Theorem 6.1 : If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. 495 BC) (on the left) and by US president James Gar eld (1831{1881) (on the right) Proof by Pythagoras: in the gure on the left, the area of the large square (which is equal to (a + b)2) is equal to the sum of the areas of the four triangles (1 2 ab each triangle) and the area of Since the HL is a postulate, we accept it as true without proof. The other congruence theorems for right triangles might be seen as special cases of the other triangle congruence postulates and theorems. Step 4: Angles in isosceles triangles Because each small triangle is an isosceles triangle, they must each have two equal angles. Sec 2.6 Geometry – Triangle Proofs Name: COMMON POTENTIAL REASONS FOR PROOFS . Proof : We are given a triangle ABC in which a line parallel to side BC intersects other two sides A B and AC at D and E respectively (see Fig. “Hy-Leg Postulate” for right triangle congruency. Definition of Angle Bisector: The ray that divides an angle into two congruent angles. We need to prove that 2 For the angle bisectors, use the angle bisector theorem: AZ ZB ¢ BX XC ¢ CY YA ˘ AC BC ¢ AB AC ¢ BC AB ˘1. Definition of Midpoint: The point that divides a segment into two congruent segments. Converse of the Triangle Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side of the triangle. Theorem 6-12 If three or more parallel lines intersect two transversals, then they divide the transversals proportionally. Triangle Sum The sum of the interior angles of a triangle is 180º. De nition 5. [2] Suppose we have an arbitrary triangle ABC. 47 Similar Triangles (SSS, SAS, AA) 48 Proportion Tables for Similar Triangles 49 Three Similar Triangles Chapter 9: Right Triangles 50 Pythagorean Theorem 51 Pythagorean Triples 52 Special Triangles (45⁰‐45⁰‐90⁰ Triangle, 30⁰‐60⁰‐90⁰ Triangle) 53 Trigonometric Functions and Special Angles A4 Appendix A Proofs of Selected Theorems THEOREM 1.7 Functions That Agree at All But One Point (page 62) Let be a real number, and let for all in an open interval containing If the limit of as approaches exists, then the limit of also exists and See for Bruce Edwards’s video of this proof. The vertex of an isosceles triangle that has an angle di erent form the two equal angles is called the apex of the isosceles triangle. Definition of Congruence: Having the exact same size and shape and there by having the exact same measures. Base Angle Converse (Isosceles Triangle) If two angles of a triangle are congruent, the sides opposite these angles are congruent. 570 BC{ca. another right triangle, then the triangles are congruent. The unit will close with creating “indirect” proofs which is a new strategy for proving theorems by stating a contradiction. Theorem 6-13 Triangle Angle Bisector Theorem 6.10). Proofs of Pythagorean Theorem 1 Proof by Pythagoras (ca. As a last step, we rotate the triangles 90 o, each around its top vertex.The right one is rotated clockwise whereas the left triangle is rotated counterclockwise. The angle that de nes the apex of the isosceles triangle is called the apex angle. a a+b b Step 5: Angles in the big triangle add up to 180° The sum of internal angles in any triangle is 180°. Obviously the resulting shape is a square with the side c and area c 2. Base Angle Theorem (Isosceles Triangle) If two sides of a triangle are congruent, the angles opposite these sides are congruent.