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Section 7.1 – Triangle Application Theorems

Section 7.1 – Triangle Application Theorems. Stephanie Lalos. Theorem 50. The sum of measures of the three angles of a triangle is 180 o. B. 60. 80. 40. C. A. o. Proof. According to the parallel postulate, there exists exactly one line through point A parallel to BC

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Section 7.1 – Triangle Application Theorems

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  1. Section 7.1 – Triangle Application Theorems Stephanie Lalos

  2. Theorem 50 • The sum of measures of the three angles of a triangle is 180o B 60 80 40 C A 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, o A 3 1 2 B C o o

  4. Other Proofs LemmaIf ABCD is a quadrilateral and <)CAB = <)DCA then AB and DC are parallel. ProofAssume 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. DefinitionTwo 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'. Right triangles are used to prove the sum of the angles of a triangle in a youtube video that can be seen here.

  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 For alternative exterior angle help visit… Regents Prep 1

  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 B 1 A C

  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) B Given: D & E are midpoints Therefore, AD DB & BE EC Prove: a. DE AC b. DE = (AC) D E A C

  8. B Extend DE through to a point F so that EF DE. F is now established, so F and C determine FC. F E D (vertical angles are congruent) BED CEF (SAS) (CPCTC) A C 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). FC DA (alt. int. Lines) FC DA (transitive)

  9. Sample Problems 80 100 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 z 55 x y 60 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. 4x 2x 6x 2x + 4x + 6x = 180 12x = 180 x = 15 2x = 30

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

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

  13. Practice Problems Find the measures of the numbered angles. 1. 0 86 o o 69 2 3 65 o 5 4 o 1 47 95 o 71 o 2 2. 1 3 25 o 40 o

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

  15. R 7. B 6. 2x+4 50 o D x+24 4x+6 A C Q S Find: Find: 8. Always, Sometimes, Never • The acute angles of a right triangle are complementary. • A triangle contains two obtuse angles. • If one angle of an isosceles triangle is 60 , it is equilateral. • 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 <http://regentsprep.org/REgents/math/triang/LExtAn g.htm>. Rhoad, Richard, George Milauskas, and Robert Whipple. Geometry for Enjoyment and Challenge. Boston: McDougal Littell, 1991. "Triangle." Apronus. 29 May 2008 <http://www.apronus.com/geometry/triangle.htm>.

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