1. FINAL EXAM REVIEW Chapter 6-7 Key Concepts

2. Vocabulary Chapter 6 inequalityinversecontrapositive logically equivalent indirect proof Chapter 7 ratiomeans/extremesproportion scale factor POP’s similar figures proportional segments

3. Exterior Angle Inequality Theorem The measure of the exterior angle of a triangle is greater than the measure of either remote interior angle. The measure of the exterior angle of a triangle is greater than the measure of either remote interior angle. A B C 1 m 1 > m A and m 1 > m C by Ext. Ineq. Thm 7777 7

4. GIVEN STATEMENT:If p, then q. CONTRAPOSITIVE:If not q, then not p. CONVERSE:If q, then p. INVERSE:If not p, then not q. Inverse negates given statement. Contrapositive negates converse.

5. An Indirect Proof Template 1) Assume temporarily that the conclusion is not true. 2) Then…Reason logically until you reach a contradiction of a known fact or a given. 3) But this contradicts that…state the contradiction. 4) Thus our assumption is false, therefore.., the conclusion is true..

6. Theorem If one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side. A CB If BC > AB, then big small m m<C.

7. Crikey, that seems easy! Longest side opposite biggest angle, shortest side opposite smallest angle, medium side opposite medium angle.

8. Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. A CB 11 914 9 + 11 = 20 20 > 14 11 + 14 = 25 25 > 9 14 + 9 = 23 23 > 11

9. Corollaries The segment from a point to a line is the shortest segment from the point to the line. The segment from a point to a plane is the shortest segment from the point to the plane. ● ●

10. SAS Inequality Theorem  If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is larger than the triangle, but the included angle of the first triangle is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle. small BIG

11. SSS Inequality Theorem  If two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is larger than the third side of the second, then the included angle of the first triangle is larger than the included angle of the second triangle. BIG small

12. P.O.P.’s (10)3 = 2(15) Given: Is equivalent to: & & Also

13. Similar Polygons Their vertices can be paired so that: Corresponding angles are congruent Corresponding sides are in proportion (their lengths have the same ratio) 4 8 6 12 8 4 3 6

14. Similarity Chart Definition: All angles congruent All sides proportional SAS Similarity Theorem SSS Similarity Theorem All Polygons AA Postulate (2 <‘s = 2 <‘s) Triangles

15. AA Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. ~

16. SAS Similarity Theorem If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar. A BC D E F 3 6 4 8 ∆ABC ~ ∆DEF m<A = m<D 3 6 = 4 8 small big

17. SSS Similarity Theorem If the sides of two triangles are in proportion, then the triangles are similar. A BC D E F 4 6 6 9 ∆ABC ~ ∆DEF 812 4 6 = 6 9 = 8 small big

18. Triangle Proportionality Thm. If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. big A small A whole A big B small B whole B = side C1 Side C2 = = ==== whole A small A whole B small B side C1 side C2 big A small A big B small B big A whole A big B whole B whole A big Abig B small A small B big A big B whole A whole B Think of it as two separate similar triangles.

19. Corollary If three // lines intersect two transversals,… then they divide the transversals proportionally. a b c d = a b c d

20. Triangle Angle-Bisector Thm. If a ray bisects an angle of a triangle,… then it divides the opposite side into segments proportional to the other two sides. a b c d = a b c d

21. Homework ► Study for final exam ► Suggestion: Work problems on pg. 281-283 (answer key will be handed out) (answer key will be handed out) Review Constructions #1-7, 10-13 in Ch. 10 Ch. 7 Multiple Choice Refresher on page 632