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Introduction to Proofs Structure and Strategy of Writing Proofs Key Ideas A proof uses a set of definitions, theorems, and postulates to explain why a statement must be true. There must be a logical connection between each statement or sentence in the proof and the next. Making use of figures is an excellent way of planning your strategy when writing a proof. What Is a Proof? A proof is a logical argument that explains why a statement is true. It is the foundation of geometry. By starting with only a small number of accepted postulates, we can prove all other theorems in geometry. When making a logical argument, we have the following tools at our disposal: definitions, postulates, and theorems. POSTULATES AND THEOREMS A postulate is an obvious statement of fact, such as “only one line can be drawn through two points on a plane.” It is such a basic truth that we don’t have a satisfactory way of proving it. A theorem is a statement that has been logically proven. Once we accept the proof of a theorem, we can use that theorem in proofs of other theorems. For example, we may use the angle sum theorem, which states the sum of the angles in a triangle equals 180˚. Since it is an accepted and proven theorem, we do not need to prove it again when using it in another proof. Two-Column Versus Paragraph Format The two proof formats we will use in this book are the two-column proof and the paragraph proof. The two columns of a two-column proof are labeled “Statements” and “Reasons.” In the statements column, we start with given information that was provided and assumed to be true. We then continue making statements that are true based on logical reasoning and the previous statements. Each statement is justified with a reason.
The reasons can be:
A paragraph proof includes the same information and degree of logical reasoning as a two-column proof. It is just presented in a more conversational tone. If you had to explain why some theorem is true to a friend, what you would say is what might be written in a paragraph proof.
As a paragraph proof, we might write:
bisects at because midpoints divide segments into two congruent segments.”
Here are some general strategies for writing proofs: Make and label a sketch. If a sketch is not provided, draw your own. Then label the sketch with any given information and any information you can infer from the sketch. Make a plan. The most concise and elegant proofs start with an outline. Think about the flow. Sometimes the order of statements matters. For example, you may not be able to conclude that two segments are congruent until you have stated a certain point is a midpoint. Don’t mix statements and reasons. Anything specific to the problem, such as named points, angles, segments, triangles, and so on, belong with the statements. Reasons should be information given in the problem, definitions, postulates, or theorems. Do not use circular reasoning—you cannot use what you are trying to prove as part of the proof. Also, do not repeat statements as reasons. Number your statements and reasons. It’s not over until it’s over. There is no predetermined number of statements and reasons needed for a proof. Your last statement should be what you want to prove. Ask yourself, “How do I know that is true?” If you don’t know why a statement is true, it might not be.
Using Key Idea Midpoints, Bisectors, and Perpendicular Lines The following table lists some of the elementary geometric relationships and how they might be used in a proof.
Using Midpoints and Bisectors Midpoints and bisectors are usually used to show that two segments or angles are congruent. Keep in mind the fact that the bisector points to the midpoint and the midpoint divides the segment into congruent segments. So if you are given a bisector, two separate statements are required to conclude that two segments are congruent. Example 1 bisects at C Solution: Step 1. It is usually a good idea to make a sketch if one is not provided. Then mark the sketch with congruent or other markings. into congruent segments and . Step 3. Write your statements and reasons, numbering each statement and reason. Remember that anything specific to the problem goes in the statement column. Only definitions, postulates, and theorems can be reasons.
Math Fact If a statement relies on some other fact, then the other fact must be presented first as a statement. For example in the previous example, this is the chain of conditionals in the reasons column: If a segment is a bisector, then it intersects a midpoint. If a point is a midpoint, then it divides a segment into two congruent segments. We start by establishing the hypothesis of the first conditional and end with the conclusion of the second conditional. We cannot switch the order of statements. When writing proofs involving perpendicular lines, keep in mind that the definition of perpendicular lines is “lines intersecting at right angles” and the definition of a right angle is “an angle measuring 90°.”
Therefore, the flow of logic in a proof would be: Example 2 Given: Line m ⊥ ray p Prove: m∠1 = m∠2
We sometimes need to classify a triangle as a right triangle. Use the definition that a right triangle is one that has a right angle. Example 3 Prove: △ABC is a right triangle, ∠ABC is a right angle because perpendicular lines form right angles. △ABC must be a right triangle, because a triangle with a right angle is a right triangle.
The special segments in triangles—altitude, median, perpendicular bisector, and angle bisector—can all show up in proofs. Angle bisectors and perpendicular bisectors are self-explanatory. For altitudes and medians, simply use the definition of the special segment as the reason to justify the relationship it creates: Altitude—a segment from a vertex perpendicular to the opposite side (or its extension) of a triangle. Median—a segment from a vertex of a triangle to the midpoint of the opposite side. Example 4 Solution:
Example 5 bisects ∠POR Prove: ∠POQ ≅ ∠QOR is given as a bisector of ∠POR. Since a bisector divides an angle into two congruent angles, ∠POQ must be congruent to ∠QOR.
Properties of Equality Besides the postulates we have already seen, the following postulates are useful tools when working out a proof. They show up frequently and you should be familiar with them.
Substitution and Transitive Properties of Equality Substitution Property of Equality—Any quantity can be substituted for an equal quantity. Transitive Property of Equality—Given quantities a, b, and c, if a = b and b = c, then a = c. The substitution and transitive properties of equality are useful when we have an indirect relationship between three different figures or quantities. It states that if two quantities are both equal to a third quantity, then the first two quantities are equal to each other. In a proof, the equality would be the statement and “substitution property” or “transitive property” would be the reason. Example Given: G is the midpoint of Prove: Solution: Label the figure with congruent markings. The midpoint tells us that . Combining that with the given information allows us to use the transitive property to link to .
Multiplication and Division Postulates The Multiplicative Property of Equality—If equal quantities are multiplied by equal quantities, then the products are equal. The Division Property of Equality—If equal quantities are divided by equal quantities, then the quotients are equal, provided the divisors are not equal to zero. In terms of quantities a, b, c, and d where a = b and c = d (with c ≠ 0 and d ≠ 0), we have:
There is a special case for the division property of equality when dividing by 2: halves of equals quantities are equal. Example Given: ∠RST is bisected by , m∠RST = (4x + 12)° Prove: m∠TSX = (2x + 6)° Solution:
In the previous example, the reason for line 6 could also have been substitution instead of the transitive property of equality. The reason for line 5 is an algebraic simplification. The algebraic operation was applying the distributive property to the factor of . We will take the shortcut of not justifying every step of an algebraic simplification unless the steps are not obvious. Addition, Subtraction, and Partition Postulates Addition/Subtraction Property of Equality If equal quantities are added (or subtracted) from equal quantities, then the sums (or differences) are equal. In terms of quantities a, b, c, and d, if a = b and c = d, then: The addition postulate states that if equal quantities are added to equal quantities, then the sums are equal. Using angle measures, for example, we might have:
This looks a lot like an algebraic system of equations. When we use these postulates in a proof, the proof will be easier to read if the two lines to be added follow one another. The quantities to be added can be segment measures, angle measures, or the figures themselves. When using the addition postulate with figures instead of measures, we often follow it up with the partition postulate of equality/congruence. The partition postulate states that a whole figure can be broken up into a sum of consecutive parts or that a sequence of consecutive parts is equivalent to a single whole figure. The segment would have to be collinear with a common endpoint, and the angles would have to be adjacent. Partition Postulate If points A, B, and C are collinear with B between A and C: AB + BC = AC If ∠ABD and ∠CBD are adjacent angles: m∠ABD + m∠CBD = m∠ABC ∠ABD + ∠CBD ≅ ∠ABC The addition symbol in the congruence statements implies the joining of two figures into a single figure, not an algebraic addition. Figure below shows the application of the partition postulate to and ∠ABC. Partition postulate applied to segments and angles Example 1 Given: Prove: DE + BC = AC Solution:
Example 2 Given: bisects ∠SRA, ∠S ≅ ∠A Prove: m∠S + m∠SRT = m∠A + m∠ART Solution: Strategy—Use the definition of angle bisector to show that m∠SRT = m∠ART
Reflexive Property of Equality Reflexive Property of Equality/Congruence Any quantity equal to itself. Any figure congruent to itself. This seems like an obvious statement. However, it is one that needs to be formally stated and justified in proofs. Example Given: and Prove: Strategy—We see from the figure that is comprised of and that is comprised of .
We want to establish two separate equalities and sum them to get . The first equality is given, but the second equality requires the use of the reflexive property.
Using Vertical Angles, Linear Pairs, and Complementary and Supplementary Angles Key Ideas Vertical angles, linear pairs, complementary angles, and supplementary angles are some of the angle relationships that are the building blocks of proofs. We can often infer these relationships directly from a figure without additional given information other than the existence of straight lines and right angles.
Linear Pairs and Supplementary Angles Linear pairs and vertical angles are two angle relationships that can be inferred from a figure. The angles are usually not specifically stated to be linear pairs, but we can determine that they are from the figure. Linear pairs let us conclude that a pair of adjacent angles are supplementary. Combining the definition of complementary or supplementary angles with the transitive property gives the following useful theorem: Complements/Supplements of Congruent Angles Theorem If two angles are supplementary or complementary to the same (or congruent) angles, then those first two angles are congruent. Example 1 and intersect at point O Prove: ∠AOC ≅ ∠BOD Solution: Strategy—Apply the transitive property of equality to supplementary linear pairs ∠AOC and ∠BOC and to ∠BOD and ∠BOC. We are not going to use the vertical angle theorem since that is what we are trying to prove.
Example 2 , ∠1 ≅ ∠3 Prove: ∠2 ≅ ∠4 Solution:
Vertical Angles As with linear pairs, given information in a proof will usually not include vertical angles. We may be given only the fact that two straight lines intersect. From that we can conclude that the opposite angles formed are congruent, vertical angles. Example Prove: ∠BAC ≅ ∠ACB Solution:
Complementary Angles The measures of complementary angles sum to 90°. Therefore, any right angle partitioned into two or more angles will represent complementary angles. Example Prove: ∠BPC and ∠DPE are complementary Solution: Strategy—show that the measures of ∠APD and ∠DPE sum to 90°, and substitute vertical angle ∠BPC for ∠APD.
Math Fact In the previous example. Without knowing is a straight line, we could not conclude that ∠BPC and ∠APD are vertical angles.
Using Parallel Lines Key Ideas If we are given parallel lines intersected by a transversal, we can conclude that the alternate interior angles are congruent, the corresponding angles are congruent, and the same side interior angles are congruent. We can also use the converses of these statements to prove that two lines are parallel.
Proving Angle Relationships Given Parallel Lines We can use the relationships between angles formed by parallel lines to prove angles are congruent or supplementary. Look for the transversal and pairs of alternate interior, corresponding, or same side interior angles. If there is more than one transversal, be careful that you work with pairs of angles formed by the same transversal. Example 1 Given: Line m || line n Prove: ∠1 and ∠3 are supplementary Solution: ∠1 and ∠2 are a linear pair, which are supplementary. ∠2 ≅ ∠3 because they are corresponding angles formed by parallel lines. Substituting ∠3 for ∠2 in the first statement gives ∠1 supplementary to ∠3. Example 2 Given: Line n || line p, Prove: ∠1 and ∠2 are supplementary
Proving Lines Are Parallel Two lines can be proven to be parallel by showing that any pair of alternate interior angles are congruent, of same side interior angles are supplementary, or of corresponding angles are congruent. You need to demonstrate the relationship for only one pair of angles. Example 1 Given: ∠1 ≅ ∠3 and ∠2 ≅ ∠4 Prove: p || q Solution: Strategy—show that the same side interior angles are supplementary.
Example 2 Given: Prove:
Solution:
Using Triangle Relationships Key Ideas The triangle angle sum theorem, exterior angle theorem, and isosceles triangle theorems can often be applied in proofs involving triangles to justify angle and segment relationships.
Exterior Angle Theorem When a side of a triangle is extended, we can state that the measure of the exterior angle equals the sum of the two nonadjacent interior angles. The reason would be “triangle exterior angle theorem.” Example Given: △ABC with side extended Prove: m∠1 − m∠2 = m∠4 − m∠3 Solution: Strategy—Apply the external triangle angle theorem to both ∠1 and ∠2. Then subtract the two relationships.
Triangle Angle Sum Theorem We can state that the sum of the measures of the angles of a triangle equals 180° using the “triangle angle sum theorem” for the reason. Example 1 Given: ∠1 ≅ ∠4 and ∠3 ≅ ∠6 Prove: bisects ∠ACB Solution:
Example 2 Prove the triangle exterior angle theorem. Given: △ABC and side extended to D Prove: m∠3 + m∠4 = m∠1 Solution: Strategy—∠1 and ∠2 are supplementary linear pairs. ∠2, ∠3, ∠4 are the interior angles of a triangle and sum to 180°. Combining these leads to the exterior angle theorem.
Example 3 Prove the triangle angle sum theorem
Given: △ABC Prove: the sum of the measures of the interior angles in △ABC equals 180˚ Solution: The strategy is to construct line through point C and parallel to . We know the sum of angles 3, 4, and 5 must be 180˚ since they comprise a straight line. Using our angle relationships in parallel lines cut by a transversal, we find two pairs of congruent angles, ∠4 ≅ ∠1 and ∠5 ≅ ∠2. Substitution yields the desired relationship.
The two-column proof is as follows.
Isosceles Triangle Theorem Whenever you have two congruent sides in a triangle, you can conclude that the opposite angles are congruent by the isosceles triangle theorem. The converse is true as well. Example 1 Given: Prove: ∠1 ≅ ∠3 Solution:
Example 2 Given: Prove: ∠BCD ≅ ∠BDC
To prove a triangle is isosceles, you need to prove that two sides are congruent. Sometimes it is easier to prove two angles are congruent and then use the converse of the isosceles angle theorem to show the opposite sides are congruent. Example Given: bisects ∠CBD Prove: △ABD is isosceles
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