Honors Math 2 Name: ________________________ Date: __________________ Hinge Theorem To begin, you must prove two theorems: 1. In any triangle the angle opposite the greater side is greater. 2. In any triangle the side opposite the greater angle is greater. To prove (1), fill in the reasons to each statement. 𝐵𝐷 is constructed such that 𝐴𝐵 ≅ 𝐴𝐷. !!!! Given: !!!!
𝐴𝐶 > 𝐴𝐵
Prove: ∠ABC > ∠BCA Statements Reasons 1. 𝐴𝐶 > 𝐴𝐵 1. 2. 𝐴𝐵 ≅ 𝐴𝐷 2. 3. ∆𝐴𝐷𝐵 is isosceles 3. 4. ∠ADB > ∠BCA 4. 5. ∠ADB ≅ ∠ABD 5. 6. ∠ABD > ∠BCA 6. 7. ∠ABC > ∠ABD 7. 8. 8. ∠ABC > ∠BCA The converse of (1) is (2). Use the theorem you just proved (“In any triangle the angle opposite the greater side is greater”) to prove the converse by contradiction: Given: ∠ABC > ∠BCA Prove: !!!!
𝐴𝐶 > !!!!
𝐴𝐵 Now that you have proven the greatest angle is opposite the greatest side, and the greatest side is opposite the greatest angle, we can move onto prove the Hinge Theorem. The Hinge Theorem states that if two triangles have two pairs of congruent sides, then the triangle with the larger included angle also has the larger third side. Given: 𝐴𝐵 ≅ 𝐷𝐸, 𝐴𝐶 ≅ 𝐷𝐹, and ∠BAC > ∠EDF Prove: 𝐵𝐶 > 𝐸𝐹. Follow the outline to prove the Hinge Theorem Step 1: Construct ∠EDG so that ∠EDG ≅ ∠BAC and 𝐴𝐶 ≅ 𝐷𝐺. Connect EG and FG. (we will do this construction together) Step 2: Prove ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐺 Step 3: Prove 𝐵𝐶 ≅ 𝐺𝐸 Step 4: Prove ∠DGF ≅ ∠DFG Step 5: Prove ∠EFG > ∠EGF Step 6: Prove EG > EF, thus BC > EF Converse of the Hinge Theorem: If two triangles have two pairs of congruent sides, the triangle with the longer third side also has the larger angle included between the first two sides. Do a proof by contradiction using the following set up: Given: 𝐴𝐵 ≅ 𝐷𝐹, 𝐴𝐶 ≅ 𝐷𝐸, BC > EF Prove: ∠BAC > ∠EDF Step 1: Suppose ∠BAC ≯∠EDF. Then what are the possible relationships between ∠BAC and ∠EDF? Step 2: Prove the other 2 possibilities false so that you can conclude its not possible that ∠BAC ≯∠EDF and thus ∠BAC > ∠EDF. Homework: page 504 #16-­‐19