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More Lessons for High School Geometry

Math Worksheets

A series of free, online High School Geometry Videos and Lessons.

Examples, solutions, videos, worksheets, and activities to help Geometry students.

In this lesson, we will learn

- how to use CPCTC
- how to use two column proofs
- how to use flowchart proofs
- how to use paragraph proofs
- how to use special isosceles triangle properties

CPCTC is an acronym for corresponding parts of congruent triangles are congruent. CPCTC is commonly used at or near the end of a proof which asks the student to show that two angles or two sides are congruent. It means that once two triangles are proven to be congruent, then the three pairs of sides that correspond must be congruent and the three pairs of angles that correspond must be congruent.

How to use the principle that corresponding parts of congruent triangles are congruent, or CPCTC?

Two column proofs are organized into statement and reason columns. Each statement must be justified in the reason column. Before beginning a two column proof, start by working backwards from the “prove” or “show” statement. The reason column will typically include “given”, vocabulary definitions, conjectures, and theorems.

How to organize a two column proof.

A brief lesson and practice on drawing diagrams and completing two column proofs from word problems

Flowchart proofs are organized with boxes and arrows; each “statement” is inside the box and each “reason” is underneath each box. Each statement in a proof allows another subsequent statement to be made. In flowchart proofs, this progression is shown through arrows. Flowchart proofs are useful because it allows the reader to see how each statement leads to the conclusion.

How to outline a flowchart proof?

Using flowcharts in proofs for Geometry

A paragraph proof is like a two-column proof written in sentences. Paragraph proofs need to be written in a chronological order showing how each statement supported by a definition, postulate, or theorem, leads to the next statement being true.

In isosceles (and equilateral) triangles, a segment drawn from the vertex angle to the opposite side is the altitude, angle bisector and median. Isosceles triangle properties are used in many proofs and problems where the student must realize that, for example, an altitude is also a median or an angle bisector to find a missing side or angle.

How to identify a segment from the vertex angle in an isosceles triangle to the opposite side?

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