Find the value of x. 1. 2. September 19, 2018 5.2 Congruent Polygons

Geometry 5.2 Congruent Polygons

Essential Question Given two congruent triangles, how can you use rigid motions to map one triangle to the other triangle? September 19, 2018 5.2 Congruent Polygons

Goals Identify Congruent Figures. Name Corresponding Parts. Prove Triangles Congruent. September 19, 2018 5.2 Congruent Polygons

Congruent Figures Two figures are congruent if they have exactly the same shape and same size. These figures are congruent: September 19, 2018 5.2 Congruent Polygons

So are these: September 19, 2018 5.2 Congruent Polygons

These are not congruent: September 19, 2018 5.2 Congruent Polygons

If two figures are congruent, then there is a correspondence between them. Corresponding Angles A  R B  S C  T Corresponding Sides R S T AB  RS BC  ST AC  RT ABC  RST Congruence Statement September 19, 2018 5.2 Congruent Polygons

ABC  RST You could also write: BAC  SRT CAB  TRS ACB  RTS BCA  STR CBA  TSR A B C R S T September 19, 2018 5.2 Congruent Polygons

B is NOT congruent to T. But you cannot write: ABC  RTS A B C B is NOT congruent to T. R S T When writing a congruence statement, corresponding parts must be matched correctly. September 19, 2018 5.2 Congruent Polygons

Example 1 NSRM BADC  SNMR ABCD  CDAB  RMNS ADCB  NMRS Write a congruence statement for these congruent figures. A M N B C S D R NSRM ABCD  BADC  SNMR CDAB  RMNS ADCB  NMRS September 19, 2018 5.2 Congruent Polygons

Example 2 In this diagram, RST  XYZ. R S  ______ Y X  ______ 85° 12 40° R S  ______ Y X  ______ YZ = ______ 12 RS = ______ XY mS = ______ 85° 40° mX = ______ September 19, 2018 5.2 Congruent Polygons

Your Turn ABCD  KJHL. Solve for x and y. 4x – 3 = 9 5y – 2 = 113 You cannot trust pictures. Always use the given congruence statement. A 9 cm B L K 91° 4x – 3 cm 86° 113° (5y – 2)° J D C H 6 cm 4x – 3 = 9 4x = 12 x = 3 5y – 2 = 113 5y = 115 y = 23 September 19, 2018 5.2 Congruent Polygons

Theorem 5.3 Prop. of Triangle Congruence Reflexive: Every Triangle is congruent to itself. ABC  ABC. Symmetric: If BAD  DOG, then DOG  BAD. Transitive: If ABC  DEF, and DEF  RST, then ABC  RST. September 19, 2018 5.2 Congruent Polygons

Theorem 5.4 Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. A B C D E F If A  D and B  E, then C  F. September 19, 2018 5.2 Congruent Polygons

Example 3 ABC  DEF. Find x. A B C 22° 87° D E F (4x + 15)° Using Triangle Angle Sum Theorem: mA + 87 + 22 = 180 mA + 109 = 180 mA = 71 Using Third Angles Theorem mA = 4x + 15 71 = 4x + 15 56 = 4x x = 14 September 19, 2018 5.2 Congruent Polygons

Proving Triangles Congruent Two triangles are congruent if you can show that all three corresponding sides are congruent, and all three corresponding angles are congruent. 2 s   3 s & 3 sides . September 19, 2018 5.2 Congruent Polygons

Example 4 BAT  MAN Is BAT  MAN? M T 58 A 58 N B Using the segment marks: Angles: B  M (same measure) BAT  MAN (why?) T  N (why?) Three angles congruent. AT  AN BT  MN AB  AM (Vert s) (3rd s theorem) Three segments congruent. BAT  MAN September 19, 2018 5.2 Congruent Polygons

What have you learned? Your Turn to Practice! 1. Given ABC  DEF. a. Name three pairs of congruent sides. b. Name three pairs of congruent angles. A C B D AB & DE, BC & EF, AC & DF F E M &  P, N & Q, O & R September 19, 2018 5.2 Congruent Polygons

Your Turn to Practice! 2. In the diagram, MKL  JET. Complete the following statements. L  ___ 𝑀𝐾  ___ mM = ___ mT = ___ ML = ___ ETJ  _____ M T 𝐽𝐸 32° K L 85° J 63° 85° 6 cm 6 cm E T KLM September 19, 2018 5.2 Congruent Polygons

Your Turn to Practice! 3. Complete the statement. a. If WRD  PLK, then 𝑊𝑅  _____. b. If BGT  DSN, then ∠ 𝑇  _____. c. If JCX  MWP, then 𝑋𝐶  _____. d. If RHK  WVO, then KRH  _____. 𝑃𝐿 ∠ 𝑁 𝑃𝑊 OWV September 19, 2018 5.2 Congruent Polygons

Summary If two figures are congruent, then… corresponding angles are congruent, corresponding sides are congruent. September 19, 2018 5.2 Congruent Polygons

Assignment September 19, 2018 5.2 Congruent Polygons