallel lines are cut by a transversal, then alternate interior angles are congruent.

Two parallel lines cut by a transversal forming two pairs of congruent angles

Based on the characteristics of the diagram, the following relations hold true.

If then and

Proof

Geometric approach

To prove that alternate interior angles are congruent, it will be shown that and are congruent.

Notice that by definition

and

are vertical angles. By the Vertical Angles Theorem, they are therefore congruent angles.

Furthermore, by definition

and

are corresponding angles. Hence, by the Corresponding Angles Theorem,

and

are also congruent angles.

Applying the Transitive Property of Congruence,

and

can be concluded to be congruent angles as well.

The same reasoning applies to the other pair of alternate interior angles. Therefore, when a pair of parallel lines is cut by a transversal, the pairs of alternate interior angles are congruent.

Two-Column Proof

The previous proof can be summarized in the following two-column table.

Statements	Reasons
and are vertical angles	Def. of vertical angles
	Vertical Angles Theorem
and are corresponding angles	Def. of corresponding angles
	Corresponding Angles Theorem
	Transitive Property of Congruence

Proof

Using Transformations

Apart from the points of intersection, consider two more points on each line.

Next,

and

will be translated parallel to the transversal until the points

and

lie on

Then,

and

will be rotated

about

It should be noted that since point

lies on the transversal, when translating it to

the point will fall into the same position as

Therefore,

will not be affected by the rotation around

After this combination of rigid motions,

and

are mapped onto

and

This means that

is mapped onto

Therefore,

and

are congruent angles.

Note that

and

share the same location. It can also be seen that

lies on

and

lies on

Because of this,

is congruent to

Applying the Transitive Property of Congruence,

is congruent to

The converse statement is also true.

Rule

Converse Alternate Interior Angles Theorem

If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel.

Based on the characteristics of the diagram, the following relation holds true.

If or then

Proof

The proof will be based on the given diagram, but it holds true for any pair of lines cut by a transversal. Consider only one pair of congruent alternate interior angles and one more angle.

It needs to be proven that

and

are parallel lines. It is already given that

is congruent to

The diagram shows that

and

are vertical angles. By the Vertical Angles Theorem, these angles are congruent.

Notice the common angle of

in both relationships. By the Transitive Property of Congruence, since

is congruent to

and

is congruent to

then

is congruent

The diagram also shows that

and

are corresponding angles. Given that relation, the Converse Corresponding Angles Theorem can be applied.

Converse Corresponding Angles Theorem
If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel.

Since and are corresponding congruent angles, then and are parallel lines. To summarize, all of the steps will be described in a two-column proof.

Statement	Reason
	Given
	Vertical Angles Theorem
	Transitive Property of Congruence
	Converse Corresponding Angles Theorem

Discussion

Alternate Exterior Angles Theorem and Its Converse

Similar properties can be discovered for alternate exterior angles.

Rule

Alternate Exterior Angles Theorem

If parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.

Based on the characteristics of the diagram, the following relations hold true.

If then and

Proof

Geometric Approach

In order to prove that alternate exterior angles are congruent, it will be shown that and are congruent.

Notice that by definition,

and

are corresponding angles. Therefore, by the Corresponding Angles Theorem, they are congruent angles.

Furthermore, by definition,

and

are vertical angles. Therefore, by the Vertical Angles Theorem,

and

are congruent angles.

Then, by applying the Transitive Property of Congruence,

and

can be concluded to be congruent angles as well.

The same reasoning applies to the other pair of alternate exterior angles. Therefore, when a pair of parallel lines is cut by a transversal, the pairs of alternate exterior angles are congruent.

Two-Column Proof

The previous proof can be summarized in the following two-column table.

Statements	Reasons
and are corresponding angles	Def. of corresponding angles
	Corresponding Angles Theorem
and are vertical angles	Def. of vertical angles
	Vertical Angles Theorem
	Transitive Property of Congruence

Proof

Using Transformations

Consider the points of intersection as well as two more points on each line.

Next,

and

will be translated in the direction of the transversal so that points

and

lie on

Then,

and

will be rotated

about

After this combination of rigid motions,

and

are mapped onto

and

This means that

is mapped onto

Therefore,

and

are congruent angles.

Since

and

share the same location,

lies on

and

lies on

Because of this,

is congruent to

Applying the Transitive Property of Congruence,

is congruent to

It has been proved that one pair of alternate exterior angles is congruent. Further, since

lies on

it can also be proven that

is congruent to

Applying the Transitive Property of Congruence again,

is congruent to

To conclude, it has been obtained that both pairs of alternate exterior angles are congruent.

and

Rule

Converse Alternate Exterior Angles Theorem

If two lines and a transversal form alternate exterior angles that are congruent, then the two lines are parallel.

Based on the properties of the diagram, the following relation holds true.

If or then

Proof

The proof will be based on the given diagram, but it holds true for any pair of lines cut by a transversal. Consider only one pair of congruent alternate exterior angles and one more angle.

It needs to be proven that

and

are parallel lines. It is already given that

is congruent to

From the diagram, it can also be noted that

and

are vertical angles. By the Vertical Angles Theorem, these angles are congruent.

By the Transitive Property of Congruence, because

is congruent to

and

is congruent to

Further,

and

are corresponding angles. Hence, the Converse Corresponding Angles Theorem can be applied.

Converse Corresponding Angles Theorem
If two lines are cut by a transversal so that the corresponding angles are congruent, then the lines are parallel.

Since and are corresponding congruent angles, and are parallel lines. Each step of the proof will now be summarized in a two-column proof.

Statement	Reason
	Given
	Vertical Angles Theorem
	Transitive Property of Congruence
	Converse Corresponding Angles Theorem

Example

Using Alternate Interior Angles to Solve Problems

In order to build Tulip Street on the south side of the Lilian river, which goes through Flowerland Village, there is a need to build a bridge. Devontay, an architect, proposed the following plan for the bridge.

It is known that the measure of is equal to and the measure of is equal to What are the measures of and

Hint

How do the measures of and relate to each other? Use the given expressions to form an equation for

Solution

By analyzing the diagram it can be noted that and are alternate interior angles.

Therefore, by the Alternate Interior Angles Theorem, these angles are congruent. Hence, the measures of these angles are the same.

By substituting

with

and

with

the equation for

can be formed.

SubstituteII

▼

Solve for

Rearrange equation

Knowing the value of

the measure of each of these angles can be calculated.

▼

Substitute

for

and evaluate

Substitute

Multiply

AddTerms

Add terms

Since the angles are congruent, it can be concluded that they both measure

Explore

Investigating Points on a Perpendicular Bisector

Up to now, some basic theorems about angles have been seen and proven through some rigid motions. Before the end of the lesson, one last theorem about segments will be learned. Consider a perpendicular bisector of a segment

Move

and the endpoints of the segment and compare the distances

and

What conjecture can be made about position of

in respect to the endpoints

and

of the segment? Does this conjecture also apply to other points on the perpendicular bisector

Discussion

Perpendicular Bisector Theorem and Its Converse

Rule

Perpendicular Bisector Theorem

Any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the line segment.

Based on the characteristics of the diagram, is the perpendicular bisector of Therefore, is equidistant from and

Proof

Geometric Approach

Suppose is the perpendicular bisector of Then is the midpoint of

Consider a triangle with vertices and and another triangle with vertices and and

Both and have a right angle and congruent legs and Since all right angles are congruent, Furthermore, by the Reflexive Property of Congruence, is congruent to itself.

By the Side-Angle-Side Congruence Theorem, the triangles are congruent. Therefore, since corresponding parts of congruent figures are congruent, their hypotenuses and are also congruent. By the definition of congruent segments, and have the same length. This means that is equidistant from and

Using this reasoning it can be proven that any point on a perpendicular bisector is equidistant from the endpoints of the segment.

Two-Column Proof

The proof can be summarized in the following two-column table.

Statements	Reasons
and are right angles	Definition of a perpendicular bisector.
	All right angles are congruent.
	Reflexive Property of Congruence.
	SAS Congruence Theorem.
	Corresponding parts of congruent figures are congruent.
	Definition of congruent segments.

Proof

Using Transformations

Suppose is the perpendicular bisector of

Using the given points and as vertices, two triangles can be formed. The resulting triangles, and can be proven to be congruent by identifying a congruence transformation that maps one triangle onto the other.

Since

and

are congruent, the distance between

and

is equal to the distance between

and

Therefore,

is the image of

after a reflection across

Since

lies on

a reflection across

maps

onto itself. The same is true for

Reflection Across
Preimage	Image

The above table shows that the images of the vertices of

are the vertices of

Therefore,

is the image of

after a reflection across

Since a reflection is a rigid motion, this proves that the triangles are congruent.

Corresponding parts of congruent figures are congruent, so

and

are congruent. By the definition of congruent segments,

and

are equal. This means that

is equidistant from

and

The same reasoning can be applied to any point on a perpendicular bisector, showing that the point is equidistant from the endpoints of the segment.

Rule

Converse Perpendicular Bisector Theorem

If a point is equidistant from the endpoints of a line segment, then it lies on the perpendicular bisector of the segment.

Point C on the perpendicular bisector equidistant from endpoints A and B

Based on the characteristics of the diagram, the following relation holds true.

Proof

Converse Perpendicular Bisector Theorem

Consider and a point equidistant from and

To prove that

lies on the perpendicular bisector of

it will be shown that the line perpendicular to

through

bisects

is the point of intersection between the line and the segment, it must be proven that

This line forms two right triangles that share a common leg

Because all right angles are congruent,

is congruent to

Also, by the Reflexive Property of Congruence,

is congruent to itself. Since

is equal to

is congruent to

By the Hypotenuse Leg Theorem,

and

are congruent triangles. Because corresponding parts of congruent figures are congruent,

is congruent to

Additionally, it was already known that and are perpendicular.

By the definition of a perpendicular bisector, is the perpendicular bisector of Therefore, lies on the perpendicular bisector of

Closure

Solving Problems Using Perpendicular Bisectors

In Flowerland Village, there are two related families, Funnystongs and Cleverstongs, who live opposite each other. Mr. Funnystong and Mr. Cleverstong want to pave a road between the houses so that every point of the road is equidistant to their houses.

If the houses are

meters away, how far from the houses and along what line should the road be paved?

Answer

Distance: meters from each house.
Direction: Along the perpendicular bisector to the segment with endpoints at the houses.

Hint

What does the the Perpendicular Bisector Theorem state?

Solution

Recall what the Perpendicular Bisector Theorem states.

Any point on a perpendicular bisector is equidistant from the endpoints of the line segment.

With this theorem in mind, the position of the road can be determined. To do so, draw a segment whose endpoints are located at the houses.

Before drawing the perpendicular bisector of this segment, its midpoint should be found. Since the distance between the houses is meters, the perpendicular bisector will pass through a point that is meters away from the houses.

Based on the theorem, it can be said that each point on the bisector is equidistant from the houses. Therefore, the road between the houses should be paved along the segment's perpendicular bisector.

	{{ 'ml-lesson-number-slides' \| message : article.intro.bblockCount }}
	{{ 'ml-lesson-number-exercises' \| message : article.intro.exerciseCount }}
	{{ 'ml-lesson-time-estimation' \| message }}

{{ article.displayTitle }}

Catch-Up and Review

Extra

Proof

Two-Column Proof

Proof

Hint

Solution

Proof

Proof

Hint

Solution

Proof

Two-Column Proof

Proof

Proof

Proof

Two-Column Proof

Proof

Proof

Hint

Solution

Proof

Two-Column Proof

Proof

Proof

Answer

Hint

Solution