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Chapter 4: Congruent Triangles

Chapter 4: Congruent Triangles. Lesson 1: Classifying Triangles. Classifying Triangle by Angles. Acute Triangle: all of the angles are acute Obtuse Triangle: one angle is obtuse, the other two are acute Right Triangle: one angle is right, the other two are acute

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Chapter 4: Congruent Triangles

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  1. Chapter 4: Congruent Triangles Lesson 1: Classifying Triangles

  2. Classifying Triangle by Angles • Acute Triangle: all of the angles are acute • Obtuse Triangle: one angle is obtuse, the other two are acute • Right Triangle: one angle is right, the other two are acute • Equiangular Triangle: all the angles are 60 degrees

  3. Classifying Triangles by Sides • Scalene Triangle: all sides are different measures • Isosceles Triangle: at least two sides have the same measure • Equilateral Triangle: all sides have the same measure 7 3 5 * vertex angle= formed by the two congruent sides of an isosceles triangle * base= the side of an isosceles triangle not congruent to the others

  4. If point Y is the midpoint of VX, and WY = 3.0 units, classify ΔVWY as equilateral, isosceles, or scalene. Explain your reasoning.

  5. ALGEBRA Find the measure of the sides of isosceles triangle KLM with base KL. __

  6. ALGEBRAFind x and the measure of each side of equilateral triangle ABC if AB = 6x – 8, BC = 7 + x, and AC = 13 – x.

  7. Find the measure of each side of Triangle JKL and classify the triangle based on its sides. • J(-3, 2) K(2, 1) L(-2, -3)

  8. Find y ___ ___

  9. Chapter 4: Congruent Triangles Lesson 2: Angles of Triangles

  10. The sum of the measures of the angles of a triangle is always 180 degrees. The acute angles of a right triangle are complementary There can be at most one right or one obtuse angle in a triangle Third Angle Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles of the triangles are also congruent. X A Y B Z C If A X, and B Y, then C Z.

  11. Interior and Exterior Angles of Triangles • Exterior angle: formed by one side of a triangle and the extension of another side • The interior angles farthest from the exterior angle are its remote interior angles.(remote interior angles are not adjacent to the exterior angle) Remote interior angles Exterior angle An exterior angle is equal to the sum of its remote interior angles. ex: 1 + 2 = 4 2 1 3 4

  12. Anticipation Guide: read each statement. State whether the sentence is true or false. If the statement is false- rewrite it with the correct term in place of the underlined word • The acute angles of a right triangle are supplementary • The sum of the measures of the angles of any triangle is 100 • A triangle can have at most one right angle or acute angle • If two angles of one triangle are congruent to two angles of another triangle, then the third angle of the triangles are congruent • The measure of an exterior angle of a triangle is equal to the difference of the measures of the two remote interior angles • If the measures of two angles of a triangle are 62 and 93, then the measure of the third angle is 35 • An exterior angle of a triangle forms a linear pair with an interior angle of the triangle

  13. SOFTBALL The diagram shows the path of the softball in a drill developed by four players. Find the measure of each numbered angle.

  14. Find the measure of each numbered angle.

  15. GARDENING Find the measure of FLW in the fenced flower garden shown.

  16. The piece of quilt fabric is in the shape of a right triangle. Find the measure of ACD.

  17. Find the measure of each numbered angle.

  18. Find m3.

  19. Chapter 4: Congruent Triangles Lesson 6: Isosceles Triangles

  20. Isosceles Triangles . Vertex Angle - If two sides of a triangle are congruent, the two angles opposite of them are also congruent leg leg -If two angles of a triangle are congruent, then two sides opposite of them are also congruent Base angles - If a triangle is equilateral, it is also equiangular

  21. A.Find mR. B. Find PR

  22. A. Find mT.

  23. ALGEBRA Find the value of each variable

  24. Chapter 4: Congruent Triangles Lesson 3: Congruent Triangles

  25. Definition of Congruent Triangles • Congruent triangles are triangles with exactly the same size and shape • CPCTC: Corresponding Parts of Congruent Triangles are Congruent • Two triangles are congruent if and only if their corresponding parts are congruent

  26. Corresponding Parts A • Corresponding parts have the same congruence markings • AB HI • AC HJ • BC IJ • A H • B I • C J B C H I J

  27. Congruence Transformations • Slide or Translation: the triangle is in the same position farther down, up, or across the page • Turn or Rotation: the triangle is spun around a point (usually one of the angles) • Flip or reflection: the triangle is shown in a mirror image across a line of symmetry

  28. Write a congruence statement for the triangles.

  29. Name the corresponding congruent angles for the congruent triangles.

  30. In the diagram, ΔITP ΔNGO. Find the values of x and y.

  31. In the diagram, ΔFHJ ΔHFG. Find the values of x and y.

  32. 1. Given 1. 2. Reflexive Property of Congruence 2. 3.Q  O, NPQ  PNO 3. Given 4. _________________ 4.QNP  ONP ? 5.ΔQNPΔOPN 5. Definition of Congruent Polygons Find the missing information in the following proof. Prove:ΔQNPΔOPN Proof:

  33. Write a two-column proof. Prove:ΔLMNΔPON

  34. Chapter 4: Congruent Triangles Lesson 4 and 5: Proving Congruence- SSS, SAS, ASA, AAS, and HL

  35. SSS • Side-Side-Side • If all three sets of corresponding sides are congruent, the triangles are congruent A M B C N O ABC MNO

  36. SAS • Side-Angle-Side • If two corresponding sides and the included angles of two triangles are congruent, then the triangles are congruent * The included angle is the angle between the congruent sides X F Y Z G H XYZ FGH

  37. ASA • Angle-Side-Angle • If two sets of corresponding angles and the included sides are congruent, then the triangles are congruent * The included side is the side between the two congruent angles J R L K T S JKL RST

  38. AAS • Angle-Angle-Side • If two sets of corresponding angles and one of the corresponding non-included sides are congruent, then the triangles are congruent T E G F V U EFG TUV

  39. HL • Hypotenuse-Leg • If the hypotenuse and one set of corresponding legs of two right triangles are congruent, then the triangles are congruent C R D H A M CDH RAM

  40. Determine if the triangles are congruent. If they are, write the congruence statement.

  41. Given: AC ABD is the midpoint of BC.Prove: ΔADC  ΔADB ___ ___

  42. Determine whether ΔABCΔDEFfor A(–5, 5), B(0, 3), C(–4, 1), D(6, –3), E(1, –1), and F(5, 1).

  43. Determine if the triangles are congruent. If they are, write the congruence statement.

  44. Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove congruence, choose not possible.

  45. Write a two column proof.

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