# Geometry Proofs Using Congruency

Set Theory
Points, Lines, Planes
Angles
Transversals
Planes and Space
Triangles
Polygons
Circles
Tangents and Secants
Chords, Power Point
Proofs
Congruence
Congruent Triangles
Congruent Polygons
Congruency Proofs
Congruent Circles, Arcs
Similarity
Similarity Proofs
Pythagorean Theorem
Symmetry and Transformations
Analytic Geometry
Coordinate Proofs
Solids
Non-Euclidean Geometry
Rules Postulates Theorems
Glossary
Morleys Theorem
Fractals

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.

Given that CAB DAE.

Prove that DAB CAE.

Statement Reason
1. CAB DAE 1. Given
2. mCAB mDAE 2. Definition of Congruent Angles
4. mCAB + mCAD = mDAB
5. mDAB = mCAE5. Substitution
6. DAB 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.

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

Prove: 1 4

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

Since congruency is transitive, it is a possible to combine steps 6 and 7 into one substitution leaving the final statement as 1 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 CD and E is a midpoint for AD and BC.

Prove: ABE CED

Reason
1. AB CD and E is a midpoint for AD and BC.1. Given
2. AE ED, BE EC2. Definition of Midpoint
3. ABE CED 3. SSS

Since ABE 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 CD and AC BD

Prove: ABE ACD

StatementReason
1. AC BD
BC CD
1. Given
2. 1, 2 are Right Angles2. Perpendicular Line Segments that Intersect form Right Angles.
3. BC CD 3. Right Angles are Congruent.
4. AC AC4. Reflexive Property of Congruency
5. ABC 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: A F, AC DF and EDA BCF, line m.

Prove: ABC ACD

StatementReason
1. A F, AC DF and EDA BCF, line m 1. Given
2. EDA EDF are supplementary
EDA EDF
2. Definition of Supplementary Angle
3. BCA EDF 3. If two angles are congruent then their supplementary angles are congruent.
4. ABC DEF 4. ASA

## AAS Theorem

Given: A F, B D, BC DE

Prove: ABC DEF

StatementReason
1. A F, B D, BC DE1. Given
2. C 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. ABC 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.

Given: ABC, midsegment DE

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

StatementReason
1. ABC, midsegment DE1. Given.
2. line DF 2. Construction, 2 points determine a line.
3. Point F, DE = EF3. Construction, Ruler Postulate
4. DE EF 4. Line segments with equal measure are congruent.
5. AED FEC5. Vertical angles are congruent
6. AE = EC6. Definition of midpoint.
7. AE EC7. Definition of Congruent Line Segments (Congruent line segments have equal measure)
8. AED FEC 8. SAS
9. BD = DA9. Definition of midpoint.
10. DA CF10. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
11. BD = CF11. Substitution
13. BA CF13. Two line segments that form alternate interior angles are congruent.
14. BD 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 = ½ DF18. Definition of Midpoint.
19. DE = ½ BC19. Substitution.

## Isosceles Triangle Theorem

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

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

Given: ABC , AB AC

Prove: B C

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

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