Geometry - Lesson 6A: Isosceles Triangle Theorem Notes PDF

GEOMETRY - LESSON 6A ISOSCELES TRIANGLE THEOREM PARTS OF AN ISOSCELES TRIANGLE vertex angle leg leg base angles base  THE ISOSCELES TRIANGLE THEOREM THEOREM If 2 sides of a triangle are...

GEOMETRY - LESSON 6A ISOSCELES TRIANGLE THEOREM PARTS OF AN ISOSCELES TRIANGLE vertex angle leg leg base angles base  THE ISOSCELES TRIANGLE THEOREM THEOREM If 2 sides of a triangle are , then the opposite those sides are. A GIVEN: AB AC PROVE: B C HINT: DRAW AUXILIARY LINE THAT BISECTS A B D C PARAGRAPH PROOF: Draw an auxiliary line that bisects A. Draw point D at the intersection of this line and BC. ABD and ACD are then congruent by SAS (using the 2 given segments, the 2 parts of the bisected A, and the common side AD). Therefore B is congruent to C by CPCTC.  ISOSCELES TRIANGLE THEOREM COROLLARIES ISOSCELES TRIANGLE THEOREM If 2 sides of a are , then the opposite those sides are. COROLLARY 1: An equilateral triangle is also equiangular. COROLLARY 2: An equilateral triangle has three 60 angles. COROLLARY 3: The bisector of the vertex angle of an isoscles triangle is perpendicular to the base at its midpoint.  THE ISOSCELES TRIANGLE THEOREM ex1) Find the value of y. (5x - 75) y (2x + 9) Since this triangle has 2 congruent sides, then the angles opposite those sides must also be congruent. Then, since the interior angles of a triangle must add to 180: CONVERSE OF ISOSCELES TRIANGLE THEOREM THEOREM If 2 of a triangle are , then the sides opposite those are. A GIVEN: B C PROVE: AB AC HINT: DRAW AUXILIARY LINE THAT BISECTS A B C PARAGRAPH PROOF: Draw an auxiliary line that bisects A. Draw point D at the intersection of this line and BC. Since 2 of ABD and ACD are congruent, then the 3rd must be congruent also. So ABD is congruent to ACD by ASA and AB AC by CPCTC.  COROLLARY OF CONVERSE OF ISOSCELES TRIANGLE THEOREM CONVERSE OF ISOSCELES TRIANGLE THEOREM If 2 of a are , then the sides opposite those are. SINGLE, SAD, An equiangular triangle is also equilateral. LONELY, COROLLARY: CONVERSE OF ISOSCELES TRIANGLE THEOREM ex2) Find the value of x. Since this triangle has 2 congruent angles, then the sides opposite those angles must also be congruent. 4) 3( + 94 x (2x -6 ) -3 43 43 4 - (5x + 10) CONVERSE OF ISOSCELES TRIANGLE THEOREM PROOF J 1 3 GIVEN: 1 2 O PROVE: OK OJ 4 K 2 STATEMENTS REASONS  GEOMETRY - LESSON 6B TRIANGLE CONGRUENCE THEOREMS B T W O E I BOW TIE AAS  AAS (angle-angle-side) Theorem: B T GIVEN: BW TE B T O I PROVE: BOW TIE W O E I STATEMENTS REASONS 1. BW TE 1. Given 2. B T 2. Given 3. O I 3. Given 4. E W 4. If 2 of one are to 2 of another , then the 3rd are also. 5. BOW TIE 5. ASA (2, 1, 4) HL (hypotenuse-leg) Theorem: If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. A S W E U M AWE SUM HL Any 2 Triangles: SSS SAS ASA AAS 2 Right Triangles: HL IMPORTANT NOTE: Although we have decided to call SSS, SAS, and ASA postulates, and AAS and HL theorems, there is no one correct way to axiomatize the facts of geometry. A number of different axiomatizations have been developed by respected mathematicians, some referring to our postulates as theorems. Our goal is not to learn a particular "correct" set of postulates, but to learn to reason using whichever postulates and theorems we have available.  GEOMETRY - LESSON 6C MEDIANS, ALTITUDES, PERPENDICULAR BISECTORS MEDIAN OF A TRIANGLE: a segment from a vertex to the midpoint of the opposite side ex1) Draw each median. C C C A B A B A B median from median from median from vertex A vertex B vertex C  A TRIANGLE a perpendicular segment from a vertex to the line that contains the opposite side ex2) Draw each altitude of this acute triangle. F F F D D D altitude from altitude from altitude from  A TRIANGLE a perpendicular segment from a vertex to the line that ex3) Draw each altitude of this obtuse triangle. G H G H G H altitude from altitude from altitude from  A TRIANGLE a perpendicular segment from a vertex to the line that ex4) Draw each altitude of this right triangle. L L L J J J altitude from altitude from altitude from PERPENDICULAR BISECTOR OF A SEGMENT a line (or ray or segment) that is perpendicular to the segment at its midpoint ex5) Draw the perpendicular bisector of this segment. M PERPENDICULAR BISECTOR THEOREMS THEOREM If a point lies on the perpendicular bisector of a segment, then GIVEN: Line m is the perp bisector of BC m B Dm ABD and ACD are congruent by SAS (using the 2 parts of the bisected BC, the 2 right and the common side AD). Therefore PERPENDICULAR BISECTOR THEOREMS THEOREM the point lies on the perpendicular bisector of the segment. THIS IS THE CONVERSE OF THE PREVIOUS THEOREM DEFINITION the length of the perpendicular segment from the point to the line PERPENDICULAR BISECTOR THEOREMS + 3 7x  THEOREMS THEOREM If a point lies on the bisector of an angle, then the point is X P B C GIVEN:  THEOREMS CONVERSE OF PREVIOUS THEOREM point lies on the bisector of the angle. X P B C GIVEN: