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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 P, 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.

∠1∠3
∠2∠4

Proof

Geometric Approach

Analyzing the diagram, it can be observed that ∠1 and ∠2 form a straight angle, so these are supplementary angles. Similarly, ∠2 and ∠3 are also supplementary angles.

Two intersecting lines that form two pairs of vertical angles

Therefore, by the Angle Addition Postulate, the sum of m∠1 and m∠2 is and the sum of m∠2 and m∠3 is also These facts can be used to express m∠2 in terms of m∠1 and also in terms of m∠3.

Angle Addition Postulate Isolate m∠2
m∠1+m∠2 = m∠2 =
m∠2+m∠3 = m∠2 =
By the Transitive Property of Equality, the expressions representing m∠2 can be set equal to each other.
Then the obtained equation can be simplified.
-m∠1=-m∠3
m∠1=m∠3
By the definition of congruent angles, this means that the vertical angles ∠1 and ∠3 are congruent angles. Using the same argumentation, ∠2 and ∠4 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
∠1 and ∠2 supplementary Definition of straight angle
Definition of supplementary angles
Subtraction Property of Equality
∠2 and ∠3 supplementary Definition of straight angle
Definition of supplementary angles
Subtraction Property of Equality
Transitive Property of Equality
m∠1=m∠3 Subtraction and Multiplication Properties of Equality

Proof

Using Transformations

Consider the points A, B, C, and D on each ray that starts at the point of intersection E of the two lines.

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

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 3x+14 and 6x5, 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 x. Identify the relationship between ∠1 and ∠3, as well as ∠2 and ∠4 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 x can be determined.
Solve for x
Now that the value of x is known, the measures of ∠1 and ∠2 can be calculated.
x=19
m∠1=3x+14 m∠2=6x5
m∠1=3(19)+14 m∠2=6(19)5
m∠1=57+14 m∠2=1145
Next, by analyzing the position of ∠1 and ∠3, as well as ∠2 and ∠4, 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 ∠1 and ∠3 are each and ∠2 and ∠4 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, A and B. 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 ∠1∠5, ∠2∠6, ∠3∠7, and ∠4∠8.

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 ∠1∠5, ∠2∠6, ∠3∠7, or ∠4∠8, 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 ∠1 and ∠2.

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 n parallel to that passes through the point of intersection between and This line forms ∠3 and ∠4.
Two parallel lines intersected by a transversal
By the Angle Addition Postulate, m∠1 is equal to the sum of m∠3 and m∠4.
Since n and are parallel lines that are cut by a transversal, by the Corresponding Angles Theorem, ∠3 and ∠2 are congruent. By the definition of congruence, these angles have the same measure.
By the Substitution Property of Equality, m∠2 can be substituted for m∠3 into the equation for m∠1.
From the above equation and since m∠4 is a positive number, it can be concluded that m∠1 is greater than m∠2.
This contradicts the given fact that ∠1 and ∠2 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 ∠1 and ∠2 are expressed as 5t2 and 4t+12, respectively.

Railings of the staircase
What are the measures of ∠1 and ∠2?

Hint

How do measures of ∠1 and ∠2 relate to each other? Use the given expressions to form an equation for t.

Solution

Analyzing the diagram, it can be noted that ∠1 and ∠2 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 m∠1 with 5t2 and m∠2 with 4t+12, the equation for t can be formed.
m∠1=m∠2
5t2=4t+12
Solve for t
t2=12
t=14
Now that the value of t is known, the measure of each of the angles can be calculated.
m∠1=5t2
Substitute 14 for t and evaluate
m∠1=5(14)2
m∠1=702
m∠1=68
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 ∠1 and ∠2 are congruent.

Two parallel lines cut by a transversal forming eight angles
Notice that by definition ∠2 and ∠5 are vertical angles. By the Vertical Angles Theorem, they are therefore congruent angles.
Furthermore, by definition ∠5 and ∠1 are corresponding angles. Hence, by the Corresponding Angles Theorem, ∠5 and ∠1 are also congruent angles.
Applying the Transitive Property of Congruence, ∠2 and ∠1 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
∠2 and ∠5 are vertical angles Def. of vertical angles
∠2∠5 Vertical Angles Theorem
∠5 and ∠1 are corresponding angles Def. of corresponding angles
∠5∠1 Corresponding Angles Theorem
∠2∠1 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, A, B, C, and D will be translated parallel to the transversal until the points A, C, and D lie on Then, A, B, and C will be rotated about F. It should be noted that since point D lies on the transversal, when translating it to the point will fall into the same position as F. Therefore, D will not be affected by the rotation around F.
Two parallel lines cut by a transversal translation
After this combination of rigid motions, A, B, C, and D are mapped onto and This means that ADB is mapped onto Therefore, ADB and are congruent angles.
Note that and F share the same location. It can also be seen that lies on and lies on Because of this, is congruent to GFH.
Applying the Transitive Property of Congruence, ADB is congruent to GFH.

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 ∠1∠2 or ∠3∠4, 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 ∠1 is congruent to ∠2.
From the diagram, it can be noted that ∠2 and are vertical angles. By the Vertical Angles Theorem, these angles are congruent.
Notice the common angle of ∠2 in both relationships. By the Transitive Property of Congruence, because ∠1 is congruent to ∠2 and ∠2 is congruent to ∠1 is congruent
It can also be noted that ∠1 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 ∠1 and are corresponding congruent angles, and are parallel lines. Finally, all the steps will be summarized in a two-column proof.

Statement Reason
∠1∠2 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 ∠1 and ∠2 are congruent.

Two parallel lines cut by a transversal forming eight angles
Notice that by definition, ∠2 and ∠8 are corresponding angles. Therefore, by the Corresponding Angles Theorem, they are congruent angles.
Furthermore, by definition, ∠8 and ∠1 are vertical angles. Therefore, by the Vertical Angles Theorem, ∠8 and ∠1 are congruent angles.
Then, by applying the Transitive Property of Congruence, ∠2 and ∠1 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
∠2 and ∠8 are corresponding angles Def. of corresponding angles
∠2∠8 Corresponding Angles Theorem
∠8 and ∠1 are vertical angles Def. of vertical angles
∠8∠1 Vertical Angles Theorem
∠2∠1 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, A, B, C, and D will be translated in the direction of the transversal so that points A, C, and D lie on Then, A, B, and C will be rotated about F.
Two parallel lines cut by a transversal forming eight angles with points
After this combination of rigid motions, A, B, C, and D are mapped onto and This means that ADB is mapped onto Therefore, ADB and are congruent angles.
Since and F share the same location, lies on and lies on Because of this, is congruent to GFH.
Applying the Transitive Property of Congruence, ADB is congruent to GFH.
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 EFH.
Applying the Transitive Property of Congruence again, CDB is congruent to EFH.
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 ∠1∠2 or ∠3∠4, 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 ∠1 is congruent to ∠2.
From the diagram, it can also be noted that ∠2 and are vertical angles. By the Vertical Angles Theorem, these angles are congruent.
By the Transitive Property of Congruence, because ∠1 is congruent to ∠2 and ∠2 is congruent to ∠1 is congruent to
Further, ∠1 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 ∠1 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
∠1∠2 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 ∠1 is equal to 4a+11 and the measure of ∠2 is equal to 8a53. What are the measures of ∠1 and ∠2?

Hint

How do the measures of ∠1 and ∠2 relate to each other? Use the given expressions to form an equation for a.

Solution

By analyzing the diagram it can be noted that ∠1 and ∠2 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 m∠1 with 4a+11 and m∠2 with 8a53, the equation for a can be formed.
m∠1=m∠2
4a+11=8a53
Solve for a
11=4a53
64=4a
16=a
a=16
Knowing the value of a, the measure of each of these angles can be calculated.
m∠1=4a+11
Substitute 16 for a and evaluate
m∠1=4(16)+11
m∠1=64+11
m∠1=75
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 AB. Move C and the endpoints of the segment and compare the distances AC and BC.
Perpendicular bisector of AB
What conjecture can be made about position of C in respect to the endpoints A and B of the segment? Does this conjecture also apply to other points on the perpendicular bisector CM?

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 AB. Therefore, C is equidistant from A and B.

AC=BC

Proof

Geometric Approach

Suppose is the perpendicular bisector of AB. Then M is the midpoint of AB.

Point C on perpendicular bisector CM of segment AB

Consider a triangle with vertices A, M, and C, and another triangle with vertices and B, M, and C.

Triangles ACM and BCM

Both ACM and BCM have a right angle and congruent legs AM and BM. Since all right angles are congruent, AMCBMC. Furthermore, by the Reflexive Property of Congruence, CM 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 AC and BC are also congruent. By the definition of congruent segments, AC and BC have the same length. This means that C is equidistant from A and B.

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
AMMB

AMC and BMC are right angles

Definition of a perpendicular bisector.
AMCBMC All right angles are congruent.
CMCM Reflexive Property of Congruence.
ACMBCM SAS Congruence Theorem.
ACBC Corresponding parts of congruent figures are congruent.
AC=BC Definition of congruent segments.

Proof

Using Transformations

Suppose is the perpendicular bisector of AB.

Point C on perpendicular bisector CM of segment AB

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

Triangles ACM and BCM

Since AM and BM are congruent, the distance between A and M is equal to the distance between B and M. Therefore, A is the image of B after a reflection across

Reflection of B across line CM.
Since C lies on a reflection across maps C onto itself. The same is true for M.
Reflection Across
Preimage Image
B A
C C
M M
The above table shows that the images of the vertices of BCM are the vertices of ACM. Therefore, ACM is the image of BCM 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 AC and BC are congruent. By the definition of congruent segments, AC and BC are equal. This means that C is equidistant from A and B.
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 AB and a point C equidistant from A and B.

Point C is equidistant from endpoints A and B
To prove that C lies on the perpendicular bisector of AB, it will be shown that the line perpendicular to AB through C bisects AB. If M is the point of intersection between the line and the segment, it must be proven that AM=MB.
Draw a line perpendicular to segment AB
This line forms two right triangles that share a common leg CM. Because all right angles are congruent, AMC is congruent to BMC. Also, by the Reflexive Property of Congruence, CM is congruent to itself. Since AC is equal to BC, AC is congruent to BC.
By the Hypotenuse Leg Theorem, AMC and BMC are congruent triangles. Because corresponding parts of congruent figures are congruent, AM is congruent to BM.
Segment AM and segment BM are congruent segments

Additionally, it was already known that and AB are perpendicular.

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

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

Answer

Distance: 8 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 16 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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