1. Stephanie Lalos

2. Theorem 50 The sum of measures of the three angles of a triangle is 180 o A B C 60 4080 o

3. Proof According to the parallel postulate, there exists exactly one line through point A parallel to BC Because of the straight angle, we know that Since and we may substitute to obtain Hence, B A C 3 21 o o o

4. Other Proofs Right triangles are used to prove the sum of the angles of a triangle in a youtube video that can be seen here.here Lemma If ABCD is a quadrilateral and <)CAB = <)DCA then AB and DC are parallel. Proof Assume to the contrary that AB and DC are not parallel. Draw a line trough A and B and draw a line trough D and C. These lines are not parallel so they cross at one point. Call this point E. Notice that <)AEC is greater than 0. Since <)CAB = <)DCA, <)CAE + <)ACE = 180 degrees. Hence <)AEC + <)CAE + <)ACE is greater than 180 degrees. Contradiction. This completes the proof. Definition Two Triangles ABC and A'B'C' are congruent if and only if |AB| = |A'B'|, |AC| = |A'C'|, |BC| = |B'C'| and, <)ABC = <)A'B'C', <)BCA = <)B'C'A', <)CAB = <)C'A'B'.

5. Definition Exterior angle – an angle of a polygon that is adjacent to and supplementary to an interior angle of the polygon Examples - 1 is an exterior angle to the below triangles 1 1 For alternative exterior angle help visit… Regents Prep

6. Theorem 51 The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles 1 C A B

7. Theorem 52 A segment joining the midpoints of two sides of a triangle is parallel to the third side, and its length is one-half the length of the third side. (Midline Theorem) A B C DE Given: D & E are midpoints Therefore, AD DB & BE EC Prove: a. DE AC b. DE = (AC)

8. A B C D E (vertical angles are congruent) BED CEF (SAS) (CPCTC) Extend DE through to a point F so that EF DE. F is now established, so F and C determine FC. F FC DA (alt. int. Lines) FC DA (transitive) DFCA is a parallelogram, one pair of opposite sides is both congruent and parallel, therefore, DF AC Opposite sides of a parallelogram are congruent, so DF=AC, since EF=DE, DE= (EF) and by substitution DE= (AC).

9. Sample Problems x + 100 + 60 = 180 55 + 80 + y = 180 x + y + z = 180 x + 160 = 180 135 + y = 180 20 + 45 + z = 180 x = 20 y = 45 z = 115 80 55 100 60y z x substitution

10. The measures of the three angles of a triangle are in the ratio 2:4:6. Find the measure of the smallest angle. 2x 4x 6x 2x + 4x + 6x = 180 12x = 180 x = 15 2x = 30

11. 80 B C A x x y y D Bisectors BD and CD meet at D Let ABC = 2x and ABC = 2y In EBC, x + y + = 180 50 + = 180 (substitution) = 130 In ABC, 2x + 2y + 80 = 180 2x + 2y = 100 x + y = 50

12. 1 A B C, and the measure of is twice that of o Let = x and = (2x) o o According to theorem 51, is equal to + Find the measure of each angle of the triangle. 150 = x + 2x 150 = 3x 50 = x = 50 o = 100 o = 30 o

13. Practice Problems 0 Find the measures of the numbered angles. 47 o 86 o 1 69 71 o 2 3 4 o 5 65 o 40 o 125 o 2 95 3 1. 2. o

14. D E F 16 Find: GH GH 3. A B C D E 70 o Find:,, and 4. Three triangles are in the ratio 3:4:5. Find the measure of the largest angle. 5.

15. A B C D 50 Find: o 7. 4x+6 2x+4 x+24 6. Find: Q R S 8. Always, Sometimes, Never a.The acute angles of a right triangle are complementary. b.A triangle contains two obtuse angles. c.If one angle of an isosceles triangle is 60, it is equilateral. d.The supplement of one of the angles in a triangle is equal in measure to the sum of the other two angles. o

16. Answer Key 1. 1 = 47 5. 75 2 = 40 3 = 93 6. 48 4 = 40 5 = 140 7. 115 2. 1 = 75 8. a. A 2 = 85 b. N 3 = 70 c. A d. A 3. GH = 8 4. = 20 = 90 = 70

17. Works Cited "Exterior Angles of a Triangle." Regents Prep. 2008. 29 May 2008. Rhoad, Richard, George Milauskas, and Robert Whipple. Geometry for Enjoyment and Challenge. Boston: McDougal Littell, 1991. "Triangle." Apronus. 29 May 2008.