Document Sample

Proving Triangles Congruent The Idea of a Congruence Two geometric figures with exactly the same size and shape. F B A C E D How much do you need to know. . . . . . about two triangles to prove that they are congruent? Corresponding Parts In Lesson 4.2, you learned that if all six pairs of corresponding parts (sides and angles) are congruent, then the triangles are congruent. 1. AB DE 2. BC EF 3. AC DF 4. A D ABC DEF 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 ABC DEF 3. AC DF Postulate 19: SSS Congruence Postulate EXAMPLE 3 Solve a real-world problem Structural Support Explain why the bench with the diagonal support is stable, while the one without the support can collapse. EXAMPLE 3 Solve a real-world problem SOLUTION The bench with a diagonal support forms triangles with fixed side lengths. By the SSS Congruence Postulate, these triangles cannot change shape, so the bench is stable. The bench without a diagonal support is not stable because there are many possible quadrilaterals with the given side lengths. EXAMPLE 2 Standardized Test Practice SOLUTION By counting, PQ = 4 and QR = 3. Use the Distance Formula to find PR. d = ( x2 – x1 ) 2 + ( y2 – y1 ) 2 EXAMPLE 2 Standardized Test Practice PR = ( – 1 – (– 5 ) )2 + ( 1 – 4 ) 2 = 4 2 + (– 3 ) 2 = 25 = 5 By the SSS Congruence Postulate, any triangle with side lengths 3, 4, and 5 will be congruent to PQR. The distance from (–1, 1) to (–1, 5) is 4. The distance from (–1, 5) to (–4, 5) is 3. The distance from (– 1, 1) to (–4, 5) is ( 5 – 1) 2 + ( (–4) – (–1) ) 2 = 4 2 + (– 3 ) 2 = 25 = 5 ANSWER The correct answer is A. EXAMPLE 1 Use the SSS Congruence Postulate Write a proof. GIVEN KL NL, KM NM PROVE KLM NLM Proof It is given that KL NL and KM NM By the Reflexive Property, LM LM. So, by the SSS Congruence Postulate, KLM NLM GUIDED PRACTICE for Example 1 Decide whether the congruence statement is true. Explain your reasoning. 1. DFG HJK SOLUTION Three sides of one triangle are congruent to three sides of second triangle then the two triangle are congruent. Side DG HK, Side DF JH,and Side FG JK. So by the SSS Congruence postulate, DFG HJK. Yes. The statement is true. GUIDED PRACTICE for Example 1 Decide whether the congruence statement is true. Explain your reasoning. 2. ACB CAD SOLUTION GIVEN : BC AD PROVE : ACB CAD PROOF: It is given that BC AD By Reflexive property AC AC, But AB is not congruent CD. Side-Angle-Side (SAS) Postulate 20 1. AB DE 2. A D ABC DEF 3. AC DF included angle Included Angle The angle between two sides G I H Included Angle Name the included angle: E YE and ES E ES and YS S Y S YS and YE Y EXAMPLE 2 Prove the AAS Congruence Theorem Prove the Angle-Angle-Side Congruence Theorem. Write a proof. GIVEN A D, C F, BC EF PROVE ABC DEF GUIDED PRACTICE for Examples 1 and 2 1. In the diagram at the right, what postulate or theorem can you use to prove that RST VUT ? Explain. SOLUTION STATEMENTS REASONS S U Given RS UV Given RTS UTV The vertical angles are congruent Do Now Eliminate the possibilities…. • Some of the measurements of ABC and DEF are given below. Can you determine if the two triangles are congruent from this information? 2.5 2.5 cm 30 30 4 cm 4cm Angle-Side-Angle (ASA) Postulate 21 1. A D 2. AB DE ABC DEF 3. B E include d side Included Side The side between two angles GI HI GH Included Side Name the included angle: E Y and E YE E and S ES Y S S and Y SY Angle-Angle-Side (AAS) Theorem 4.6 1. A D 2. B E ABC DEF 3. BC EF Non-included side Angle-Angle-Side (AAS) • If two angles and a nonincluded side of a triangle are congruent to two angles and a nonincluded side of another triangle, then the two triangles are congruent. O A Y B M N Angle-Side-Side • THERE IS NO SUCH THING AS ANGLE SIDE SIDE BECAUSE YOU CAN’T USE THAT KIND OF LANGUAGE AT SCHOOL. Warning: No SSA Postulate There is no such thing as an SSA postulate! B E F A C D NOT CONGRUENT Warning: No AAA Postulate There is no such thing as an AAA postulate! E B A C F D NOT CONGRUENT Tell whether you can use the given information at determine whether ABC DEF A D, ABDE, ACDF AB EF, BC FD, AC DE The Congruence Postulates SSS correspondence ASA correspondence SAS correspondence AAS correspondence SSA correspondence AAA correspondence Name That Postulate (when possible) SAS ASA SSA SSS Name That Postulate (when possible) AAA ASA SAS SSA Name That Postulate (when possible) Vertical Angles Reflexive Property SAS SAS Vertical Reflexive Angles SAS Property SSA HW: Name That Postulate (when possible) HW: Name That Postulate (when possible) 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 Closure Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: For SAS: For AAS: Now For The Fun Part… J Given: JO SH; O is the midpoint of SH Prove: SOJ HOJ S 0 H Write a two column Proof Given: BC bisects AD and A D Prove: AB DC A C E B D

DOCUMENT INFO

Shared By:

Categories:

Tags:
Congruent Triangles, Right Triangles, two sides, included angle, congruent figures, Triangle congruence, three sides, equilateral triangles, congruent angles, congruent sides

Stats:

views: | 37 |

posted: | 10/4/2010 |

language: | English |

pages: | 39 |

OTHER DOCS BY chenshu

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.