Consider line \begin{align*}l\end{align*}
Watch This
http://www.youtube.com/watch?v=svprkO5bM88 James Sousa: Proof: Alternate Interior Angles are Congruent
https://www.youtube.com/watch?v=ttZvFhEq6cg James Sousa: Proof: Perpendicular Bisector Theorem
Guidance
Consider two parallel lines that are intersected by a third line. (Remember that tick marks \begin{align*}(\gg)\end{align*}
This third line is called a transversal. Note that four angles are created where the transversal intersects each line. Each angle created by the transversal and the top line has a corresponding angle with an angle create by the transversal and the bottom line. These corresponding angle pairs are shown color-coded below. How do you think these corresponding angles are related?
Your intuition and knowledge of translations might suggest that these angles are congruent. Imagine translating one of the angles along the transversal until it meets the second parallel line. It will match its corresponding angle exactly. This is known as the corresponding angle postulate:
If two parallel lines are cut by a transversal, then the corresponding angles are congruent.
Remember that a postulate is a statement that is accepted as true without proof. Your knowledge of translations should convince you that this postulate is true.
Example A
Recall that vertical angles are a pair of opposite angles created by intersecting lines. Prove that vertical angles are congruent.
Solution: For this proof, you are not given a specific picture. When not given a picture, it helps to create a generic picture to reference in your proof. It's important that the picture does not include any information that you cannot assume. Below is a generic picture of intersecting lines with angles numbered for reference.
In this picture, \begin{align*}\angle 1\end{align*}
Statements |
Reasons |
\begin{align*}m\angle 1+m\angle 3=180^\circ\end{align*} |
Two angles that form a line are supplementary |
\begin{align*}m\angle 1+m\angle 3=m\angle 2+m\angle 3\end{align*} |
Algebraic substitution |
\begin{align*}m\angle 1=m\angle 2\end{align*} |
Subtraction property of equality |
\begin{align*}\angle 1 \cong \angle 2\end{align*} |
If two angles have the same measure, they are congruent. |
Vertical angles are congruent is a theorem. Now that it has been proven, you can use it in future proofs without proving it again.
Example B
When two parallel lines are cut by a transversal, two pairs of alternate interior angles are formed. In the diagram below, \begin{align*}\angle 3\end{align*}
Prove that if two parallel lines are cut by a transversal, alternate interior angles are congruent.
Solution: Use the diagram above, and prove that \begin{align*}\angle 3 \cong \angle 5\end{align*}
\begin{align*}\angle 1 \cong \angle 3\end{align*}
*Note: The transitive property states that if two objects are equal/congruent to the same third object, then they are equal/congruent to each other. The transitive property is a form of substitution. You can use it in any proof.
The statement “if two parallel lines are cut by a transversal, then alternate interior angles are congruent” is a theorem. Now that it has been proven, you can use it in future proofs without proving it again.
Example C
Prove that points on the perpendicular bisector of a line segment are equidistant from the endpoints of the line segment.
Solution: Start by exploring this claim. Draw a picture of a line segment and a perpendicular bisector of this segment. Remember that a perpendicular bisector is perpendicular to the line segment (meets it at a right angle) and bisects the line segment (cuts it in half).
The claim is that any point on the perpendicular bisector (such as point \begin{align*}C\end{align*}
To prove the original statement, it will suffice to prove that if \begin{align*}\overleftrightarrow{C D}\end{align*}
The statement “points on a perpendicular bisector of a line segment are equidistant from the segment's endpoints” is a theorem. Now that it has been proven, you can use it in future proofs without proving it again.
Concept Problem Revisited
How many lines exist that are parallel to \begin{align*}l\end{align*}
Common sense should tell you that there is only one line through \begin{align*}P\end{align*}
Interestingly, for hundreds of years people have attempted to prove this statement from simpler statements with no luck. Eventually, it was accepted that it simply must be a postulate, a statement that is assumed to be true without proof. This is known as the parallel postulate.
Vocabulary
The parallel postulate states that given a line and a point, there is exactly one line through the point parallel to the line.
Vertical angles are opposite angles created by intersecting lines. A theorem states that vertical angles are always congruent.
A transversal is a line that intersects multiple other lines.
The transitive property states that if two objects are equal/congruent to the same object, they are equal/congruent to each other.
In the diagram below, \begin{align*}\angle 3\end{align*}
In the diagram below, \begin{align*}\angle 1\end{align*}
In the diagram below, \begin{align*}\angle 4\end{align*}
In the diagram below, \begin{align*}\angle 1\end{align*}
In the diagram below, \begin{align*}\angle 2\end{align*}
Two angles are complementary if the sum of their measures is \begin{align*}90^\circ\end{align*}
Two angles are supplementary if the sum of their measures is \begin{align*}180^\circ\end{align*}.
A postulate is a statement that is assumed to be true without proof.
A theorem is a true statement that must/can be proven.
A proof is a mathematical argument that shows step by step why a statement must be true. All proofs must contain statements and reasons.
A paragraph proof is a proof that is written out in words/sentences.
A two-column proof organizes statements and reasons into columns.
A flow diagram proof organizes statements in boxes with reasons underneath. Arrows show the flow of logic from the original assumptions and given statements to the conclusion.
The reflexive property states that anything is congruent to itself.
CPCTC is an abbreviation for “corresponding parts of congruent triangles are congruent”. It is used to show that two angles or line segments are congruent after it has been shown that two triangles are congruent.
The converse of a statement switches the “if” part of the statement (known as the hypothesis) with the “then” part of the statement (known as the conclusion).
Guided Practice
1. If corresponding angles are congruent, then are lines parallel?
2. Prove that if lines are parallel, then same side interior angles (such as \begin{align*}\angle 3\end{align*} and \begin{align*}\angle 6\end{align*}) are supplementary.
3. Prove that if alternate interior angles are congruent, then lines are parallel.
Answers:
1. This is known as the converse of the corresponding angles postulate. The original postulate said:
Original: If lines are parallel, then corresponding angles are congruent.
Here, the “if” part of the statement (known as the hypothesis) is switched with the “then” part of the statement (known as the conclusion).
Converse: If corresponding angles are congruent, then lines are parallel.
In general, just because a statement is true doesn't necessarily mean its converse is true. In this case, the converse does happen to be true. The only way for corresponding angles to be congruent is for the lines to be parallel. The corresponding angles converse is also a postulate, which means it is accepted as true without proof.
2. In general you can use any style of proof that you prefer. Here, use a paragraph proof.
\begin{align*}\angle 3 \cong \angle 5\end{align*} because they are alternate interior angles created by parallel lines and alternate interior angles are congruent when lines are parallel. \begin{align*}m\angle 3 = m\angle 5\end{align*} because congruent angles have the same measure. \begin{align*}m\angle 5+m\angle 6=180^\circ\end{align*} because two angles that form a line are supplementary. By substitution, \begin{align*}m\angle 3+m\angle 6=180^\circ\end{align*}. \begin{align*}\angle 3\end{align*} and \begin{align*}\angle 6\end{align*} are supplementary because two angles with measures that add to \begin{align*}180^\circ\end{align*} are supplementary.
The statement “if two parallel lines are cut by a transversal, then same side interior angles are supplementary” is a theorem. Now that it has been proven, you can use it in future proofs without proving it again.
3. This is the converse of the alternate interior angles theorem.
Original: If lines are parallel, then alternate interior angles are congruent.
Here, the “if” part of the statement (known as the hypothesis) is switched with the “then” part of the statement (known as the conclusion).
Converse: If alternate interior angles are congruent, then lines are parallel.
To prove this statement, start with a picture of alternate interior angles that are assumed to be congruent, but don't assume the lines are parallel. In the picture below, assume \begin{align*}\angle 1 \cong \angle 2\end{align*}. Prove that \begin{align*}m \ \| \ n\end{align*} (two parallel bars indicate parallel lines).
Statements |
Reasons |
\begin{align*}\angle 1 \cong \angle 2\end{align*} |
Given |
\begin{align*}\angle 1 \cong \angle 3\end{align*} |
Vertical angles are congruent |
\begin{align*}\angle 2 \cong \angle 3\end{align*} |
Transitive property of congruence |
\begin{align*}m \ \| \ n\end{align*} |
If corresponding angles are congruent then lines are parallel. |
Practice
1. In Example A, the theorem “vertical angles are congruent” was proved with a two-column proof. Rewrite this proof in a paragraph format.
2. In Example B, the theorem “if lines are parallel then alternate interior angles are congruent” was proved with a paragraph proof. Rewrite this proof with a flow diagram.
3. In Example C, the theorem “points on a perpendicular bisector of a line segment are equidistant from the segment's endpoints” was proved with a flow diagram. Rewrite this proof in a two-column format.
4. In Guided Practice #2, the theorem “if lines are parallel then same side interior angles are supplementary” was proved with a paragraph proof. Rewrite this proof in a two-column format.
5. In Guided Practice #3, the theorem “if alternate interior angles are congruent then lines are parallel” was proved with a two-column proof. Rewrite this proof with a flow diagram.
6. Alternate exterior angles are outside a pair of lines and on opposite sides of a transversal. \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 8\end{align*} are an example of alternate exterior angles. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 7\end{align*} are another example of alternate exterior angles.
The theorem “if lines are parallel then alternate exterior angles are congruent” is partially proved below. Fill in the blanks to complete the proof. Note, the angles referenced are from the above picture.
Statements |
Reasons |
Two parallel lines are cut by a transversal |
_________ |
\begin{align*}\angle 2 \cong \angle 6\end{align*} |
_________ |
\begin{align*}\angle 6 \cong\end{align*}_________ |
Vertical angles are congruent. |
_________ |
Transitive property. |
7. What is the converse of the theorem “if lines are parallel, then same side interior angles are supplementary”?
8. Prove the converse that you wrote in #7. Use any style of proof you prefer. Hint: Look at Guided Practice #3 for help.
9. In Example C, the theorem “points on a perpendicular bisector of a line segment are equidistant from the segment's endpoints” was proved. This theorem could be rewritten as “if a point is on the perpendicular bisector of a line segment, then the point is equidistant from the endpoints of the line segment”. What is the converse of this theorem? To check your answer, look at #10.
10. The converse of the theorem in #9 is “if a point is equidistant from the endpoints of a line segment, then the point is on the perpendicular bisector of the line segment”. To prove this new theorem you can use the picture below.
Assume point \begin{align*}C\end{align*} is a random point that is equidistant from endpoints \begin{align*}A\end{align*} and \begin{align*}B\end{align*}. Point \begin{align*}D\end{align*} is the midpoint of line segment \begin{align*}\overline{AB}\end{align*}. Your goal is to show that \begin{align*}\overline{CD}\end{align*} must be perpendicular to \begin{align*}\overline{AB}\end{align*} \begin{align*}(\overline{CD} \perp \overline{AB})\end{align*}. This new theorem is partially proved below. Fill in the blanks to complete the proof.
In 11-13, you will prove that if two angles are complementary to the same angle, then the two angles are congruent.
11. Draw a generic picture of this situation and label the three angles.
12. What are the “givens” from your picture? What are you trying to prove?
13. Write a proof of the statement using whatever proof style you prefer.
14. Using your work from 11-13 to help, prove that if two angles are supplementary to the same angle, then the two angles are congruent.
15. Give at least 3 methods for proving that lines are parallel.