Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Mtgnt_04 trans wk bk 9/29/03 2:49 PM 4.2 Page 75 Congruence and Triangles Goals p Identify congruent figures and corresponding parts. p Prove that two triangles are congruent. VOCABULARY Congruent Two geometric figures are congruent if they have exactly the same size and shape. Corresponding angles When two figures are congruent, the corresponding angles are the angles that are in corresponding positions and are congruent. Corresponding sides When two figures are congruent, the corresponding sides are the sides that are in corresponding positions and are congruent. Example 1 Naming Congruent Parts Write a congruence statement for the triangles. Identify all pairs of congruent corresponding parts. C A Q B P R Solution The diagram indicates that ! ABC c ! PQR . The congruent angles and sides are as follows. Angles: aA c aP , aB c aQ , aC c aR Sides: AB *& c PQ **& , BC *& c QR **& , CA *& c RP **& Lesson 4.2 • Geometry Notetaking Guide 75 Mtgnt_04 trans wk bk 9/29/03 2:49 PM Page 76 Example 2 Using Properties of Congruent Figures In the diagram, DEFG c KLMN. a. Find the value of x. E 8 ft b. Find the value of y. D N F 111# K 12 ft 93# 5x " 2 77# Solution a. You know that FG *& c MN **&. M G L (5y " 6)# So, FG ! MN. 12 ! 5x " 2 Substitute for FG and MN . 10 ! 5x Subtract 2 from each side. 2 !x Divide each side by 5 . b. You know that aE c aL. So, maE ! maL. 111# ! (5y " 6)# Substitute for maE and maL . 105 ! 5y Subtract 6 from each side. 21 ! y Divide each side by 5 . THEOREM 4.3: THIRD ANGLES THEOREM If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. If aA c aD and aB c aE, then aC c aF . 76 Geometry Notetaking Guide • Chapter 4 B A C E D F Mtgnt_04 trans wk bk 9/29/03 2:49 PM Page 77 Using the Third Angles Theorem Example 3 Find the value of x. V 67# Solution U 54# In the diagram, aV c aY and aU c aZ. From the Third Angles Theorem, you know that aW c aX. So, maW ! maX. From the Triangle Sum Theorem, maW ! 180# $ 54# $ 67# ! 59 #. maW ! maX 59 # ! (4x $ 5)# Z W Y Third Angles Theorem Substitute. 64 ! 4x Add 5 to each side. 16 ! x Divide each side by 4 . Example 4 (4x $ 5)# X Determining Whether Triangles are Congruent Decide whether the triangles are congruent. Justify your reasoning. D G 81# E F 81# H Solution Paragraph Proof From the diagram, you are given that all three pairs of corresponding sides are congruent. DE *& c HG **& , EF *& c GF *&, DF *& c HF *& Because aD and aH have the same measure, aD c aH. By the Vertical Angles Theorem, you know that aDFE c aHFG . By the Third Angles Theorem, aE c aG . Answer So, all three pairs of corresponding sides and all three pairs of corresponding angles are congruent . By the definition of congruent triangles, !DEF c !HGF . Lesson 4.2 • Geometry Notetaking Guide 77 Mtgnt_04 trans wk bk 9/29/03 2:49 PM Page 78 THEOREM 4.4: PROPERTIES OF CONGRUENT TRIANGLES Reflexive Property of Congruent Triangles B Every triangle is congruent to itself . A C E Symmetric Property of Congruent Triangles If !ABC c !DEF, then !DEF c !ABC . D F K Transitive Property of Congruent Triangles If !ABC c !DEF and !DEF c !JKL, then !ABC c !JKL . J L Checkpoint Complete the following exercises. 1. Find the value of x. 10 (5x $ 2)# K M A 58# 74# C L B 2. Decide whether the triangles are congruent. Justify your reasoning. D E Z Y F X From the diagram, it is given that DE *& c XY **&, EF *& c YZ **&, DF *& c XZ **&, aD c aX, and aE c aY. By the Third Angles Theorem, aDFE c aXZY. By the definition of congruent triangles, !DEF c !XYZ. 78 Geometry Notetaking Guide • Chapter 4