Topics
1. Lines and Angles
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United States of America
Grades 9-12

1.05 Direct proof

Lesson
Practice

Introduction

Logic, in combination with definitions, postulates, and theorems, can be used to draw conclusions about geometric figures. We use transformations from 8th grade as a foundation for writing proofs and learn about more formal proofs in geometry.

Informal proof

In mathematics, we have postulates which are accepted facts. If we apply deductive reasoning to postulates and other starting hypotheses, we can construct logical arguments for new facts of mathematics. A new fact is called a theorem and its argument is called its proof.

Theorem

A true statement that follows as a result of other true statements.

Proof

An argument that uses logic to show that some given information leads to the conclusion of the proof goal.

We can write informal proofs in a variety of ways including: narrative paragraphs, bulleted lists, and annotated diagrams.

Examples

Example 1

Consider the vertical angles theorem, which states that vertical angles are congruent.

Use transformations to prove the following:

Given: \angle ABC and \angle DBE are vertical angles

Prove: \angle ABC \cong \angle DBE

Line A E and C D intersecting at point B. Angles A B C and D B E are congruent, as well as angles A B D and C B E.
Worked Solution
Create a strategy

We can use translations, reflections, rotations, and dilations to demonstrate congruence between figures.

Apply the idea

We are given that \angle ABC and \angle DBE are vertical angles. Since \angle ABC and \angle DBE are vertical angles, they share vertex B. We can use tracing paper to rotate \angle ABC clockwise 180\degree around vertex B and verify that its image is \angle DBE. Therefore, \angle ABC \cong \angle DBE.

Line A E and C D intersecting at point B. Angles A B C and D B E are congruent, as well as angles A B D and C B E. Angle A B C is highlighted. A circular arrow labeled 180 degrees, pointing from A B C to D B E.
Reflect and check

This is an example of an informal paragraph proof that uses an annotated diagram.

Idea summary

We can use narrative paragraphs, bulleted lists, and annotated figures to informally write proofs.

Formal proof

Formal proofs have a more rigid structure and can be helpful when we want to present our justification in a very organized way.

Flow chart proof

A type of proof that uses arrows to show the flow of a logical argument. Statements are connected by arrows to show how each statement comes from the ones before it. Each arrow should have a reason with it.

A flow chart proof can have various forms. An example of the flow chart proof structure follows:

A flow chat proof with the statements in symbols inside empty boxes and reason below each box. Starting with 2 empty boxes, Given. For each box, an arrow pointing down the next box, an empty box, justification. For each box, an arrow pointing down the next box, an empty box, justification. An arrow pointing down the next box, an empty box, conclusion.
Two column proof

A type of proof written as numbered rows which have the statement in one column and the reason in the other column. Each statement must follow from a row above or be given.

The structure of a two column proof follows:

To prove: Proof goal
StatementsReasons
1.StatementGiven
2.StatementGiven
3.StatementJustification
4.StatementJustification
5.StatementJustification
6.StatementConclusion

Examples

Example 2

Consider the congruent complements theorem, which states that If two angles are complementary to the same angle then they are congruent.

Construct a flow chart proof to prove the following:

Given:

  • \angle A and \angle C are complementary
  • \angle B and \angle C are complementary

Prove: \angle A \cong \angle B

Two right angles. Each right angle has a ray in the interior of the angle and its endpoint on the angle's vertex. For top right angle, the ray divides the angle into two angles labeled A and C. For bottom right angle, the ray divides the angle into two angles labeled B and C. Angle A and B are congruent, as well as both angle C.
Worked Solution
Apply the idea
A flow chart proof with 5 levels and the reason below each step. The first and second levels have 2 steps. On the first level, the left step is labeled angle A and angle C are complementary, with reason Given, and the right step is labeled angle B and angle C are complementary, with reason Given. On the second level, the left step is labeled measure of angle A plus measure of angle C equals 90 degrees, with reason Definition of complementary angles, and the right step is labeled measure of angle B plus measure of angle C equals 90 degrees, with reason Definition of complementary angles. On the third level, the left and right step from the second level now converge into a single step. The step is labeled measure of angle A plus measure of angle C equals measure of angle B plus measure of angle C, with reason Transitive property of equality. On the fourth level, the step is labeled measure of angle A equals measure of angle B, with reason Addition property of equality. On the fifth level, the step is labeled angle A and angle B are congruent, with reason Definition of congruent angles.
Reflect and check

Using a diagram of the congruent complements theorem, we can informally prove that \angle A \cong \angle B by using translation of one figure to its image to show their congruence.

Example 3

Consider the congruent supplements theorem, which states that if two angles are supplementary to the same angle then they are congruent.

Construct a two column proof to prove the following:

Given:

  • \angle A and \angle C are supplementary
  • \angle B and \angle C are supplementary

Prove: \angle A \cong \angle B

Two straight angles. Each straight angle has a ray in the interior of the angle and its endpoint on the angle's vertex. For top straight angle, the ray divides the angle into two angles labeled A and C. For bottom straight angle, the ray divides the angle into two angles labeled B and C. Angle A and B are congruent, as well as both angle C.
Worked Solution
Apply the idea
To prove: \angle A \cong \angle B
StatementsReasons
1.\angle A and \angle C are supplementaryGiven
2.\angle B and \angle C are supplementaryGiven
3.m\angle A +m\angle C=180\degreeDefinition of supplementary angles
4.m\angle B +m\angle C=180\degreeDefinition of supplementary angles
5.m\angle A +m\angle C=m\angle B +m\angle CTransitive property of equality
6.m\angle A=m\angle BAddition property of equality
7.\angle A \cong \angle BDefinition of congruent angles
Reflect and check

We can see that the proofs for the congruent supplements theorem and the congruent complements theorem are very similar. The flow chart proof and two column proof use the same steps but display them differently.

Idea summary

Flow chart proofs and two column proofs can use the same reasoning with different formats. These are formal proof writing methods.

Outcomes

G.CO.C.9

Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

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