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Chapter 4

Chapter 4. Congruent Triangles. Chapter Objectives. Classification of Triangles by Sides Classification of Triangles by Angles Exterior Angle Theorem Triangle Sum Theorem Adjacent Sides and Angles Parts of Specific Triangles 5 Congruence Theorems for Triangles. Lesson 4.1.

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Chapter 4

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  1. Chapter 4 Congruent Triangles

  2. Chapter Objectives • Classification of Triangles by Sides • Classification of Triangles by Angles • Exterior Angle Theorem • Triangle Sum Theorem • Adjacent Sides and Angles • Parts of Specific Triangles • 5 Congruence Theorems for Triangles

  3. Lesson 4.1 Triangles and Angles

  4. Lesson 4.1 Objectives • Identify the parts of a triangle • Classify triangles according to their sides • Classify triangles according to their angles • Calculate angle measures in triangles

  5. Classification of Triangles by Sides

  6. Classification of Triangles by Angles

  7. Example 1 • You must classify the triangle as specific as you possibly can. • That means you must name • Classification according to angles • Classification according to sides • In that order! • Example Obtuse isosceles

  8. Vertex • The vertex of a triangle is any point at which two sides are joined. • It is a corner of a triangle. • There are 3 in every triangle

  9. Adjacent sides are those sides that intersect at a common vertex of a polygon. These are said to be adjacent to an angle. Adjacent angles are those angles that are right next to each other as you move inside a polygon. These are said to be adjacent to a specific side. Adjacent Sides and Adjacent Angles

  10. Special Parts in a Right Triangle • Right triangles have special names that go with it parts. • For instance: • The two sides that form the right angle are called the legs of the right triangle. • The side opposite the right angle is called the hypotenuse. • The hypotenuse is always the longest side of a right triangle. hypotenuse legs

  11. Special Parts of an Isosceles Triangle • An isosceles triangle has only two congruent sides • Those two congruent sides are called legs. • The third side is called the base. legs base

  12. More Parts of Triangles • If you were to extend the sides you will see that more angles would be formed. • So we need to keep them separate • The three original angles are called interior angles because they are inside the triangle. • The three new angles are called exterior angles because they lie outside the triangle.

  13. Example 2 Classify the following triangles by their sides and their angles. Scalene Scalene Obtuse Right Isosceles Acute

  14. Theorem 4.1:Triangle Sum Theorem • The sum of the measures of the interior angles of a triangle is 180o. B mA + mB + mC = 180o C A

  15. Example 3 Solve for x and then classify the triangle based on its angles. 75 Acute 50 3x + 2x + 55 = 180 Triangle Sum Theorem 5x + 55 = 180 Simplify 5x = 125 SPOE x = 25 DPOE

  16. B A 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 nonadjacent interior angles. C m A +m B = m C

  17. Example 4 Solve for x x = 50 + 70 Exterior Angle Theorem x = 120 Simplify

  18. A B C Corollary to theTriangle Sum Theorem • A corollary to a theorem is a statement that can be proved easily using the original theorem itself. • This is treated just like a theorem or a postulate in proofs. • The acute angles in a right triangle are complementary. mA + mB = 90o

  19. Homework 4.1 • In Class • 1-9 • p199-201 • In HW • 10-26, 31-39, 41-47, 49, 50, 52-68 • Due Tomorrow

  20. Lesson 4.2 Congruence and Triangles

  21. Lesson 4.2 Objectives • Identify congruent figures and their corresponding parts. • Prove two triangles are congruent. • Apply the properties of congruence to triangles.

  22. Congruent Triangles • When two triangles are congruent, then • Corresponding angles are congruent. • Corresponding sides are congruent. • Corresponding, remember, means that objects are in the same location. • So you must verify that when the triangles are drawn in the same way, what pieces match up?

  23. B AB  DE BC  EF AC  DF A C Naming Congruent Parts • Be sure to pay attention to the proper notation when naming parts. •  ABC  DEF • This is called a congruence statement. D F • A  D • B  E • C  F and E

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

  25. Prove Triangles are Congruent • In order to prove that two triangles are congruent, we must • Show that ALLcorresponding angles are congruent, and • Show that ALL corresponding sides are congruent. • We must show all 6 are congruent!

  26. Example 5 Complete the following statements. • Segment EF  ___________ • segment OP • P  ________ • F • G  ________ • Q • mO = ________ • 110o • QO = ________ • 7 km • GFE  __________ • QPO • Yes, the order is important!

  27. Theorem 4.4:Properties of Congruent Triangles • Reflexive Property of Congruent Triangles •  ABC  ABC • Reflexive Property of  • Symmetric Property of Congruent Triangles • If  ABC  DEF, then  DEF  ABC. • Symmetric Property of  • Transitive Property of Congruent Triangles • If  ABC  DEF and  DEF  JKL, then ABC  JKL. • Transitive Property of 

  28. Homework 4.2 • None!

  29. Lesson 4.3 Proving Triangles are Congruent: SSS & SAS

  30. Lesson 4.3 Objectives • Prove triangles are congruent using the SSS Congruence Postulate • Prove triangles are congruent using the SAS Congruence Postulate

  31. Postulate 19:Side-Side-Side Congruence Postulate • If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.

  32. Postulate 20:Side-Angle-Side Congruence Postulate • If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

  33. Which One Do I Use? • Remember there are 6 parts to every triangle. • Identify which parts of the triangle do you know (100% sure) are congruent. • Rotate around the triangle keeping one thing in mind. • Cannot rotate so that 2 parts in a row are missed! • That means as you rotate by counting angle, then side, then angle, then side, then angle, and then side you cannot miss two pieces in a row! • You can skip 1, but not 2!! • Be sure the pattern that you find fits the same pattern in the same way from the other triangle. • If it fits, they are congruent.

  34. Example 6 Decide whether or not the congruence statement is true. Explain your reasoning. The statement is not true because the vertices are out of order. Reflexive Property of Congruence Reflexive Property of Congruence Because the segment is shared between two triangles, and yet it is the same segment The statement is not true because the vertices are out of order. The statement is true because of SSS Congruence

  35. Example 7 Decide whether or not there is enough information to conclude SAS Congruence. No Yes! Reflexive Property of Congruence Yes!

  36. Homework 4.3 • In Class • 1-5 • p216-218 • HW • 6-20 • Due Tomorrow

  37. Lesson 4.4 Proving Triangles are Congruent: ASA & AAS

  38. Lesson 4.4 Objectives • Prove that triangles are congruent using the ASA Congruence Postulate • Prove that triangles are congruent using the AAS Congruence Theorem

  39. Postulate 21:Angle-Side-Angle Congruence • If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.

  40. Theorem 4.5:Angle-Angle-Side Congruence • If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of the second triangle, then the two triangles are congruent.

  41. Example 8 Complete the proof Given Given Reflexive POC SSS Congruence

  42. Homework 4.4 • In Class • 1-7 • p223-227 • HW • 8-18 evens • Due Tomorrow

  43. Lesson 4.5 Using Congruent Triangles

  44. Lesson 4.5 Objectives • Observe that corresponding parts of congruent triangles are congruent

  45. Showing Triangles are Congruent • You only have 4 shortcuts right now to show that two triangles are congruent to each other. • SSS Congruence • SAS Congruence • ASA Congruence • AAS Congruence • Otherwise you need to show all 6 parts of a triangle have matching congruent parts to another triangle. • If you can use one of the above 4 shortcuts to show triangle congruency, then we can assume that all corresponding parts of the trianglesare congruent as well.

  46. Surveying • MNP  MKL • Given • Segment NM  Segment KM • Definition of a midpoint • LMK PMN • Vertical Angles Theorem • KLM NPM • ASA Congruence • Segment LK  Segment PN • Corresponding Parts of Congruent Triangles

  47. Example 9 Tell which triangles you show to be congruent in order to prove the statement is true. What postulate or theorem would help you show the triangles are congruent. Show: STV  UTV Show: Segment XY  Segment ZW Alternate Interior Angles Theorem (Parallel Lines) Reflexive Property of Congruence Reflexive Property of Congruence WXZ  YZX STV  UTV SSS Congruence ASA Congruence • Corresponding Parts of Congruent Triangles • Corresponding Parts of Congruent Triangles

  48. Homework 4.5 • In Class • 1-3 • p232-235 • HW • 4-18, 25-36 • Due Tomorrow

  49. Lesson 4.6 Isosceles, Equilateral, and Right Triangles

  50. Lesson 4.6 Objectives • Use properties of isosceles and equilateral triangles. • Identify more properties based on the definitions of isosceles and equilateral triangles. • Use properties of right triangles.

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