Inequalities in Two Triangles

Inequalities in Two Triangles In your own words, write the meaning of each vocabulary term. indirect proof when you assume the opposite, prove that definitely can not happen... proving the original statement true (aka. contradiction) inequality an equation with an inequality symbol (≤ - less than or equal) (≥ - more than or equal) Theorems Theorem 6.12 Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second.   think about this two exact same sides..... smaller angle bigger angle 5 5 Side vs Side 7 7 Theorem 6.13 Converse of the Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second.   converse = reverse think..... which angle will need to be bigger to account for the longer side? angle vs angle 6.6 Practice Think: hungry smaller and bigger ⇄ smaller and bigger ⇄ smaller In Exercises 1-9, complete the statement with <, >, or =. Explain your reasoning. BC < EF Reasoning: Bigger angle means bigger side BC = EF Reasoning: Both angles are opposite to sides and in ∆ ∆ BC > EF Reasoning: 90° is bigger m∠A = m∠D Reasoning: Both angles are opposite to sides and in ∆ ∆ m∠A ≤ m∠D Reasoning: Bigger side means there is a bigger angle m∠A > m∠D Reasoning: 2 > 1 AB > AC Reasoning: 145° > 135° AB = CD Reasoning: These are ∆∆ and vertical angles m∠1 ≥ m∠2 Reasoning: 17 > 15 12. Loop a rubber band around the blade ends of a pair of scissors. Describe what happens to the rubber band as you open the scissors. How does that relate to the Hinge Theorem?   The pair of SCISSORS doesn't change As you open the scissors; the band has to be stretched out to compensate for the bigger angle made 13. Starting from a point 10 miles north of Crow Valley, a crow flies northeast for 5 miles. Another crow, starting from a point 10 miles south of Crow Valley, flies due west for 5 miles. Which crow is farther from Crow Valley? Explain.   The first crow is further away since that one is flying at a 135° from Crow Valley while the other crow is flying at a 90° from Crow Valley *It helps to do a picture. Handwritten Text: The entire text is handwritten in a neat and legible style, likely with a dark pen or pencil on white paper. Diagrams: Two diagrams are included: A pair of scissors with a rubber band looped around the blades. A map-like diagram illustrating the flight paths of the two crows. Headings: The main section is labeled with the question numbers (12 and 13). Page Layout: The text, diagrams, and headings are organized in a clear and concise manner.