An included side is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side.
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1 Page 1 of 9 Attendance Problems. 1. What are sides AC and BC called? Side AB? 2. Which side is between RA and? RC 3. Given VDEF and VGHI, if RD RG and RE RH, why is RF RI? I can apply ASA, AAS, and HL to construct triangles and to solve problems. I can prove triangles congruent by using ASA, AAS, and HL. Vocabulary: included side Common Core CC.9-12.G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. CC.9-12.G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. CC.9-12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and prove relationships in geometric figures. Participants in an orienteering race use a map and a compass to find their way to checkpoints along an unfamiliar course. Directions are given by bearings, which are based on compass headings. For example, to travel along the bearing S 43 E, you face south and then turn 43 to the east. An included side is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side.
2 Page 2 of 9 Video Example 1. If a triangle has mra = 90, mrb = 20, and AB = 4 cm. Is the triangle unique? Explain why or why not.
3 Page 3 of 9 1 Problem-Solving Application Organizers of an orienteering race are planning a course with checkpoints A, B, and C. Does the table give enough information to determine the location of the checkpoints? Bearing Distance A to B N 55 E 7.6 km B to C N 26 W C to A S 20 W 1 Understand the Problem The answer is whether the information in the table can be used to find the position of checkpoints A, B, and C. List the important information: The bearing from A to B is N 55 E. From B to C is N 26 W, and from C to A is S 20 W. The distance from A to B is 7.6 km. 2 Make a Plan Draw the course using vertical lines to show north-south directions. Then use these parallel lines and the alternate interior angles to help find angle measures of ABC. 3 Solve C km m CAB = = 35 A m CBA = ( ) = 99 You know the measures of CAB and CBA and the length of the included side AB. Therefore by ASA, a unique triangle ABC is determined. 4 Look Back B One and only one triangle can be made using the information in the table, so the table does give enough information to determine the location of all the checkpoints.
4 Page 4 of 9 Example 1. A mailman has to collect mail from mailboxes at A and B and drop it off at the post office at C. Does the table give enough information to determine the location of the mailboxes and the post office? 4. Guided Practice. If 7.6 km is the distance from B to C, is there enough information to determine the location of all the checkpoints? Explain. Video Example 2. Determine if you can use ASA to prove triangles congruent. Explain why or why not. 2 Applying ASA Congruence Determine if you can use ASA to prove UVX WVX. Explain. UXV WXV as given. Since WVX is X a right angle that forms a linear pair with UVX, WVX UVX. Also VX VX by the Reflexive Property. Therefore UVX WVX by ASA. U V W
5 Page 5 of 9 Example 2. Determine if you can use ASA to prove the triangles congruent. Explain why or why not. 5. Guided Practice. Determine if you can ASA to prove VNKL VLMN. Explain why or why not. You can use the Third Angles Theorem to prove another congruence relationship based on ASA. This theorem is Angle-Angle-Side (AAS).
6 Page 6 of 9 Video Example 3. RQ RS Given: RQRP & RSRP are right angles. Prove: VPQR VPSR 3 Using AAS to Prove Triangles Congruent Use AAS to prove the triangles congruent. Given: AB ED, BC DC Prove: ABC EDC Proof: A B C D E BC DC Given AB ED Given B D Alt. Int. Thm. A E ABC EDC AAS Alt. Int. Thm.
7 Page 7 of 9 Example 3. RX RV Given: RYZW RYWZ XY VY Prove: VXYZ VVYW
8 Page 8 of 9 6. Guided Practice. JL bisects RKLM Given: RK RM Prove: VJKL VJML Video Example 4. Determine if the two triangle are congruent. Explain why or why not. A. VPQR VSTU B. VACB VACD
9 4 Applying HL Congruence Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. A VWX and YXW According to the diagram, VWX and YXW are right triangles that share hypotenuse WX. WX XW by the Reflexive Property. It is given that WV XY, therefore VWX YXW by HL. W V Z Y Page 9 of 9 X B VWZ and YXZ This conclusion cannot be proved by HL. According to the diagram, VWZ and YXZ are right triangles, and WV XY. You do not know that hypotenuse WZ is congruent to hypotenuse XZ. Example 4. Determine if the two triangle are congruent. Explain why or why not. A. B. 7. Guided Practice. Determine if VABC VDCB. Explain why or why not. 4-6 Triangle Congruence: ASA, AAS, & HL (pp 265) 9-16, 22.
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Stop signs would be examples of congruent shapes. Since a stop sign has 8 sides, they would be congruent octagons.
hapter 5 ongruence Theorems -! s In math, the word congruent is used to describe objects that have the same size and shape. When you traced things when you were a little kid, you were using congruence.