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4.1 Apply Triangle Sum Properties

4.1 Apply Triangle Sum Properties. Objectives. Identify and classify triangles by angles or sides Apply the Angle Sum Theorem Apply the Exterior Angle Theorem. A triangle is a 3-sided polygon The sides of ∆ABC are AB, BC, and AC The vertices of ∆ABC are A, B, and C

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4.1 Apply Triangle Sum Properties

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  1. 4.1 Apply Triangle Sum Properties

  2. Objectives • Identify and classify triangles by angles or sides • Apply the Angle Sum Theorem • Apply the Exterior Angle Theorem

  3. A triangle is a 3-sided polygon The sides of ∆ABC are AB, BC, and AC The vertices of ∆ABC are A, B, and C Two sides sharing a common vertex are adjacent sides The third side is called the opposite side All sides can be adjacent or opposite (it just depends which vertex is being used) A adjacent adjacent B C Parts of a Triangle Side opposite A

  4. Classifying Triangles by Angles One way to classify triangles is by their angles… Acuteall 3 angles are acute (measure < 90°) Obtuse1 angle is obtuse (measure > 90°) Right1 angle is right(measure = 90°) An acute ∆ with all angles  is an equiangular ∆ .

  5. Example 1: ARCHITECTUREThe triangular truss below is modeled for steel construction. Classify JMN, JKO, and OLNas acute, equiangular, obtuse, or right. 60° 60°

  6. Example 1: Answer: JMN has one angle with measure greater than 90, so it is an obtuse triangle. JKO has one angle with measure equal to 90, so it is a right triangle. OLN is an acute triangle with all angles congruent, so it is an equiangular triangle.

  7. Classifying Triangles by Sides Another way to classify triangles is by their sides… Equilateral3 congruent sides Isosceles2 congruent sides Scaleneno congruent sides

  8. Identify the isosceles triangles in the figure if Example 2a: Isosceles triangles have at least two sides congruent. Answer:UTX and UVX are isosceles.

  9. Identify the scalene triangles in the figure if Example 2b: Scalene triangles have no congruent sides. Answer:VYX, ZTX, VZU, YTU, VWX, ZUX, and YXU are scalene.

  10. Example 2c: Identify the indicated triangles in the figure. a. isosceles triangles Answer: ADE, ABE b. scalene triangles Answer: ABC, EBC, DEB, DCE, ADC, ABD

  11. ALGEBRAFind d and the measure of each side of equilateral triangleKLMif and Since KLM is equilateral, each side has the same length. So Example 3: 5 = d

  12. Answer: For KLM, and the measure of each side is 7. Example 3: Next, substitute to find the length of each side. LM = 7 KM = 7 KL = 7

  13. Example 4: COORDINATE GEOMETRY Find the measures of the sides of RST. Classify the triangle by sides.

  14. Answer: ; since all 3 sides have different lengths, RST is scalene. Example 4: Use the distance formula to find the lengths of each side.

  15. 2 1 3 4 Exterior Angles and Triangles • An exterior angle is formed by one side of a triangle and the extension of another side (i.e. 1 ). • The interior angles of the triangle not adjacent to a given exterior angle are called the remote interior angles (i.e. 2 and 3).

  16. X Y Z Theorem 4.1 – Triangle Sum Theorem The sum of the measures of the interior angles of a triangle is 180°. mX + mY + mZ = 180°

  17. Find first because the measure of two angles of the triangle are known. Example 5: Find the missing angle measures. Angle Sum Theorem Simplify. Subtract 117 from each side.

  18. Answer: Example 5: Angle Sum Theorem Simplify. Subtract 142 from each side.

  19. Your Turn: Find the missing angle measures. Answer:

  20. 2 1 3 4 Theorem 4.2 – Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. m 1 = m2 + m 3

  21. Example 6: Find the measure of each numbered angle in the figure. Exterior Angle Theorem Simplify. If 2 s form a linear pair, they are supplementary. Substitution Subtract 70 from each side.

  22. Example 6: Exterior Angle Theorem Substitution Subtract 64 from each side. If 2 s form a linear pair, they are supplementary. Substitution Simplify. Subtract 78 from each side.

  23. Answer: Example 6: Angle Sum Theorem Substitution Simplify. Subtract 143 from each side.

  24. Your Turn: Find the measure of each numbered angle in the figure. Answer:

  25. Corollaries • A corollary is a statement that can be easily proven using a theorem. • Corollary 4.1 – The acute s of a right ∆ are complementary. • Corollary 4.2 – There can be at most one right or obtuse  in a∆.

  26. GARDENING The flower bed shown is in the shape of a right triangle. Find if is 20. Example 3: Corollary 4.1 Substitution Subtract 20 from each side. Answer:

  27. The piece of quilt fabric is in the shape of a right triangle. Find if is 62. Your Turn: Answer:

  28. Assignment • Pre-AP Geometry:Pgs. 221-224 #1 – 6, 14 – 19, 21 – 26, 31 – 37

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