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Triangle Sum Properties

Triangle Sum Properties. 4.1. To Clarify*******. A triangle is a polygon with three sides. A triangle with vertices A, B, and C is called triangle ABC. Triangles on a plane. We can find the side lengths √(-1 – 0) 2 + (2 – 0) 2 = √5 ≈ 2.2

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Triangle Sum Properties

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  1. Triangle Sum Properties 4.1

  2. To Clarify******* • A triangle is a polygon with three sides. • A triangle with vertices A, B, and C is called triangle ABC

  3. Triangles on a plane • We can find the side lengths • √(-1 – 0)2 + (2 – 0)2 = √5 ≈ 2.2 • √(6– 0)2 + (3 – 0)2 = √45 ≈ 6.7 • √(6 – -1)2 + (3 – 2)2 = √50 ≈ 7.1 • This is a scalene triangle • We can also determine if it is a right triangle. (hint, look for perpendicular angles) • Slope of OP = (2-0)/(-1-0) = -2 • Slope of OQ = (3-0)/(6-0) = ½ • The lines are perpendicular and form a right angle so this is a right scalene triangle

  4. Try it out • Triangle ABC has the vertices A(0,0), B(3,3) and C (-3,3). Classify it by its sides. Then determine if it is a right triangle.

  5. Extending sides • When you extend the sides of a polygon there are new angles formed. • The original angles (on the inside) are called interior*angles. • The new angles formed are called exterior*angles.

  6. Triangle Sum Theorem • 4.1: The sum of the measures of the interior angles of a triangle is 180°

  7. b Prove it 4 5 2 • Given: Triangle ABC • Prove: m<1 + m<2 + m<3 = 180° 1 3 c a

  8. Exterior Angle Theorem • 4.2: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles

  9. Apply theorem 4.2 • Find m<JKM • Step 1: Write an equation • Step 2: Plug in x (2x – 5) = 70 + x 2(75) -5 = 145

  10. Corollary* • A corollary to a theorem is a statement that can be proved easily by using the theorem. • Corollary to the triangle sum theorem: The acute angles of a right triangle are complementary

  11. Apply Congruence 4.2

  12. Congruent Figures • Two figures are congruent if they have exactly the same size and shape. • All of the parts of one figure are congruent to the corresponding parts* of the other figure.

  13. Use properties of Congruent Figures • DEFG c= SPQR • Find x • Find y 8 10

  14. Third Angles Theorem • 4.3: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

  15. Properties of Congruent triangles Theorem • Reflexive property: ABC is congruent to ABC • Symmetric property: if ABC is congruent to DEF then DEF is congruent to ABC • Transitive Property: If ABC is congruent to DEF and DEF is congruent to JKL, • then ABC is congruent to JKL

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