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Still Wandering through Caves Welcome back, young adventurer. After wandering through Lord Flathead IV’s Great Underground Empire, you have discovered the following pieces of information. A) B) C) D) E) F) G) H) Cavern Cavern Cavern Cavern Cavern Cavern Cavern Cavern 1 connects to caverns 2, 5, 6, 7. 2 connects to caverns 1, 14, 6. 4 connects to caverns 3, 9, 10, 15. 8 connects to caverns 3, 11. 9 connects to caverns 4, 5. 10 connects to caverns 3, 4. 12 connects to caverns 11, 13. 13 leads to freedom. Given: Now, you find yourself at a dead end in Cavern 14. Goal: Leave Lord Flathead’s domain. Instructions: Find a path from Cavern 14 to freedom and justify how you are able to move from one cavern to the next using the connections you have discovered during your recent wanderings. Location Justification Cavern 14 Given Given: When you enter Cavern 8, you discover a golden torch hidden in a deep well. Unfortunately, you left your rope in a dead end cavern that emptied into the only cavern with four exits. Goal: Find your way back from Cavern 8 to the proper cavern to fetch your rope so that you will be able to retrieve the golden torch from the well, again justifying how you are able to move from one cavern to the next using the connections you have discovered during your recent wanderings. Location Justification Cavern 8 Given Still Wandering through Geometry From your informal investigations into geometry thus far, you have discovered the following geometric tools (among others). A) B) C) D) E) F) G) H) I) J) K) L) M) N) O) P) Q) R) S) T) The Definition of a Right Angle The Definition of Perpendicular Lines The Definition of a Midpoint The Definition of an Angle Bisector The Definition of Linear Pair The Definition of Vertical Angles The Definition of Complementary Angles The Definition of Supplementary Angles Transitive Property Substitution Property Addition Property Subtraction Property Reflexive Property Distributive Property Segment Addition Postulate Angle Addition Postulate Linear Pair Postulate Congruent Complements Theorem Congruent Supplements Theorem Vertical Angles are congruent. U) V) W) Perpendicular lines intersect to form four right angles All right angles are congruent If exterior sides of two adjacent angles are , then the angles are complementary. X) If two lines are parallel, then the corresponding angles are congruent. Y) If two lines are parallel, then the alternate interior angles are congruent. Z) If two lines are parallel, then the alternate exterior angles are congruent. AA) If two lines are parallel, then the same side interior angles are congruent. BB) The sum of the angles of a triangle equals 180°. CC) Third Angle Theorem DD) Side-Side-Side Postulate EE) Side-Angle-Side Postulate FF) Angle-Side-Angle Postulate GG) Angle-Angle-Side Theorem HH) Hypotenuse-Leg Theorem CPCTC Corresponding Parts of s are Show that the following geometric propositions are true using the given information and the logical tools listed above. P O I N T Given: I is the midpoint of PT O is the midpoint of PI N is the midpoint of IT Show That: PO = NT Geometric Statement Justification N B Given: BAG and TAG are complementary AB is the angle bisector of NAG Show That: BAN and TAG are complementary Geometric Statement G A Justification T Given: 2 4; AC = AF Prove: CAB FAD D Geometric Statement Justification Geometric Statement Justification Geometric Statement Justification B E 5 C 6 1 2 3 4 F A Given: 2 4; EC = EF Prove: CED FEB D B E 5 C 6 1 2 3 4 F A Given: 2 1; GO = TO Prove: HOT HOG Then Prove: 3 6 O 12 3 G 4 5 H 6 T Given: HO GT; GH TH Prove: HOT HOG Geometric Statement Justification Geometric Statement Justification Geometric Statement Justification Then Prove: GO TO O 12 3 4 G 5 6 H T Given: HE = MO ; HO = ME Prove: HOM MEH Then Prove: 1 4 ; 2 3 H O 2 1 4 3 E M Given: HE║MO ; HO║ME Prove: HOM MEH Then Prove: HE MO ; HO ME H O 2 1 4 3 E M