Proving Triangles Congruent
Two geometric figures with exactly the same size and shape. The Idea of a Congruence A C B DE F
How much do you need to know... need to know about two triangles to prove that they are congruent?
In Lesson 4.2, you learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. Corresponding Parts ABC DEF B A C E D F 1.AB DE 2.BC EF 3.AC DF 4. A D 5. B E 6. C F
Do you need all six ? NO ! SSS SAS ASA AAS
Side-Side-Side (SSS) 1. AB DE 2. BC EF 3. AC DF ABC DEF B A C E D F
Side-Angle-Side (SAS) 1. AB DE 2. A D 3. AC DF ABC DEF B A C E D F included angle
The angle between two sides Included Angle G G I I H H
Name the included angle: YE and ES ES and YS YS and YE Included Angle SY E E E S S Y Y
Angle-Side- Angle (ASA) 1. A D 2. AB DE 3. B E ABC DEF B A C E D F include d side
The side between two angles Included Side GI HI GH
Name the included angle: Y and E E and S S and Y Included Side SY E YE ES SY
Angle-Angle-Side (AAS) 1. A D 2. B E 3. BC EF ABC DEF B A C E D F Non-included side
Warning: No SSA Postulate A C B D E F NOT CONGRUENT There is no such thing as an SSA postulate!
Warning: No AAA Postulate A C B D E F There is no such thing as an AAA postulate! NOT CONGRUENT
The Congruence Postulates SSS correspondence ASA correspondence SAS correspondence AAS correspondence SSA correspondence AAA correspondence
Name That Postulate SAS ASA SSS SSA (when possible)
Name That Postulate (when possible) ASA SAS AAA SSA
Name That Postulate (when possible) SAS SAS SAS Reflexive Property Vertical Angles Reflexive Property SSA
CW: Name That Postulate (when possible)
CW: Name That Postulate
Let’s Practice Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: For SAS: B D For AAS: A F A F AC FE
HW Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: For SAS: For AAS:
Statements: 1.BD BC 2.AD ║ EC 3. D C 4. ABD EBC 5.∆ABD ∆EBC Reasons: 1.
Proof: Statements: 1.BD BC 2.AD ║ EC 3. D C 4. ABD EBC 5.∆ABD ∆EBC Reasons: 1. Given
Proof: Statements: 1.BD BC 2.AD ║ EC 3. D C 4. ABD EBC 5.∆ABD ∆EBC Reasons: 1.Given 2.Given
Proof: Statements: 1.BD BC 2.AD ║ EC 3. D C 4. ABD EBC 5.∆ABD ∆EBC Reasons: 1.Given 2.Given 3.Alternate Interior Angles
Proof: Statements: 1.BD BC 2.AD ║ EC 3. D C 4. ABD EBC 5.∆ABD ∆EBC Reasons: 1.Given 2.Given 3.Alternate Interior Angles 4.Vertical Angles Theorem
Proof: Statements: 1.BD BC 2.AD ║ EC 3. D C 4. ABD EBC 5.∆ABD ∆EBC Reasons: 1.Given 2.Given 3.Alternate Interior Angles 4.Vertical Angles Theorem 5.ASA Congruence Theorem
Given: A is the midpoint of MT, A is the midpoint of SR. Prove: MS ║ TR. Statements: 1.A is the midpoint of MT, A is the midpoint of SR. 2.MA ≅ TA, SA ≅ RA 3. MAS ≅ TAR 4.∆MAS ≅ ∆TAR 5. M ≅ T 6.MS ║ TR Reasons: 1.Given 2.Definition of a midpoint 3.Vertical Angles Theorem 4.SAS Congruence Postulate 5.Corres. parts of ≅ ∆’s are ≅ 6.Alternate Interior Angles Converse 2.
Statements:Reasons: Given An angle bisector is a ray whose endpoint is the vertex of the angle and divides the angle into two congruent angles. Reflexive Property (a quantity is congruent to itself) (SAS) If two sides and the included angle of one triangle are congruent to the corresponding parts of a second triangle, the triangles are congruent.
Statements:Reasons: Perpendicular lines meet to form right angles. Given Right angles are congruent. Midpoint of a line segment is the point on that line segment that divides the segment into two congruent segments. (SAS) If two sides and the included angle of one triangle are congruent to the corresponding parts of a second triangle, the triangles are congruent.
Statements:Reasons:
Statements:Reasons: Given Perpendicular lines meet to form right angles. Right angles are congruent. (ASA) If two angles and the included side of one triangle are congruent to the corresponding parts of a second triangle, the triangles are congruent. (CPCTC) Corresponding parts of congruent triangles are congruent.
ASA – Postulate Given: HA || KS Prove: HA || KS, Given Alt. Int. Angles are congruent Vertical Angles are congruent ASA Postulate 10.
Note: is not SSS, SAS, or ASA. Identify the Congruent Triangles. Identify the congruent triangles (if any). State the postulate by which the triangles are congruent. by SSS by SAS
Example #4 – Paragraph Proof Given: Prove: is isosceles with vertex bisected by AH. Sides MA and AT are congruent by the definition of an isosceles triangle. Angle MAH is congruent to angle TAH by the definition of an angle bisector. Side AH is congruent to side AH by the reflexive property. Triangle MAH is congruent to triangle TAH by SAS. Side MH is congruent to side HT by CPCTC.
Column Proof A line to one of two || lines is to the other line. Given: Prove: Given has midpoint N Perpendicular lines intersect at 4 right angles. Substitution, Def of Congruent Angles Definition of Midpoint SAS CPCTC
Given: 1 ≅ 2, 3 ≅ 4. Prove ∆BCE ≅ ∆DCE Statements: 1. 1≅ 2, 3≅ 4 2.AC ≅ AC 3.∆ABC ≅ ∆ADC 4.BC ≅ DC 5.CE ≅ CE 6.∆BCE≅∆DCE 1.Given 2.Reflexive property of Congruence 3.ASA Congruence Postulate 4.Corres. parts of ≅ ∆’s are ≅ 5.Reflexive Property of Congruence 6.SAS Congruence Postulate
Given: AB ≅ DE, AC ≅ DF, BC ≅ EF Prove CAB ≅ FDE Statements: 1.AB ≅ DE 2.AC ≅ DF 3.BC ≅ EF 4.∆CAB ≅ ∆FDE 5. CAB ≅ FDE Reasons: 1.Given 2.Given 3.Given 4.SSS Congruence Post 5.Corres. parts of ≅ ∆’s are ≅. 12.
Write a two-column proof. Prove:ΔLMN ΔPON 2. LNM PNO 2. Vertical Angles Theorem StatementsReasons 3. M O 3. Third Angles Theorem 4.ΔLMN ΔPON 4. CPCTC 1. Given
Find the missing information in the following proof. Prove:ΔQNP ΔOPN ReasonsStatements 3. Q O, NPQ PNO 3. Given 5. Definition of Congruent Polygons 5. ΔQNP ΔOPN 4. _________________ 4. QNP ONP ? Reflexive Property of Congruence Given Third angle theorem 14.