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Congruence, Proof, and Constructions

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

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.

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.

Two intersecting lines that form two pairs of vertical angles

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


Proof

Geometric Approach

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

Two intersecting lines that form two pairs of vertical 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 also 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 obtained equation can be simplified.
By the definition of congruent angles, this means that the vertical angles and are congruent angles. Using the same argumentation, and can also be proven to be 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.

Two intersecting lines that form two pairs of vertical angles with some points
Suppose that the points and are rotated about point
Vertical Angles Proof Rotation About Intersection Point
The points and are mapped onto the points and after the rotation. This means that is mapped onto Since rotations are a rigid motions, and are congruent angles. Since the point lies on and point lies on is congruent to Applying the Transitive Property of Congruence, 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
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 and are each

A crossroad after the construction

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?

Two parallel lines cut by transversal
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.
Two parallel lines intersected by a transversal forming four pairs of angles
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.
Two parallel lines intersected by a transversal forming four pairs of corresponding angles
Based on the characteristics of the diagram, the following relation holds true.

If or then

Proof

Converse Corresponding Angles Theorem

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

Two parallel lines intersected by a transversal

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

Two parallel lines intersected by a transversal

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.

Railings of the staircase
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.
Solve for
Now that the value of is known, the measure of each of the angles can be calculated.
Substitute for and evaluate
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.

Two parallel lines cut by a transversal forming eight angles

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.

Two parallel lines cut by a transversal forming eight angles with points
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
Two parallel lines cut by a transversal translation
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.
Two parallel lines cut by a transversal forming two pairs of congruent angles
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.

One pair of alternate exterior angles

It needs to be proven that and are parallel lines. It is already given that is congruent to From the diagram, it can be noted 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, because is congruent to and is congruent to is congruent It can also be noted that and are corresponding angles. Hence, the Converse Corresponding Angles Theorem will 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. Finally, all the steps will be summarized 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.

"Two
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.

Two parallel lines cut by a transversal forming eight angles

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.

Two parallel lines cut by a transversal forming eight angles with points
Next, and will be translated in the direction of the transversal so that points and lie on Then, and will be rotated about
Two parallel lines cut by a transversal forming eight angles with points
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.
Two parallel lines cut by a transversal forming two pairs of congruent angles
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.

One pair of alternate exterior angles

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. Notice the common angle of in both relationships. By the Transitive Property of Congruence, because is congruent to and is congruent to is congruent It can also be noted that 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.

A plan of a 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.

Two parallel lines intersected by a transversal
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.
Solve for
Knowing the value of the measure of each of these angles can be calculated.
Substitute for and evaluate
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
Perpendicular bisector of AB
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.

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

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

Point C on perpendicular bisector CM of segment AB

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

Triangles ACM and BCM

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.

Triangles ACM and BCM

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

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

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

Point C on perpendicular bisector CM of segment AB

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.

Triangles ACM and BCM

Since and are congruent, the distance between and is equal to the distance between and Therefore, is the image of after a reflection across

Reflection of B across line CM.
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.
Reflection of triangle BCM across line CM.
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

Point C is equidistant from endpoints A and B
To prove that lies on the perpendicular bisector of it will be shown that the line perpendicular to through bisects If is the point of intersection between the line and the segment, it must be proven that
Draw a line perpendicular to segment AB
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
Segment AM and segment BM are congruent segments

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.

The locating of two houses opposite to each other
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.

The locating of two houses opposite to each other

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.

The locating of two houses opposite to each other

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.

The locating of two houses opposite to each other
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