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Wize High School Geometry Textbook (Common Core) > Reasoning and Proofs

Theorems and Proofs

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Theorems and Proofs

Now its time to dig into why things work the way they do in Geometry. For this we'll be making poofs and constructions to build theorems.

Theorem

A theorem is a statement that, using a series of logical steps, we can show to be true. In Geometry we might start with postulates, which are assumed to be true, or previously proven theorems in our steps.

Hypothesis and Conclusion

Often statements that need to be proven will have two parts and follow an If/Then structure. The given information is called the hypothesis and the result is called the conclusion. It is our job to fill in the logical pieces in between.

Proofs

One useful way to keep track of the information we know, and the reasons that connect them is using a two column proof. In this structure the information we know is keep on the left side, and the reasoning is written out on the right. Note that proofs are not always written in this way, but this method does provide an organized why to get started.

Example
Prove the following statement using a two column proof.
Given that ∠ABC\angle ABC is congruent to ∠ACB\angle ACB, prove that ∠ABC\angle ABC and ∠ACD\angle ACD are supplementary.
ANSWER:

StatementReasoning∠ABC≅∠ACBGiven∠BCD is a straight angleGiven in diagramm∠ACB+m∠ACD=m∠BCDAddition of angles postulatem∠ACB+m∠ACD=180Measure of straight anglem∠ABC+m∠ACD=180 because congruent angles have the same measure∠ABC and ∠ACD are supplementaryDefinition of supplementary angles\begin{array}{c|c} Statement & Reasoning \\ \hline \\ \angle ABC \cong \angle ACB & \text{Given} \\ \angle BCD \text{ is a straight angle} & \text{Given in diagram} \\ m\angle ACB + m\angle ACD = m\angle BCD & \text{Addition of angles postulate} \\ m\angle ACB + m\angle ACD = 180 & \text{Measure of straight angle} \\ m\angle ABC + m\angle ACD = 180 & \text{ because congruent angles have the same measure} \\ \angle ABC \text{ and } \angle ACD \text{ are supplementary} & \text{Definition of supplementary angles} \\ \hline \end{array}



Theorems about Segments and Angles

  • Every segment is congruent to itself. ∠A≅∠A\angle A \cong \angle A
  • If ∠A≅∠B\angle A \cong \angle B then ∠B≅∠A\angle B \cong \angle A
  • If AB‾≅CD‾\overline{AB} \cong \overline{CD} and CD‾≅EF‾\overline{CD} \cong \overline{EF} then AB‾≅EF‾\overline{AB} \cong \overline{EF}
  • Every angle is congruent to itself. ∠A≅∠A\angle A \cong \angle A
  • If∠A≅∠B\angle A \cong \angle B then ∠B≅∠A\angle B \cong \angle A
  • If ∠A≅∠B\angle A \cong \angle B and ∠B≅∠A\angle B \cong \angle A then ∠A≅∠C\angle A \cong \angle C

Useful Postulates and Theorems

  • Linear pair postulate - If two angles form a linear pair, then the are supplementary.
  • Right Angles Congruent Theorem - All right angles are congruent.
  • Congruent Supplements Theorem - If two angles are supplementary to a third angle, then they are congruent.
  • Congruent Complements Theorem - If two angles are both complementary to a third angle, then they are congruent.
  • Vertical Angles Congruence Theorem - Vertical angles are congruent

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Writing a proof


The key for writing a good proof is understanding the connections among the statements, or reasoning.


Use the diagram to write a proof that angle ∠AEB≅∠CED\angle{AEB} \cong \angle{CED}.
It can be assumed that
  • ∠AEC\angle{AEC} is a right angle.
  • ∠BED\angle{BED} is a right angle.
ANSWER:
StatementReasoning∠AEC is a right angle  Given ∠BED is a right angle  ∠AEB and ∠BEC are complementary Definition of complementary angles∠BEC and ∠CED are complementary Definition of complementary angles∠AEB≅∠CEDCongruent complements theorem\begin{array}{c|c} Statement & Reasoning \\ \hline \\ \angle{AEC} \text{ is a right angle } & \text{ Given } \\ \angle{BED} \text{ is a right angle } & \text{ } \\ \\ \angle{AEB} \text{ and } \angle{BEC} \text{ are complementary} & \text { Definition of complementary angles} \\ \\ \angle{BEC} \text{ and } \angle{CED} \text{ are complementary} & \text { Definition of complementary angles} \\ \\ \angle{AEB} \cong \angle{CED} & \text{Congruent complements theorem} \\ \\ \hline \end{array}


Matching


Match the name of each postulate or theorem to its statement.

A.
If two angles form a linear pair, then they are supplementary.
B.
Vertical angles are congruent.
C.
If two angles are both supplementary to a third angle, then they are congruent.
D.
Right angles are congruent.
Linear pair postulate
Vertical angle congruent theorem
Right angles congruence theorem
Congruent supplements theorem

Complete the proof

One important theorem is that pairs of vertical angles are congruent. Complete the proof of this theorem by filling in the missing reasoning.


StatementReasoning∠1 and ∠2 are vertical angles  ? ∠1 and ∠3 are a linear pair ? ∠3 and ∠2 are a linear pair ∠1 and ∠3 are supplementary  ? ∠3 and ∠2 are supplementary ∠1≅∠2 ? \begin{array}{c|c} Statement & Reasoning \\ \hline \\ \angle{1} \text{ and } \angle{2} \text{ are vertical angles } & \text{ ? } \\ \\ \angle{1} \text{ and } \angle{3} \text{ are a linear pair} & \text{ ? } \\ \angle{3} \text{ and } \angle{2} \text{ are a linear pair} & \text{ } \\ \\ \angle{1} \text{ and } \angle{3} \text{ are supplementary } & \text{ ? } \\ \angle{3} \text{ and } \angle{2} \text{ are supplementary} & \text{ } \\ \\ \angle{1} \cong \angle{2} & \text{ ? } \\ \\ \hline \end{array}
A.
Congruent supplements theorem
B.
Given
C.
Definition of a linear pair
D.
Linear pair postulate
∠1 and ∠2 are vertical angles \angle{1} \text{ and } \angle{2} \text{ are vertical angles }
∠1 and ∠3 are a linear pair ∠3 and ∠2 are a linear pair \angle{1} \text{ and } \angle{3} \text{ are a linear pair } \\ \angle{3} \text{ and } \angle{2} \text{ are a linear pair }
∠1 and ∠3 are supplementary ∠3 and ∠2 are supplementary \angle{1} \text{ and } \angle{3} \text{ are supplementary } \\ \angle{3} \text{ and } \angle{2} \text{ are supplementary }
∠1≅∠2\angle1\cong\angle2
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Proof about congruent angles


Write a proof that argues that two angles that are complementary to the same angle, are congruent to each other.


Specifically the proof should start with the given information:
  • ∠1\angle{1} and ∠2\angle{2} are complementary
  • ∠3\angle{3}and ∠2\angle{2} are complementary
and conclude that
  • ∠1≅∠3\angle{1} \cong \angle{3}
Don't forget to use definitions, postulates, or known theorems in your reasoning.