Geometry 4-4 Study Guide: Congruent Triangles (pp ) Page 1 of 13
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1 Page 1 of 13 Attendance Problems. 1. Name all sides and angles of! VFGH. 2. What is true about! RK and! RL? Explain why. 3. What does it mean for two segments to be congruent? I can use properties of congruent triangles. I can prove triangles congruent by using the definition of congruence. Vocabulary corresponding angles corresponding sides congruent polygons CC.9-12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and prove relationships in geometric figures. Geometric figures are congruent if they are the same size and shape. Corresponding angles and corresponding sides are in the same position in polygons with an equal number of sides. Two polygons are congruent polygons if and only if their corresponding sides are congruent. Thus triangles that are the same size and shape are congruent. Q: What quantity is represented by three congruent dirty trees? A: 99.
2 Page 2 of 13 Helpful Hint Two vertices that are the endpoints of a side are called consecutive vertices. For example, P and Q are consecutive vertices. To name a polygon, write the vertices in consecutive order. For example, you can name polygon PQRS as QRSP or SRQP, but not as PRQS. In a congruence statement, the order of the vertices indicates the corresponding parts.
3 Page 3 of 13 Helpful Hint When you write a statement such as <ABC <DEF, you are also stating which parts are congruent. Video Example 1.! VABC VDEF Identify all pairs of congruent corresponding parts. 1 Naming Congruent Corresponding Parts S RST and XYZ represent the triangles of the space station s support structure. If RST XYZ, identify all pairs of congruent corresponding parts. Angles: R X, S Y, T Z Sides: RS XY, ST YZ, RT XZ R Z Y T X "He didn't run for reelection. `Politics brings you into contact with all the people you'd give anything to avoid,' he said. `I'm staying home.'" -- Garrison Keillor, Lake Wobegone Days
4 Page 4 of 13 Example 1. Given:! VPQR VSTW parts. Identify all pairs of congruent corresponding 4. Guided Practice. If polygon LMNP polygon EFGH, identify all pairs of corresponding congruent parts. Video Example 2.! VABD VCBD. A. Find the value of x. B. Find! mrbdc.
5 Page 5 of 13 2 Using Corresponding Parts of Congruent Triangles Given: EFH GFH A Find the value of x. FHE and FHG are rt.. FHE FHG m FHE = m FHG (6x - 12) = 90 6x = 102 x = 17 E Def. of lines Rt. Thm. Def. of 21.6 F (6x - 12) H Substitute values for m FHE and m FHG. Add 12 to both sides. Divide both sides by 6. G B Find m GFH. m EFH + m FHE + m E = 180 m EFH = 180 m EFH = 180 m EFH = 68.4 GFH EFH m GFH = m EFH m GFH = 68.4 Sum Thm. Substitute values for m FHE and m E. Simplify. Subtract from both sides. Corr. of are. Def. of Trans. Prop. of = Example 2.! VABC VDBC. A. Find the value of x. B. Find! mrdbc.
6 Page 6 of 13 Guided Practice.! VABC VDEF. 5. Find the value of x. 6. Find! mrf. Video Example 3. RACD & RACB are right Rs. C is the midpoint of BD. Given:! AD AB AC bisects RA. Prove:! VACD VACB
7 Page 7 of 13 3 Proving Triangles Congruent Given: P and M are right angles. R is the midpoint of PM. PQ MN, QR NR Prove: PQR MNR Proof: Q P R M N Statements Reasons 1. P and M are rt. 2. P M 3. PRQ MRN 4. Q N 5. R is the mdpt. of PM. 6. PR MR 7. PQ MN ; QR NR 8. PQR MNR 1. Given 2. Rt. Thm. 3. Vert. Thm. 4. Third Thm. 5. Given 6. Def. of mdpt. 7. Given 8. Def. of
8 Page 8 of 13 Example 3. RYWX & RYWZ are right angles. YW bisects RXYZ. Given:! W is the midpoint of XZ. XY YZ. Prove:! VXYW VZYW
9 Page 9 of Guided Practice. AD bisects BE. BE bisects AD. Given:! AB DE RA RD Prove:! VABC VDEC (p 243) 13, 15,
10 Page 10 of 13 Video Example 4. AD DC BC DC Given:! RDAC RCBD AD BC BD AC Prove:! VACD VBDC
11 Page 11 of 13 4 Engineering Application The bars that give structural support to a roller coaster form triangles. Since the angle measures and the lengths of the corresponding sides are the same, the triangles are congruent. Given: JK KL, ML KL, KLJ LKM, JK ML, JL MK Prove: JKL MLK Proof: J N K Statements 1. JK KL, ML KL 2. JKL and MLK are rt.. 3. JKL MLK 4. KLJ LKM 5. KJL LMK 6. JK ML, JL MK 7. KL LK 8. JKL MLK Reasons 1. Given 2. Def. of lines 3. Rt. Thm. 4. Given 5. Third Thm. 6. Given 7. Reflex. Prop. of 8. Def. of M L
12 Page 12 of 13 Example 4. The diagonal bars across a gate give it support. Since the angle measures and the lengths of the corresponding sides are the same, the triangles are congruent. PR and QT bisect each other. Given:! RPQS RRTS QP RT Prove:! VQPS VTRS
13 Page 13 of Guided Practice. MK bisects JL. JL bisects MK Given:! JK ML JK ML Prove:! VJKN VLMN 4-4 Congruent triangles (p 243) 13, 15, 17-20, 24. 4A Ready to Go On pretest & posttests.
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