Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download lg_ch04_06 Prove Triangles Congruent by ASA and AAS_teacher
Document related concepts
History of geometry wikipedia , lookup
Rational trigonometry wikipedia , lookup
Euler angles wikipedia , lookup
Riemann–Roch theorem wikipedia , lookup
Noether's theorem wikipedia , lookup
Trigonometric functions wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
Four color theorem wikipedia , lookup
History of trigonometry wikipedia , lookup
Integer triangle wikipedia , lookup
Transcript
4.6 Prove Triangles Congruent by ASA and AAS • You will use two more methods to prove congruences. • Essential Question: If a side of one triangle is congruent to a side of another triangle, what information about the angles would allow you You will learn how to answer this to prove the triangles question by learning about the are congruent? ASA Postulate and the AAS Theorem. Warm-Up1Exercises EXAMPLE Identify congruent triangles Can the triangles be proven congruent with the information given in the diagram? If so, state the postulate or theorem you would use. SOLUTION a. The vertical angles are congruent, so two pairs of angles and a pair of non-included sides are congruent. The triangles are congruent by the AAS Congruence Theorem. Warm-Up1Exercises EXAMPLE Identify congruent triangles b. There is not enough information to prove the triangles are congruent, because no sides are known to be congruent. c. Two pairs of angles and their included sides are congruent. The triangles are congruent by the ASA Congruence Postulate. Warm-Up2Exercises EXAMPLE Prove the AAS Congruence Theorem Prove the Angle-Angle-Side Congruence Theorem. Write a proof. GIVEN PROVE A D, ABC C DEF F, BC EF Warm-Up Exercises GUIDED PRACTICE 1. for Examples 1 and 2 In the diagram at the right, what postulate or theorem can you use to RST VUT ? Explain. prove that SOLUTION STATEMENTS S U Given RS UV Given RTS AAS; REASONS RTS UTV UTV The vertical angles are congruent because they are vertical angles. Warm-Up Exercises GUIDED PRACTICE for Examples 1 and 2 Rewrite the proof of the Triangle Sum Theorem on page 219 as a flow proof. 2. ABC GIVEN PROVE m 1+m 2+m 3 = 180° STATEMENTS 1. Draw BD parallel to AC . 2. m 4 + m 2 + m 5 = 180° REASONS 1. Parallel Postulate 2. Angle Addition Postulate and definition of straight angle 3. 1 4, 3 4. m 1= m 4,m 5. m 1+m 2+m 3. Alternate Interior Angles 5 3= m 5 3 = 180° Theorem 4. Definition of congruent angles 5. Substitution Property of Equality Warm-Up3Exercises EXAMPLE Write a flow proof In the diagram, CE BD and CAB Write a flow proof to show GIVEN PROVE CE BD, CAB ABE ADE ABE CAD CAD. ADE Warm-Up4Exercises EXAMPLE Standardized Test Practice Warm-Up4Exercises EXAMPLE Standardized Test Practice The locations of tower A, tower B, and the fire form a triangle. The dispatcher knows the distance from tower A to tower B and the measures of A and B. So, the measures of two angles and an included side of the triangle are known. By the ASA Congruence Postulate, all triangles with these measures are congruent. So, the triangle formed is unique and the fire location is given by the third vertex. Two lookouts are needed to locate the fire. Warm-Up4Exercises EXAMPLE Standardized Test Practice ANSWER The correct answer is B. Warm-Up Exercises GUIDED PRACTICE 3. for Examples 3 and 4 In Example 3, suppose ABE ADE is also given. What theorem or postulate besides ASA ABE ADE? can you use to prove that ANSWER AAS Congruence Theorem. Daily Homework Quiz Warm-Up Exercises Tell whether each pair of triangle are congruent by SAS, ASA, SSS, AAS or HL. If it is not possible to prove the triangle congruent, write not necessarily congruent. 1. ANSWER ASA . Daily Homework Quiz Warm-Up Exercises Tell whether each pair of triangle are congruent by SAS, ASA, SSS, AAS or HL. If it is not possible to prove the triangle congruent, write not necessarily congruent. 2. ANSWER not necessarily congruent . Daily Homework Quiz Warm-Up Exercises Write flow proof. Given : BD bisects ABC, Prove : ABD CBD 3. A C Daily Homework Quiz Warm-Up Exercises ANSWER • You will use two more methods to prove congruences. • Essential Question: If a side of one triangle is congruent to a side of another triangle, what information about the angles would allow you to prove the triangles are congruent? • Triangles are congruent by the ASA Congruence Postulate. • Triangles are congruent by the AAS Congruence Theorem. • Another format for proofs is the flow proof. The triangles will be congruent if the conditions of the ASA Congruence Postulate or of the AAS Congruence Theorem are met.
