# Chapter 4: Reasoning and Proof

**At Grade**Created by: CK-12

Here you will learn how to write a mathematical proof. You will consider three styles of proofs: paragraph proofs, two-column proofs and flow diagram proofs. You will then practice writing proofs as you discover theorems about lines, angles, triangles, and quadrilaterals. You will also practice applying the theorems that you have proved.

- 4.1.
## Theorems and Proofs

- 4.2.
## Theorems about Lines and Angles

- 4.3.
## Applications of Line and Angle Theorems

- 4.4.
## Theorems about Triangles

- 4.5.
## Theorems about Concurrence in Triangles

- 4.6.
## Applications of Triangle Theorems

- 4.7.
## Theorems about Quadrilaterals

- 4.8.
## Applications of Quadrilateral Theorems

### Chapter Summary

You learned that a proof is a formal explanation for why a statement is true. Three styles of proofs are paragraph proofs, two-column proofs, and flow diagram proofs. You learned that once a theorem is proved, you can use that theorem in future proofs without proving it again. The converse of a theorem switches the hypothesis (the “if” part) with the conclusion (the “then” part). The converse of a theorem is not necessarily also a theorem, but it can be.

You learned theorems about parallel lines and angles. You learned that when lines are parallel, corresponding angles, alternate interior angles, and alternate exterior angles are all congruent. Same side interior and exterior angles are supplementary.

You also learned theorems about triangles. You learned how to prove that the sum of the interior angles of a triangle is \begin{align*}180^\circ\end{align*} and you learned that the measure of an exterior angle of a triangle is the same as the sum of the measures of the remote interior angles. You learned about the four points of concurrency within triangles and special properties that they have.

Finally, you learned about special quadrilaterals. You proved properties of parallelograms, rectangles, rhombuses, and kites based on the definitions of these quadrilaterals. You then used these definitions and properties to help you to solve problems.

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