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CONGRUENT TRIANGLES

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19 diapositives · 517 Views · Aug 20, 2014

Explore the concept of congruent triangles with Jim Smith from JCHS. This guide explains the nature of congruence, emphasizing that all corresponding angles and sides must be equal. Learn about five key methods to prove triangle congruence: SSS, SAS, ASA, AAS, and HL (for right triangles). The guide provides clear reasoning and prerequisites for each method, along with practical examples. Enhance your understanding of triangle properties and the application of essential geometric principles in proofs.

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  1. CONGRUENT TRIANGLES Sections 4-2, 4-3, 4-5 Jim Smith JCHS

  2. When we talk about congruent triangles, we mean everything about them Is congruent. All 3 pairs of corresponding angles are equal…. And all 3 pairs of corresponding sides are equal

  3. For us to prove that 2 people are identical twins, we don’t need to show that all “2000” body parts are equal. We can take a short cut and show 3 or 4 things are equal such as their face, age and height. If these are the same I think we can agree they are twins. The same is true for triangles. We don’t need to prove all 6 corresponding parts are congruent. We have 5 short cuts or methods.

  4. SSS If we can show all 3 pairs of corr. sides are congruent, the triangles have to be congruent.

  5. SAS Non-included angles Included angle Show 2 pairs of sides and the included angles are congruent and the triangles have to be congruent.

  6. This is called a common side. It is a side for both triangles. We’ll use the reflexive property.

  7. Which method can be used to prove the triangles are congruent

  8. Common side SSS Vertical angles SAS Parallel lines alt int angles Common side SAS

  9. PART 2

  10. ASA, AAS and HL A ASA – 2 angles and the included side S A AAS – 2 angles and The non-included side A A S

  11. HL ( hypotenuse leg ) is used only with right triangles, BUT, not all right triangles. ASA HL

  12. PROOFS When Starting A Proof, Make The Marks On The Diagram Indicating The Congruent Parts. Use The Given Info, Properties, Definitions, Etc. We’ll Call Any Given Info That Does Not Specifically State Congruency Or Equality A PREREQUISITE

  13. SOME REASONS WE’LL BE USING • DEF OF MIDPOINT • DEF OF A BISECTOR • VERT ANGLES ARE CONGRUENT • DEF OF PERPENDICULAR BISECTOR • REFLEXIVE PROPERTY (COMMON SIDE) • PARALLEL LINES ….. ALT INT ANGLES

  14. Given: AB = BD EB = BC Prove: ∆ABE ˜ ∆DBC A C = B 1 2 Our Outline P rerequisites S ides A ngles S ides Triangles ˜ SAS E D =

  15. A C Given: AB = BD EB = BC Prove: ∆ABE ˜ ∆DBC B 1 2 = SAS E D STATEMENTS REASONS P S A S ∆’s none AB = BD Given 1 = 2 Vertical angles EB = BC Given ∆ABE ˜ ∆DBC SAS =

  16. C Given: CX bisects ACB A ˜ B Prove: ∆ACX˜ ∆BCX = 2 1 = AAS B A X P A A S ∆’s CX bisects ACB Given 1 = 2 Def of angle bisc A = B Given CX = CX Reflexive Prop ∆ACX ˜ ∆BCX AAS =

  17. Can you prove these triangles are congruent?