JRLeon Discovering Geometry Chapter HGSH C H A P T E R 4 O B J E C T I V E S  Discover and explain sums of the measures of two and three interior angles of a triangle  Discover properties of the base angles and the vertex angle bisector in isosceles triangles  Discover inequalities among sides and angles in triangles  Investigate SSS, SAS, SSA, ASA, SAA, and AAA as potential shortcuts to proving triangle congruence  Show that pairs of angles or pairs of sides are congruent by identifying related triangles and proving them congruent, then applying CPCTC  Create flowchart proofs  Review algebraic properties and methods for solving linear equations  Develop logical and visual thinking skills  Develop inductive reasoning, problem solving skills, and cooperative behavior  Practice using geometry tools

JRLeon Discovering Geometry Chapter 4.7 HGSH You have been making many discoveries about triangles. As you try to explain why the new conjectures are true, you build upon definitions and conjectures you made before. So far, you have written your explanations as deductive arguments or paragraph proofs. First, we’ll look at a diagram and explain why two angles must be congruent, by writing a paragraph proof, in Example A. Then we’ll look at a different tool for writing proofs, and use that tool to write the same proof, in Example B.

JRLeon Discovering Geometry Chapter 4.7 HGSH

Were you able to follow the logical steps in Example A? Sometimes a logical argument or a proof is long and complex, and a paragraph might not be the clearest way to present all the steps. A flowchart is a visual way to organize all the steps in a complicated procedure in proper order. Arrows connect the boxes to show how facts lead to conclusions. Flowcharts make your logic visible so that others can follow your reasoning. To present your reasoning in flowchart form, create a flowchart proof. Place each statement in a box. Write the logical reason for each statement beneath its box. For example, you would write “RC  RC, because it is the same segment,” as

JRLeon Discovering Geometry Chapter 4.7 HGSH Here is the same logical argument that you created in Example A in flowchart proof format.

JRLeon Discovering Geometry Chapter 4.7 HGSH Given 2. CD  AD3. BD  BD Same Segment 1. BA  BC Given SSS Congruence Conjecture 4. BAD  BCD   CPCTC 5.  1   2 So,  1 +  2 =  ABC Therefore BD is the angle bisector of  ABC

JRLeon Discovering Geometry Chapter 4.7 HGSH CONCLUSION: Sometimes a flowchart can better demonstrate relationships among the ideas in a proof than can a paragraph. The flowchart can be horizontal (from left to right) or vertical (from top to bottom). To make a flowchart proof, you can write down a box for the “Given” at the left or the top and a box for the “Show” at the right or the bottom. Then fill in the other boxes, moving both forward and backward. by asking : What do I need to know in order to claim the conclusion is true? What must I show to prove that intermediate result? You can plan a proof by reasoning backward.

JRLeon Discovering Geometry Chapter 4.7 HGSH Homework: Lesson 4.7 – Pages 239 – 240, Prob. 1 thru 4