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Geometry Proofs Using Congruency





 



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Congruency is used often in proving statements in geometry. These geometric proofs can vary in complexity from simple statements such as an angle being congruent to itself to more advanced concepts. The basic idea behind doing a proof in geometry is to start with a given statement and using various rules of geometry in a logical sequence to prove a concluding statement. Each rule is a link in the chain of logic that connects the given statement to the conclusion.

When doing a two-column proof, the statement is placed in the left hand column while the rule justifying the statement is placed in the right hand column. Many proofs using congruency are used to show that geometric figures such as triangles, quadrilaterals and polygons are congruent.

Proof Involving Three Angles

The following proof shows that two angles are congruent when each is adjacent and supplementary to one of two given congruent angles the other of which is included as a part.

Two angles are congruent if they are supplementary to the same angle or congruent angles.

Given that angle.CAB congruent to angle.DAE.

Prove that angle.DAB congruent to angle.CAE.

Statement Reason
1. angle.CAB congruent to angle.DAE 1. Given
2. mangle.CAB congruent to mangle.DAE 2. Definition of Congruent Angles
3. mangle.CAB + mangle.CAD = mangle.DAE + mangle.CAD 3. Addition
4. mangle.CAB + mangle.CAD = mangle.DAB
    mangle.DAE + mangle.CAD = mangle.CAE
4. Angle Addition Postulate
5. mangle.DAB = mangle.CAE5. Substitution
6. angle.DAB congruent to angle.CAE 6. Definition of Congruent Angles

Proof Involving A Transversal With Three Parallel Lines

When two parallel lines are cut by a transversal, the corresponding angles as well as the alternate interior and exterior angles are congruent.

This can be extended to three parallel lines.

The congruence properties of two parallel lines cut by a transversal can be extended to three parallel lines.

Given: j || k and k || l (j, k and l parallel to each other), transversal m

Prove: angle.1 congruent to angle.4

StatementReason
1. j || k (j parallel to k)1. Given
2. angle.1 congruent to angle.22. Corresponding Angles of a transversal are congruent.
3. angle.2 congruent to angle.33. Vertical Angles are Congruent.
4. k || l4. Given
5. angle.3 congruent to angle.45. Corresponding Angles of a Transversal are Congruent.
6. angle.1 congruent to angle.36. Substitution
7. angle.1 congruent to angle.47. Substitution

Since congruency is transitive, it is a possible to combine steps 6 and 7 into one substitution leaving the final statement as angle.1 congruent to angle.4.

Geometry Proofs For Congruent Triangles

When doing congruency proofs for triangles the three main congruency postulates are used, namely, the SSS, SAS and ASA postulates.

AAS or Angle - Angle - Side can be used indirectly using the sum of angles in a triangle property.

SSS Postulate

Showing that three sides of one triangle are congruent to another proves that both triangles are congruent.

Given: AB congruent to CD and E is a midpoint for AD and BC.

Prove: triangle. ABE congruent to triangle. CED

Diagram for a geometry proof involving sss congruent triangles.
Reason
1. AB congruent to CD and E is a midpoint for AD and BC.1. Given
2. AE congruent to ED, BE congruent to EC2. Definition of Midpoint
3. triangle. ABE congruent to triangle. CED 3. SSS

Since angle.ABE congruent to angle.CED are vertical angles and therefore congruent, it is also possible to use the SAS postulate.

SAS Postulate

When using SAS it is important to remember that the angle is included between two sides of the triangle. For this reason the letters SAS and ASA are written in exact order to indicate the corresponding parts being compared. It is also important to distinguish from SSA which does not prove triangles to be congruent.

Given: BC congruent to CD and AC perpendicular to BD

Prove: triangle. ABE congruent to triangle. ACD

Diagram for a geometry proof involving sas congruent triangles.
StatementReason
1. AC perpendicular to BD
    BC congruent to CD
1. Given
2. angle.1, angle.2 are Right Angles2. Perpendicular Line Segments that Intersect form Right Angles.
3. BC congruent to CD 3. Right Angles are Congruent.
4. AC congruent to AC4. Reflexive Property of Congruency
5. triangle. ABC congruent to triangle. ACD 5. SAS

ASA Postulate

If two angles with an included side is congruent to the corresponding two angles and included side of another triangle then the two triangles are congruent.

Given: angle.A congruent to angle.F, AC congruent to DF and angle.EDA congruent to angle.BCF, line m.

Prove: triangle. ABC congruent to triangle. ACD

Diagram for a geometry proof involving asa congruent triangles.
StatementReason
1. angle.A congruent to angle.F, AC congruent to DF and angle.EDA congruent to angle.BCF, line m 1. Given
2. angle.EDA congruent to angle.EDF are supplementary
    angle.EDA congruent to angle.EDF
2. Definition of Supplementary Angle
3. angle.BCA congruent to angle.EDF 3. If two angles are congruent then their supplementary angles are congruent.
4. triangle. ABC congruent to triangle. DEF 4. ASA

AAS Theorem

Given: angle.A congruent to angle.F, angle.B congruent to angle.D, BC congruent to DE

Prove: triangle. ABC congruent to triangle. DEF

Diagram for a geometry proof involving aas congruent triangles.
    StatementReason
1. angle.A congruent to angle.F, angle.B congruent to angle.D, BC congruent to DE1. Given
2. angle.C congruent to angle.F 2. If two angles in one triangle are congruent to two angles in another then the third angle in each triangle are congruent.
3. triangle. ABC congruent to triangle. DEF 3. ASA

Midsegment Theorem

A midsegment of a triangle is parallel to the side not including the any of the endpoints and is half the length of that side.

Diagram for a geometry proof involving the midsegment of a triangle.

Given: triangle.ABC, midsegment DE

Prove: DE || BC, DE = 1/2BC

The midsegment is extended by a line segment equal to its length.
StatementReason
1. triangle. ABC, midsegment DE1. Given.
2. line DF 2. Construction, 2 points determine a line.
3. Point F, DE = EF3. Construction, Ruler Postulate
4. DE congruent to EF 4. Line segments with equal measure are congruent.
5. angle.AED congruent to angle.FEC5. Vertical angles are congruent
6. AE = EC6. Definition of midpoint.
7. AE congruent to EC7. Definition of Congruent Line Segments (Congruent line segments have equal measure)
8. triangle. AED congruent to triangle. FEC 8. SAS
9. BD = DA9. Definition of midpoint.
10. DA congruent to CF10. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
11. BD = CF11. Substitution
12. angle.ADE congruent to angle.EFC 12. CPCTC
13. BA congruent to CF13. Two line segments that form alternate interior angles are congruent.
14. BD congruent to CF14. BA contains BD
15. BCFD is a parallelogram15. A quadrilateral is a parallelogram if opposite sides are parallel and congruent.
16. DF || BC and DF = BC16. Definition of a Parallelogram.
17. DE || BC 17. DF contains DE
18. DE = DF18. Definition of Midpoint.
19. DE = BC19. Substitution.

Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angles opposite these sides are congruent.

Isosceles triangle diagram for a geometry proof.

This proof involves a correspondence of ABC ↔ ACB as if there were two separate isosceles triangles.

Given: triangle. ABC , AB congruent to AC

Prove: angle.B congruent to angle.C

StatementReason
1. AB congruent to AC 1. Given
2. angle.A congruent to angle.A2. Reflexive Property
3. triangle. ABC congruent to triangle. ACB 3. SAS
4. angle.B congruent to angle.C 4. CPCTC
 




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