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4.4 & 4.5 & 5.2 proving triangles congruent

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Mary Angeline Molabola
Mary Angeline Molabola

This document provides information about proving triangles congruent using various postulates and theorems: 1. It describes the SSS, SAS, ASA, and AAS postulates that can be used to prove any triangles congruent, as well as the HL postulate that applies only to right triangles. 2. Examples are given to demonstrate applying each of the postulates to determine if two triangles are congruent and to find missing corresponding parts. 3. Additional theorems like CPCTC (corresponding parts of congruent triangles are congruent) and properties of vertical angles, isosceles triangles, and midpoints are also explained.

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4.4-4.5 & 5.2: Proving Triangles Congruent p. 206-221, 245-251 Adapted from: http://jwelker.lps.org/lessons/ppt/geod_4_4_congruent_triangles.ppt
SSS - Postulate If all the sides of one triangle are congruent to all of the sides of a second triangle, then the triangles are congruent. (SSS)
Example #1 – SSS – Postulate Use the SSS Postulate to show the two triangles are congruent. Find the length of each side. AC = 5 BC = 7 2 2 AB = 5 + 7 = 74 MO = 5 NO = 7 2 2 MN = 5 + 7 = 74 ∆ACB ≅ ∆MON By SSS
Definition – Included Angle J ∠ K is the angle between JK and KL. It is called the included angle of sides JK and KL. K L What is the included angle for sides KL and JL? J ∠L K L
SAS - Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent. (SAS) L S Q P A S A J S S K ∆JKL ≅ ∆PQR by SAS R
Example #2 – SAS – Postulate K L Given: N is the midpoint of LW N is the midpoint of SK N Prove: ∆LNS ≅ ∆WNK W S Statement Reason N is the midpoint of LW N is the midpoint of SK 1 Given 2 LN ≅ NW , SN ≅ NK 2 Definition of Midpoint 3 ∠LNS ≅∠WNK 3 Vertical 4 ∆LNS ≅ ∆WNK 1 4 SAS Angles are congruent
Definition – Included Side J JK is the side between ∠ J and ∠ K. It is called the included side of angles J and K. K L J What is the included side for angles K and L? KL K L
ASA - Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent. (ASA) J X Y K L ∆JKL ≅ ∆ZXY by ASA Z
Example #3 – ASA – Postulate H A W Given: HA || KS AW ≅WK K Prove: ∆HAW ≅ ∆SKW S Reasons Statement HA || KS, AW ≅ WK 1 Given 2 ∠HAW ≅∠SKW 2 Alt. Int. Angles are congruent 3 ∠HWA ≅∠SWK 3 Vertical 4 ∆HAW ≅ ∆SKW 1 4 Angles are congruent ASA Postulate
Identify the Congruent Triangles. Identify the congruent triangles (if any). State the postulate by which the triangles are congruent. A J R B C H I S K M O L P VABC ≅VSTR by SSS VPNO ≅VVUW by SAS N V T U W Note: VJHI is not SSS, SAS, or ASA.
Example ∆AMT is isosceles with vertex ∠MAT Given: ∠MAT is bisected by AH Prove: MH ≅ HT Statement 1) Reason ∆AMT is isosceles with vertex ∠MAT 1) Given ∠MAT is bisected by AH
AAS (Angle, Angle, Side) • If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding non- C included side of another triangle, . . . then the 2 triangles are CONGRUENT! A D B F E
Example Given: AW || TB E is the midpoint of WB Prove: AW ≅ TB Statement 1) AW || TB E is the midpoint of WB 2) Reason 1) 2) Given
HL (Hypotenuse, Leg) only used with right triangles**** • If both hypotenuses and a pair of legs of two RIGHT triangles are congruent, . . . then the 2 triangles are CONGRUENT! A C B D F E
Given: AB ≅ CB Example ∠BDA and ∠BDC are right ∆ABD and ∆CBD are right triangles Prove: ∠A ≅ ∠C Statement 1) AB ≅ CB ∠BDA and ∠BDC are right ∆ABD and ∆CBD are right triangles 2) Reason 1) 2) Given
The Triangle Congruence Postulates &Theorems FOR ALL TRIANGLES SSS SAS ASA AAS FOR RIGHT TRIANGLES ONLY HL Only this one is new LL HA LA
Summary • Any Triangle may be proved congruent by: (SSS) (SAS) (ASA) (AAS) • Right Triangles may also be proven congruent by HL ( Hypotenuse Leg) • Parts of triangles may be shown to be congruent by Congruent Parts of Congruent Triangles are Congruent (CPCTC).
Example 1 Given the information in the diagram, is there any way to determine if A CB ≅ DF ? C B D YES!! ∆CAB ≅ ∆DEF by SAS so CB ≅ DF by CPCTC E F
Example 2 A C B • Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? D E F No ! SSA doesn’t work
Example 3 A C D • Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? B YES ! Use the reflexive side CB, and you have SSS
Name That Postulate (when possible) SAS SSA ASA SSS
Name That Postulate (when possible) AAA SAS ASA SSA
Name That Postulate (when possible) Reflexive Property SAS Vertical Angles SAS Vertical Angles SAS Reflexive Property SSA
Let’s Practice Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: ∠B ≅ ∠D For SAS: AC ≅ FE For AAS: ∠A ≅ ∠F
Homework Assignment

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4.4 & 4.5 & 5.2 proving triangles congruent

  • 1. 4.4-4.5 & 5.2: Proving Triangles Congruent p. 206-221, 245-251 Adapted from: http://jwelker.lps.org/lessons/ppt/geod_4_4_congruent_triangles.ppt
  • 2. SSS - Postulate If all the sides of one triangle are congruent to all of the sides of a second triangle, then the triangles are congruent. (SSS)
  • 3. Example #1 – SSS – Postulate Use the SSS Postulate to show the two triangles are congruent. Find the length of each side. AC = 5 BC = 7 2 2 AB = 5 + 7 = 74 MO = 5 NO = 7 2 2 MN = 5 + 7 = 74 ∆ACB ≅ ∆MON By SSS
  • 4. Definition – Included Angle J ∠ K is the angle between JK and KL. It is called the included angle of sides JK and KL. K L What is the included angle for sides KL and JL? J ∠L K L
  • 5. SAS - Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent. (SAS) L S Q P A S A J S S K ∆JKL ≅ ∆PQR by SAS R
  • 6. Example #2 – SAS – Postulate K L Given: N is the midpoint of LW N is the midpoint of SK N Prove: ∆LNS ≅ ∆WNK W S Statement Reason N is the midpoint of LW N is the midpoint of SK 1 Given 2 LN ≅ NW , SN ≅ NK 2 Definition of Midpoint 3 ∠LNS ≅∠WNK 3 Vertical 4 ∆LNS ≅ ∆WNK 1 4 SAS Angles are congruent
  • 7. Definition – Included Side J JK is the side between ∠ J and ∠ K. It is called the included side of angles J and K. K L J What is the included side for angles K and L? KL K L
  • 8. ASA - Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent. (ASA) J X Y K L ∆JKL ≅ ∆ZXY by ASA Z
  • 9. Example #3 – ASA – Postulate H A W Given: HA || KS AW ≅WK K Prove: ∆HAW ≅ ∆SKW S Reasons Statement HA || KS, AW ≅ WK 1 Given 2 ∠HAW ≅∠SKW 2 Alt. Int. Angles are congruent 3 ∠HWA ≅∠SWK 3 Vertical 4 ∆HAW ≅ ∆SKW 1 4 Angles are congruent ASA Postulate
  • 10. Identify the Congruent Triangles. Identify the congruent triangles (if any). State the postulate by which the triangles are congruent. A J R B C H I S K M O L P VABC ≅VSTR by SSS VPNO ≅VVUW by SAS N V T U W Note: VJHI is not SSS, SAS, or ASA.
  • 11. Example ∆AMT is isosceles with vertex ∠MAT Given: ∠MAT is bisected by AH Prove: MH ≅ HT Statement 1) Reason ∆AMT is isosceles with vertex ∠MAT 1) Given ∠MAT is bisected by AH
  • 12. AAS (Angle, Angle, Side) • If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding non- C included side of another triangle, . . . then the 2 triangles are CONGRUENT! A D B F E
  • 13. Example Given: AW || TB E is the midpoint of WB Prove: AW ≅ TB Statement 1) AW || TB E is the midpoint of WB 2) Reason 1) 2) Given
  • 14. HL (Hypotenuse, Leg) only used with right triangles**** • If both hypotenuses and a pair of legs of two RIGHT triangles are congruent, . . . then the 2 triangles are CONGRUENT! A C B D F E
  • 15. Given: AB ≅ CB Example ∠BDA and ∠BDC are right ∆ABD and ∆CBD are right triangles Prove: ∠A ≅ ∠C Statement 1) AB ≅ CB ∠BDA and ∠BDC are right ∆ABD and ∆CBD are right triangles 2) Reason 1) 2) Given
  • 16. The Triangle Congruence Postulates &Theorems FOR ALL TRIANGLES SSS SAS ASA AAS FOR RIGHT TRIANGLES ONLY HL Only this one is new LL HA LA
  • 17. Summary • Any Triangle may be proved congruent by: (SSS) (SAS) (ASA) (AAS) • Right Triangles may also be proven congruent by HL ( Hypotenuse Leg) • Parts of triangles may be shown to be congruent by Congruent Parts of Congruent Triangles are Congruent (CPCTC).
  • 18. Example 1 Given the information in the diagram, is there any way to determine if A CB ≅ DF ? C B D YES!! ∆CAB ≅ ∆DEF by SAS so CB ≅ DF by CPCTC E F
  • 19. Example 2 A C B • Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? D E F No ! SSA doesn’t work
  • 20. Example 3 A C D • Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? B YES ! Use the reflexive side CB, and you have SSS
  • 21. Name That Postulate (when possible) SAS SSA ASA SSS
  • 22. Name That Postulate (when possible) AAA SAS ASA SSA
  • 23. Name That Postulate (when possible) Reflexive Property SAS Vertical Angles SAS Vertical Angles SAS Reflexive Property SSA
  • 24. Let’s Practice Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: ∠B ≅ ∠D For SAS: AC ≅ FE For AAS: ∠A ≅ ∠F