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Name ________________________ Worksheet 2.3 Triangle Congruency and SAS 1. Review of Triangle Terminology. Write a robust definition (in conditional form) of the following and draw a labeled sketch. Recall definitions are bi-conditional. Name Definition Sketch D Scalene Triangle If a triangle has no congruent sides, then it is scalene. E F Isosceles Triangle Equilateral Triangle Acute Triangle Obtuse Triangle Right Triangle 2. Which of the following triangles are possible? Draw a sketch of all possible triangles. a. An obtuse isosceles triangle b. A right equilateral triangle c. A scalene acute triangle d. An isosceles right triangle 3. Congruent Triangles. Since triangles are simply enclosed figures made up of only three sides and three angles, then the following definition should follow: Definition: If each of three sides of a triangle is congruent to the corresponding sides of another triangle and the corresponding angles are also congruent, then the triangles are congruent. Suppose we are given two triangles (such as the ones below): W B C U V A Specifically, we know ABC WVU if all corresponding angles and corresponding sides are congruent. To prove that all sides and angles are congruent every time we want to prove two triangles are congruent would be needlessly time consuming. We therefore need to understand that there are certain MINIMUM conditions which guarantee congruence. We will study these minimum conditions in their entirety, but we will begin by looking at two of these scenarios, each stated as a postulate: Side-Angle-Side (SAS): -- Postulate If two sides of one triangle are congruent to two sides of a second triangle AND the included angles are also congruent, then the triangles are congruent. Place tick marks for this scenario. C Given: AB DH A D CA DG D A B G H Therefore: _______________ ________________ by the SAS postulate. (Be sure that corresponding vertices are correct in your statement) 4. Practice. Use the given information to complete each statement. If triangles cannot be shown to be congruent from the information given, write “cannot be determined.” a. C is the midpoint of BE & AD. ABC Reason (briefly explain in your own words); b. KI IT ; IE bisects TEK KIE Reason (briefly explain in your own words); 5. Extension from Congruent Triangles. Suppose we look at a scenario where we have been given or have shown that AB WV , B V , and BC UV . Make tick markings on the triangles below that reflect this information. W B C U V A a. What conclusion can we draw about these triangles and why? b. Write down all the conclusions that can be drawn concerning all remaining corresponding parts of these triangles. c. The justification we will use for the statements in part b is the acroynm C P C T C. What do you think CPCTC stands for? 6. Complete the proof (once you prove the first statement, simply continue your proof to reach the second statement). Q Given: AQ BC 1 2 AP AR Prove: P 1) APQ ARQ 2) AQP AQR 1 B Statements R 2 A C Reasons Please notice that, in order to use CPCTC in your proof, you need to have a statement about congruent triangles listed in your proof BEFORE the CPCTC step! 7. Write a valid proof involving a pair of congruent triangles. Remember, to prove congruent triangles you need THREE pieces of information regarding congruent parts (hence, SAS). S Given: SZ TX SY TY T Y Prove: SX TZ X Statements Z Reasons 8. Consider a triangle whose sides measure 2t 1 , t 5 , and 3t 8 meters. a. Determine all possible value(s) for t , making the triangle isosceles. b. Is it possible to find a value for t which will make the triangle equilateral? If so, determine its value.