10. Theorems About Lines and Angles
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Chapter 1
10.
Theorems About Lines and Angles
Various theorems provide insights into the relationships between lines and angles. One such theorem is the alternate interior angles theorem, which states that when two parallel lines are intersected by a transversal, the alternate interior angles are congruent. Similarly, the corresponding angles theorem emphasizes the congruence of angles in specific positions when parallel lines are cut by a transversal. These theorems, along with others related to lines and angles, offer valuable tools for understanding geometric structures and their properties. By grasping these concepts, one can better interpret and solve real-world problems that involve lines, angles, and their intricate relationships.
Lesson Settings & Tools
| | 13 Theory slides |
| | 12 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Theorems About Lines and Angles
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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 P, move the corresponding slider.
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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.
∠ 1 ≅ ∠ 3
∠ 2 ≅ ∠ 4
Proof
Geometric ApproachAnalyzing the diagram, it can be seen that ∠ 1 and ∠ 2 form a straight angle, so these are supplementary angles. Similarly, ∠ 2 and ∠ 3 are also supplementary angles.
Therefore, by the Angle Addition Postulate, the sum of m∠ 1 and m∠ 2 is 180^(∘), and the sum of m∠ 2 and m∠ 3 is also 180^(∘). These facts can be used to express m∠ 2 in terms of m∠ 1 and in terms of m∠ 3.
| Angle Addition Postulate | Isolate m∠ 2 |
|---|---|
| m∠ 1+m∠ 2 = 180^(∘) | m∠ 2 = 180^(∘)-m∠ 1 |
| m∠ 2+m∠ 3 = 180^(∘) | m∠ 2 = 180^(∘)-m∠ 3 |
By the Transitive Property of Equality, the expressions representing m∠ 2 can be set equal to each other. m∠ 2= 180^(∘)-m∠ 1 m∠ 2= 180^(∘)-m∠ 3 ⇓ 180^(∘)-m∠ 1= 180^(∘)-m∠ 3 Then the equation can be simplified.
180^(∘)-m∠ 1=180^(∘)-m∠ 3
SubEqn
LHS-180^(∘)=RHS-180^(∘)
- m∠ 1=- m∠ 3
MultEqn
LHS * (- 1)=RHS* (- 1)
m∠ 1=m∠ 3
By the definition of congruent angles, this means that the vertical angles ∠ 1 and ∠ 3 are congruent angles. The same process can be used to prove ∠ 2 and ∠ 4 congruent.
Two-Column Proof
The previous proof can be summarized in the following two-column table.
Statements
|
Reasons
|
1. l_1 and l_2 lines
|
1. Given
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2. ∠ 1 and ∠ 2 supplementary
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2. Definition of straight angle
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3. m∠ 1+m∠ 2=180^(∘)
|
3. Definition of supplementary angles
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4. m∠ 2=180^(∘)-m∠ 1
|
4. Subtraction Property of Equality
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5. ∠ 2 and ∠ 3 supplementary
|
5. Definition of straight angle
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6. m∠ 2+m∠ 3=180^(∘)
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6. Definition of supplementary angles
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7. m∠ 2=180^(∘)-m∠ 3
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7. Subtraction Property of Equality
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8. 180^(∘)-m∠ 1=180^(∘)-m∠ 3
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8. Transitive Property of Equality
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9. m∠ 1=m∠ 3
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9. Subtraction and Multiplication Properties of Equality
|
Proof
Using TransformationsConsider the points A, B, C, and D on each ray that starts at the point of intersection E of the two lines.
Suppose that points A and B are rotated 180^(∘) about point E.
The points A and B are mapped onto the points A' and B' after the rotation. This means that ∠ AEB is mapped onto ∠ A'EB'. Since rotations are a rigid motion, ∠ AEB and ∠ A'EB' are congruent angles. ∠ AEB ≅ ∠ A'EB' Since the point A' lies on EC and point B' lies on ED, ∠ A'EB' is congruent to ∠ CED. ∠ A'EB' ≅ ∠ CED By applying the Transitive Property of Congruence, it can be confirmed that ∠ AEB is congruent to ∠ CED. ∠ AEB ≅ ∠ A'EB' ∠ A'EB' ≅ ∠ CED ⇓ ∠ AEB ≅ ∠ CED
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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 6x-5, respectively.
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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.
6x-5+3x+14=180^(∘)
By solving it, the value of x can be determined.
6x-5+3x+14=180^(∘)
▼
Solve for x
x=19^(∘)
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=6x-5 |
| m∠ 1=3( 19)+14 | m∠ 2=6( 19)-5 |
| m∠ 1=57+14 | m∠ 2=114-5 |
| m∠ 1= 71^(∘) | m∠ 2= 109^(∘) |
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. lm∠ 1=m∠ 3 m∠ 2=m∠ 4 ⇒ lm∠ 3= 71^(∘) m∠ 4= 109^(∘) In this way, it was obtained that ∠ 1 and ∠ 3 are each 71^(∘), and ∠ 2 and ∠ 4 are each 109^(∘).
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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?
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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.
If l_1 ∥ l_2, then ∠ 1 ≅ ∠ 5, ∠ 2 ≅ ∠ 6, ∠ 3 ≅ ∠ 7, and ∠ 4 ≅ ∠ 8.
Proof
It is given that l_1 and l_2 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.
If ∠ 1 ≅ ∠ 5, ∠ 2 ≅ ∠ 6, ∠ 3 ≅ ∠ 7, or ∠ 4 ≅ ∠ 8, then l_1 ∥ l_2.
Proof
This theorem can be proven by an indirect proof. Let l_1 and l_2 be two lines intersected by a transversal line l_3 forming corresponding congruent angles ∠1 and ∠2.
Since the goal is to prove that l_1 is parallel to l_2, it will be temporarily assumed that l_1 and l_2 are not parallel. Temporary Assumption l_1 ∦ l_2 By the Parallel Postulate, there exists a line n parallel to l_2 that passes through the point of intersection between l_1 and l_3. This line forms ∠3 and ∠4.
By the Angle Addition Postulate, m∠1 is equal to the sum of m∠3 and m∠4. m∠1=m∠3+m∠4 Since n and l_2 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. ∠3 ≅ ∠2 ⇕ m∠3 = m∠2 By the Substitution Property of Equality, m∠2 can be substituted for m∠3 into the equation for m∠1. m∠1=m∠3+m∠4 ↓ m∠1= m∠2+m∠4 From the above equation and since m∠ 4 is a positive number, it can be concluded that m∠1 is greater than m∠2. m∠1>m∠2 This contradicts the given fact that ∠1 and ∠2 are congruent. The contradiction came from assuming that l_1 and l_2 are not parallel lines. Therefore, l_1 and l_2 must be parallel lines.
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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 5t-2 and 4t+12, respectively.
What are the measures of ∠ 1 and ∠ 2?
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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.
∠ 1&≅ ∠ 2 &⇓ m∠ 1&=m∠ 2
By substituting m∠ 1 with 5t-2 and m∠ 2 with 4t+12, the equation for t can be formed.
Now that the value of t is known, the measure of each of the angles can be calculated.
m∠ 1=5t-2
▼
Substitute 14 for t and evaluate
m∠ 1=68
Since the angles are congruent, it can be concluded that they both measure to be 68^(∘). l m ∠ 1 = m ∠ 2 m ∠ 1 = 68^(∘) ⇒ m ∠ 2 = 68^(∘)
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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.
If l_1 ∥ l_2, then ∠ 1 ≅ ∠ 2 and ∠ 3 ≅ ∠ 4.
Proof
Geometric approachTo prove that alternate interior angles are congruent, it will be shown that ∠ 1 and ∠ 2 are congruent.
Notice that by definition ∠ 2 and ∠ 5 are vertical angles. By the Vertical Angles Theorem, they are therefore congruent angles. ∠ 2 ≅ ∠ 5 Furthermore, by definition ∠ 5 and ∠ 1 are corresponding angles. Hence, by the Corresponding Angles Theorem, ∠ 5 and ∠ 1 are also congruent angles. ∠ 5 ≅ ∠ 1 Applying the Transitive Property of Congruence, ∠ 2 and ∠ 1 can be concluded to be congruent angles as well. ∠ 2 ≅ ∠ 5 ∠ 5 ≅ ∠ 1 ⇒ ∠ 2 ≅ ∠ 1 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
|
1. ∠ 2 and ∠ 5 are vertical angles
|
1. Def. of vertical angles
|
2. ∠ 2≅ ∠ 5
|
2. Vertical Angles Theorem
|
3. ∠ 5 and ∠ 1 are corresponding angles
|
3. Def. of corresponding angles
|
4. ∠ 5≅ ∠ 1
|
4. Corresponding Angles Theorem
|
5. ∠ 2 ≅ ∠ 1
|
5. Transitive Property of Congruence
|
Proof
Using TransformationsApart from the points of intersection, consider two more points on each line.
Next, A, B, C, and D will be translated parallel to the transversal until the points A, C, and D lie on l_2. Then, A, B, and C will be rotated 180^(∘) about F. It should be noted that since point D lies on the transversal, when translating it to l_2 the point will fall into the same position as F. Therefore, D will not be affected by the rotation around F.
After this combination of rigid motions, A, B, C, and D are mapped onto A', B', C', and D'. This means that ∠ ADB is mapped onto ∠ A'D'B'. Therefore, ∠ ADB and ∠ A'D'B' are congruent angles. ∠ ADB ≅ ∠ A'D'B' Note that D' and F share the same location. It can also be seen that A' lies on FG and B' lies on FH. Because of this, ∠ A'D'B' is congruent to ∠ GFH. ∠ A'D'B' ≅ ∠ GFH Applying the Transitive Property of Congruence, ∠ ADB is congruent to GFH. ∠ ADB ≅ ∠ A'D'B' ∠ A'D'B' ≅ ∠ GFH ⇓ ∠ ADB ≅ ∠ 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.
If ∠ 1 ≅ ∠ 2 or ∠ 3 ≅ ∠ 4, then l_1 ∥ l_2.
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 l_1 and l_2 are parallel lines. It is already given that ∠ 1 is congruent to ∠ 2. ∠ 1 ≅ ∠ 2 The diagram shows that ∠ 2 and ∠ α are vertical angles. By the Vertical Angles Theorem, these angles are congruent. ∠ 2 ≅ ∠ α Notice the common angle of ∠ 2 in both relationships. By the Transitive Property of Congruence, since ∠ 1 is congruent to ∠ 2 and ∠ 2 is congruent to ∠ α, then ∠ 1 is congruent ∠ α. ∠ 1 ≅ ∠ 2 ∠ 2 ≅ ∠ α ⇓ ∠ 1 ≅ ∠ α The diagram also shows that ∠ 1 and ∠ α 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 ∠ 1 and ∠ α are corresponding congruent angles, then l_1 and l_2 are parallel lines. To summarize, all of the steps will be described in a two-column proof.
Statement
|
Reason
|
1. ∠ 1 ≅ ∠ 2
|
1. Given
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2. ∠ 2 ≅ ∠ α
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2. Vertical Angles Theorem
|
3. ∠ 1 ≅ ∠ α
|
3. Transitive Property of Congruence
|
4. l_1 ∥ l_2
|
4. Converse Corresponding Angles Theorem
|
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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.
If l_1 ∥ l_2, then ∠ 1 ≅ ∠ 2 and ∠ 3 ≅ ∠ 4.
Proof
Geometric ApproachIn order to prove that alternate exterior angles are congruent, it will be shown that ∠ 1 and ∠ 2 are congruent.
Notice that by definition, ∠ 2 and ∠ 8 are corresponding angles. Therefore, by the Corresponding Angles Theorem, they are congruent angles. ∠ 2 ≅ ∠ 8 Furthermore, by definition, ∠ 8 and ∠ 1 are vertical angles. Therefore, by the Vertical Angles Theorem, ∠ 8 and ∠ 1 are congruent angles. ∠ 8 ≅ ∠ 1 Then, by applying the Transitive Property of Congruence, ∠ 2 and ∠ 1 can be concluded to be congruent angles as well. ∠ 2 ≅ ∠ 8 ∠ 8 ≅ ∠ 1 ⇒ ∠ 2 ≅ ∠ 1 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
|
1. ∠ 2 and ∠ 8 are corresponding angles
|
1. Def. of corresponding angles
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2. ∠ 2≅ ∠ 8
|
2. Corresponding Angles Theorem
|
3. ∠ 8 and ∠ 1 are vertical angles
|
3. Def. of vertical angles
|
4. ∠ 8≅ ∠ 1
|
4. Vertical Angles Theorem
|
5. ∠ 2 ≅ ∠ 1
|
5. Transitive Property of Congruence
|
Proof
Using TransformationsConsider the points of intersection as well as two more points on each line.
Next, A, B, C, and D will be translated in the direction of the transversal so that points A, C, and D lie on l_2. Then, A, B, and C will be rotated 180^(∘) about F.
After this combination of rigid motions, A, B, C, and D are mapped onto A', B', C', and D'. This means that ∠ ADB is mapped onto ∠ A'D'B'. Therefore, ∠ ADB and ∠ A'D'B' are congruent angles. ∠ ADB ≅ ∠ A'D'B' Since D' and F share the same location, A' lies on FG and B' lies on FH. Because of this, ∠ A'D'B' is congruent to ∠ GFH. ∠ A'D'B' ≅ ∠ GFH Applying the Transitive Property of Congruence, ∠ ADB is congruent to ∠ GFH. ∠ ADB ≅ ∠ A'D'B' ∠ A'D'B' ≅ ∠ GFH ⇓ ∠ ADB ≅ ∠ GFH It has been proved that one pair of alternate exterior angles is congruent. Further, since C' lies on EF, it can also be proven that ∠ C'D'B' is congruent to ∠ EFH. ∠ C'D'B' ≅ ∠ EFH Applying the Transitive Property of Congruence again, ∠ CDB is congruent to ∠ EFH. ∠ CDB ≅ ∠ C'D'B' ∠ C'D'B' ≅ ∠ EFH ⇓ ∠ CDB ≅ ∠ EFH To conclude, it has been obtained that both pairs of alternate exterior angles are congruent.
∠ ADB ≅ ∠ GFH and ∠ CDB ≅ ∠ EFH
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.
If ∠ 1 ≅ ∠ 2 or ∠ 3 ≅ ∠ 4, then l_1 ∥ l_2.
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 l_1 and l_2 are parallel lines. It is already given that ∠ 1 is congruent to ∠ 2. ∠ 1 ≅ ∠ 2 From the diagram, it can also be noted that ∠ 2 and ∠ α are vertical angles. By the Vertical Angles Theorem, these angles are congruent. ∠ 2 ≅ ∠ α By the Transitive Property of Congruence, because ∠ 1 is congruent to ∠ 2 and ∠ 2 is congruent to ∠ α, ∠ 1 is congruent to ∠ α. ∠ 1 ≅ ∠ 2 ∠ 2 ≅ ∠ α ⇓ ∠ 1 ≅ ∠ α Further, ∠ 1 and ∠ α 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 ∠ α are corresponding congruent angles, l_1 and l_2 are parallel lines. Each step of the proof will now be summarized in a two-column proof.
Statement
|
Reason
|
1. ∠ 1 ≅ ∠ 2
|
1. Given
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2. ∠ 2 ≅ ∠ α
|
2. Vertical Angles Theorem
|
3. ∠ 1 ≅ ∠ α
|
3. Transitive Property of Congruence
|
4. l_1 ∥ l_2
|
4. Converse Corresponding Angles Theorem
|
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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 ∠ 1 is equal to 4a+11 and the measure of ∠ 2 is equal to 8a-53. What are the measures of ∠ 1 and ∠ 2?
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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.
Therefore, by the Alternate Interior Angles Theorem, these angles are congruent. Hence, the measures of these angles are the same. ∠ 1&≅ ∠ 2 &⇓ m∠ 1&=m∠ 2 By substituting m∠ 1 with 4a+11 and m∠ 2 with 8a-53, the equation for a can be formed.
m∠ 1=m∠ 2
SubstituteII
m∠ 1= 4a+11, m∠ 2= 8a-53
4a+11= 8a-53
▼
Solve for a
SubEqn
LHS-4a=RHS-4a
11=4a-53
AddEqn
LHS+53=RHS+53
64=4a
DivEqn
.LHS /4.=.RHS /4.
16=a
RearrangeEqn
Rearrange equation
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=75
Since the angles are congruent, it can be concluded that they both measure 75^(∘). l m∠ 1= m ∠ 2 m ∠ 1 = 75 ^(∘) ⇒ m ∠ 2 = 75 ^(∘)
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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.
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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, CM is the perpendicular bisector of AB. Therefore, C is equidistant from A and B.
AC=BC
Proof
Geometric ApproachSuppose CM is the perpendicular bisector of AB. Then M is the midpoint of AB.
Consider a triangle with vertices A, M, and C, and another triangle with vertices and B, M, and C.
Both △ ACM and △ BCM have a right angle and congruent legs AM and BM. Since all right angles are congruent, ∠ AMC≅ ∠ BMC. Furthermore, by the Reflexive Property of Congruence, CM 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 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.
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
|
1. AM ≅ MB ∠ AMC and ∠ BMC are right angles |
1. Definition of a perpendicular bisector.
|
2. ∠AMC≅ ∠BMC
|
2. All right angles are congruent.
|
3. CM≅ CM
|
3. Reflexive Property of Congruence.
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4. △ ACM ≅ △ BCM
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4. SAS Congruence Theorem.
|
5. AC ≅ BC
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5. Corresponding parts of congruent figures are congruent.
|
6. AC = BC
|
6. Definition of congruent segments.
|
Proof
Using TransformationsSuppose CM is the perpendicular bisector of 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.
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 CM.
Since C lies on CM, a reflection across CM maps C onto itself. The same is true for M.
| Reflection Across CM | |
|---|---|
| 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 CM. Since a reflection is a rigid motion, this proves that the triangles are congruent.
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. AC=BC 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.
AC=CB ⇓ CM⊥ AB and AM=MB
Proof
Converse Perpendicular Bisector TheoremConsider AB and a point C equidistant from 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.
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. ∠ AMC ≅ ∠ BMC CM≅CM AC≅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.
Additionally, it was already known that CM and AB are perpendicular.
CM⊥ AB and AM=MB
By the definition of a perpendicular bisector, CM is the perpendicular bisector of AB. Therefore, C lies on the perpendicular bisector of AB.
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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 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.
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 16 meters, the perpendicular bisector will pass through a point that is 162=8 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.
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Theorems About Lines and Angles
Exercises
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Examining the diagram, we see that our angles have the same position in relation to the transversal and their respective lines, so they are corresponding angles. Also, since the two lines cut by the transversal are parallel, we know by the Corresponding Angles Theorem that the angles are congruent.
Let's solve this equation for x by performing inverse operations.
To make things more clear, we will color the expressions for the given angle measures.
We have three relationships to consider. Angles:& 10x-10^(∘) & 3x+32^(∘) [-0.2em] Relationship:& Vertical angles [1em] Angles:& 10x-10^(∘) & 12x-20^(∘) [-0.2em] Relationship:& Alternate interior angles [1em] Angles:& 12x-20^(∘) & 3x+32^(∘) [-0.2em] Relationship:& Corresponding angles Vertical angles are always congruent. This is also true for alternate interior angle and corresponding angles if the two lines cut by the transversal are parallel. In this case, we do not have this information. Therefore, we can only claim that the vertical angles are congruent. 10x-10^(∘)=3x+32^(∘) Let's solve this equation for x by performing inverse operations.
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Solution
Determine the measure of α.
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Let's mark a pair of corresponding angles in the diagram. Since the two lines cut by the transversal that create the corresponding angles are parallel, we know by the Corresponding Angles Theorem that they are congruent angles.
Now we can find the adjacent angles to the right of the 35^(∘)-angle in the original diagram.
Notice that α and the 75^(∘)-angle are vertical angles. By the Vertical Angles Theorem, we know that they are congruent. Therefore, we have that α=75^(∘).
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Find x from the following diagram.
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To relate the angles, we will introduce a third angle ∠ θ. For clarity purposes, we will also color code the given angle measures.
Notice that both of our expressions represent corresponding angles to ∠ θ. Since the two lines cut by the transversal are parallel, we know that they are congruent by the Corresponding Angles Theorem. Let's identify both pairs of corresponding angles.
Since both angles are congruent to ∠ θ, we know by the Transitive Property of Congruence that they are themselves congruent. m∠ θ = 15x-30^(∘) m∠ θ = 10x+5^(∘) ⇓ 10x+5^(∘) = 15x-30^(∘) Let's solve this equation using inverse operations.
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The Converse of the Perpendicular Bisector Theorem states that when a point is equidistant from the endpoints of a segment, then it is on the segment's perpendicular bisector. From the diagram, we see that DC is in fact the perpendicular bisector of AB. Therefore, AD and DB must be congruent segments.
With this information, we can write the following equation. AD= DB ⇓ 6x= 4x+4 Let's perform inverse operations to find the value of x.
Finally, to find the length of AB, we will substitute 2 for x in the expression 5x and multiply by 2 to account for AD and DB.
From the diagram, we see that E is the midpoint of AC and that DB is the perpendicular bisector to AC through E. By the Perpendicular Bisector Theorem, we conclude that AB= BC.
Since AB= BC, we can write an equation to find x by substituting the expressions for AB and BC. AB= BC ⇓ 10x+1= 7x+13 Let's solve the equation for x.
We can now determine AB by substituting 4 for x in the corresponding expression.
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Kriz says that ∠ XVW is congruent to ∠ RWS. Is there enough information to confirm what Kriz says? Explain.
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Since both ∠ XVW and ∠ RWS are below their respective lines and to the left of the transversal, they must be corresponding angles.
Corresponding angles are congruent by the Corresponding Angles Theorem if the two lines cut by the transversal are parallel. Even though the lines appear to be parallel, we have not been given this information explicitly.
Therefore, we cannot conclude that ∠ XVW and ∠ RWS have equal measures and, therefore, are congruent. Kriz is not necessarily correct.
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Examining the diagram, we can see that K is the midpoint of GJ. We also see that HK is perpendicular to GJ through K. Therefore, HK must be the perpendicular bisector to GJ.
According to the Perpendicular Bisector Theorem, if a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Since H is a point on the perpendicular bisector, we can conclude that HG and HJ are congruent segments. HG≅ HJ ⇒ HG=GJ Let's show this on the diagram.
Since HJ is 4 units long, we conclude that HG is also 4 units long.
In this exercise, we are given in the diagram that KH=KJ. According to the Converse of the Perpendicular Bisector Theorem, if a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment. This tells us that K lies on the perpendicular bisector of HJ.
Notice that G also lies on the perpendicular bisector. Therefore, it is also equidistant from the endpoints of HJ. Since we know the length of GJ, we can also determine the length of HG.
Therefore, HG=1 unit.
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Theorems About Lines and Angles
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