GeometryChapter 5: Geometric Proofsby: Emily Zogas : GeometryChapter 5: Geometric Proofsby: Emily Zogas What You Will Learn : What You Will Learn What is a Geometric Proof?
Facts to Know before constructing a proof
Congruent Angles that WORK or DO NOT WORK
How to construct a Geometric Proof
Other Terms (call something different)
About the Author
Resources QUIT Slide 3: Geometric Proof-a step-by-step explanation that take you from the given that uses definitions, axioms, postulates, and previously proved theorems to draw a conclusion or proved statement about a geometric statement.
Two types of proofs:
Indirect Proof - A proof in which a statement is shown to be true because the assumption that its negation is true leads to a contradiction.
Direct Proof - A proof in which the conclusion is drawn directly from previous conclusions, starting with the first statement. QUIT Facts to know before constructing a proof : Facts to know before constructing a proof Two-Column Method - A kind of proof in which the statements (conclusions) are listed in one column, and the reasons for each statement's truth are listed in another column. Identical in content, but different in form, from a paragraph proof The basic structure of a proof is a series of statements, each one being either an assumption or a conclusion, clearly following from an assumption or previously proved result. QUIT Congruent Angles: that work : Congruent Angles: that work AAS, Angle Angle Side- Triangles are congruent if two pairs of corresponding angles and a pair of opposite sides are equal in both triangles
ASA, Angle Side Angle- Triangles are congruent if any two angles and their included side are equal in both triangles.
SAS, Side Angle Side- Triangles are congruent if any pair of corresponding sides and their included angles are equal in both triangles.
SSS, Side Side Side- Triangles are congruent if all three sides in one triangle are congruent to the corresponding sides in the other.
HL, Hypotenuse and leg of a right triangle- Two right triangles are congruent if the hypotenuse and one corresponding leg are equal in both triangles. QUIT Congruent Angles that DO NOT WORK : Congruent Angles that DO NOT WORK AAA, Angle Angle Angle- DOES NOT WORK because having all three corresponding angles equal is not enough to prove congruence
It means that just because two triangles have congruent corresponding angles, it does not prove the triangles are congruent SSA, Side Side Angle- DOSE NOT WORK because two sides and a non-included angle is enough to prove congruence.
There are two triangles possible that have the same values but using SSA will not prove congruency. QUIT Steps to construct a Geometric Proof : Steps to construct a Geometric Proof (Click on the picture for a video explanation) You will receive a Given and statement. This is where you will start your proof. There will also be a statement that you have to prove.
Use the two column method.
In the Statements column list the given and list the proved which will be the conclusion or at the bottom of your statements column.
If there is no figure provided then look at all the information that's provided and draw a figure.
List the steps it took to reach the proof. Some of these steps may include isosceles, median of triangle, perpendicular, congruent sides, angles, etc. QUIT Other Terms : Other Terms Isosceles Triangle- A triangle which has two sides that are equal.
Median of triangle- A line joining at a vertex to the midpoint to the opposite side. ex. A triangle has 3 medians.
Perpendicular- Meeting a given line or surface at a right angle.
Postulate - statements assumed to be true
is the symbol for congruent terms.
Angle BAC Angle YXZ. QUIT Example #1Given: Segment AB is congruent to segment CB.Angle BAD is congruent to angle BCD.Prove: Segment AB is congruent to segment CD : Example #1Given: Segment AB is congruent to segment CB.Angle BAD is congruent to angle BCD.Prove: Segment AB is congruent to segment CD QUIT Example #2 : Example #2 Given:
Triangle ABC is an isosceles triangle with vertex angle A
Angles 1 and 2 are congruent QUIT Practice : Practice Given:Angle A Angle D. Angle B Angle E.Prove: triangle ABC congruent to triangle DEF? QUIT Answer : Answer QUIT About the Author : About the Author Emily Zogas
I am a student at Grand Valley State University. I am a Math major and an Earth Science minor. Other things I am involved in at GVSU are, Rowing, V.P. Finance for Spotlight: Your Student Programming Board, and Student Council For Exceptional Students.
Send me an E-mail QUIT Resources : Resources Table of Contents - Math Open Reference
Mastering the Formal Geometry Proof - For Dummies
SparkNotes: Geometric Proofs: Terms
How To Write Proofs
VoiceThread - Group conversations around images, documents, and videos
Congruent Triangles - Free Math Help QUIT