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Lesson 3-1

Triangle Fundamentals. Lesson 3-1. Polygon. Polygon - closed figure, in a plane (2-D), made of segments intersecting only at their endpoints. EX). NOT EX). Triangles. 3. Triangle - sided polygon- ABC. Vertices-. Sides of a -. A. A B C. AB BC AC. B. C. B. C. A.

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Lesson 3-1

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  1. Triangle Fundamentals Lesson 3-1 Lesson 3-1: Triangle Fundamentals

  2. Polygon Polygon - closed figure, in a plane (2-D), made of segments intersecting only at their endpoints EX) NOT EX) Lesson 3-1: Triangle Fundamentals

  3. Triangles 3 • Triangle- sided polygon- ABC Vertices- Sides of a - A A B C AB BC AC B C Lesson 3-1: Triangle Fundamentals

  4. B C A Naming Triangles Triangles are named by using its vertices. For example, we can call the following triangle: ∆ABC ∆ACB ∆BAC ∆BCA ∆CAB ∆CBA Lesson 3-1: Triangle Fundamentals

  5. Opposite Sides and Angles Opposite Sides: Side opposite to B : Side opposite to A : Side opposite to C : Opposite Angles: Angle opposite to : A Angle opposite to : B Angle opposite to : C Lesson 3-1: Triangle Fundamentals

  6. A triangle in which all angles are less than 90˚. G ° 70 ° ° 50 60 H I A ° 45 ° 100 ° 35 C B Classifying Triangles by Angles 3 Acute: Obtuse: 1 1 • A triangle in which and only angle is greater than 90˚& less than 180˚ Lesson 3-1: Triangle Fundamentals

  7. Classifying Triangles by Angles 1 1 Right: • A triangle in which and only angle is 90˚ Equiangular: • A triangle in which all angles are the same measure. Lesson 3-1: Triangle Fundamentals

  8. Equilateral: A A B C C BC = 3.55 cm B BC = 5.16 cm G H I HI = 3.70 cm Classifying Triangles by Sides No 2 sides are congruent A triangle in which all 3 sides are different lengths. Scalene: AC = 3.47 cm AB = 3.47 cm AB = 3.02 cm AC = 3.15 cm Isosceles: A triangle in which at least 2 sides are equal. • A triangle in which all 3 sides are equal. GI = 3.70 cm GH = 3.70 cm Lesson 3-1: Triangle Fundamentals

  9. polygons triangles scalene isosceles equilateral Classification by Sides with Flow Charts & Venn Diagrams Polygon Triangle Scalene Isosceles Equilateral Lesson 3-1: Triangle Fundamentals

  10. polygons triangles right acute equiangular obtuse Classification by Angles with Flow Charts & Venn Diagrams Polygon Triangle Right Obtuse Acute Equiangular Lesson 3-1: Triangle Fundamentals

  11. Parts of a right HYPOTENUSE LEG LEG Lesson 3-1: Triangle Fundamentals

  12. Parts of an Isoceles A The congruent sides are called legs and the third side is called the base Vertex Angle LEG LEG Base Angles C B BASE Lesson 3-1: Triangle Fundamentals

  13. Theorems & Corollaries Angle Sum Theorem: The sum of the interior angles in a triangle is 180˚. A line added to a picture to help prove something Auxillary Line: Third Angle Theorem: If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent. Corollary 1: Each angle in an equiangular triangle is 60˚. Corollary 2: Acute angles in a right triangle are complementary. There can be at most one right or obtuse angle in a triangle. Corollary 3: Lesson 3-1: Triangle Fundamentals

  14. A B C ° 50 ° x Isosceles Triangle Theorems If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Example: Find the value of x. By the Isosceles Triangle Theorem, the third angle must also be x. Therefore, x + x + 50 = 180 2x + 50 = 180 2x = 130 x = 65 Lesson 3-2: Isosceles Triangle

  15. A B C A 3x - 7 x+15 ° ° B 50 50 C Isosceles Triangle Theorems If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Example: Find the value of x. Since two angles are congruent, the sides opposite these angles must be congruent. 3x – 7 = x + 15 2x = 22 X = 11 Lesson 3-2: Isosceles Triangle

  16. Exterior Angle Theorem The measure of the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. Remote Interior Angles A Exterior Angle D Example: Find the mA. B C 3x - 22 = x + 80 3x – x = 80 + 22 2x = 102 mA = x = 51° Lesson 3-1: Triangle Fundamentals

  17. Congruent Triangles Lesson 4-2 Lesson 4-2: Congruent Triangles

  18. Congruent Figures Congruent figures are two figures that have the same size and shape. IF two figures are congruent THEN they have the same size and shape. IF two figures have the same size and shape THEN they are congruent. Two figures have the same size and shape IFF they are congruent. Lesson 4-2: Congruent Triangles

  19. R N D M F E Congruent Triangles  D  M  ____ =  E  N  _____ ≡ │ ≡ │  F  R  ______ = DE Note: EF ∆ DEF DF ∆MNR  ______ ∆MNR ∆FED MN  ___ NR  ___ MR  ___ Lesson 4-2: Congruent Triangles

  20. A C B P R Q Congruent Triangles - CPCTC Corresponding Parts of Congruent Triangles are Congruent CPCTC: Two triangles are congruent IFF their corresponding parts (angles and sides) are congruent. = │ Vertices of the 2 triangles correspond in the same order as the trianglesare named. ≡ A ↔ P; B ↔ Q; C ↔ R Corresponding sides and angles of the two congruent triangles: = │ ≡ Lesson 4-2: Congruent Triangles

  21. Example………… When referring to congruent triangles (or polygons), we must name corresponding vertices in the same order. S U N R A Y R SUN  RAY Y A N Also NUS  YAR U Also USN  ARY S Lesson 4-2: Congruent Triangles

  22. Example ……… If these polygons are congruent, how do you name them ? P O U N M E S A T R • Pentagon MONTA Pentagon PERSU • Pentagon ATNOM Pentagon USREP • Etc. Lesson 4-2: Congruent Triangles

  23. Included Angles & Sides Included Angle: * * * Included Side: Lesson 4-3: SSS, SAS, ASA

  24. Proving Triangles Congruent Lesson 4-3 (SSS, SAS, ASA) Lesson 4-3: SSS, SAS, ASA

  25. A A D D B C B C F F E E Postulates If two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle, then the triangles are congruent. ASA If two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent. SAS Lesson 4-3: SSS, SAS, ASA

  26. A D F B C E Postulates If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. SSS Included Angle: In a triangle, the angle formed by two sides is the included angle for the two sides. Included Side: The side of a triangle that forms a side of two given angles. Lesson 4-3: SSS, SAS, ASA

  27. Steps for Proving Triangles Congruent • Mark the Given. • Mark … Reflexive Sides/Vertical Angles • Choose a Method. (SSS , SAS, ASA) • List the Parts … in the order of the method. • Fill in the Reasons … why you marked the parts. • Is there more? Lesson 4-3: SSS, SAS, ASA

  28. A B @ AB CD 1. @ BC DA 2. @ AC CA 3. C D Problem 1  Step 1: Mark the Given Step 2: Mark reflexive sides SSS Step 3: Choose a Method (SSS /SAS/ASA ) Step 4: List the Parts in the order of the method Step 5: Fill in the reasons Step 6: Is there more? Given Given Reflexive Property SSS Postulate Lesson 4-3: SSS, SAS, ASA

  29. Problem 2  Step 1: Mark the Given Step 2: Mark vertical angles SAS Step 3: Choose a Method (SSS /SAS/ASA) Step 4: List the Parts in the order of the method Step 5: Fill in the reasons Step 6: Is there more? Given Vertical Angles. Given SAS Postulate Lesson 4-3: SSS, SAS, ASA

  30. X W Y Z Problem 3 Step 1: Mark the Given Step 2: Mark reflexive sides ASA Step 3: Choose a Method (SSS /SAS/ASA) Step 4: List the Parts in the order of the method Step 5: Fill in the reasons Step 6: Is there more? Given Reflexive Postulate Given ASA Postulate Lesson 4-3: SSS, SAS, ASA

  31. A D D A B C F E B C F E Postulates If two angles and a non included side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent. AAS If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent. HL Lesson 4-4: AAS & HL Postulate

  32. Problem 1  Step 1: Mark the Given Step 2: Mark vertical angles AAS Step 3: Choose a Method (SSS /SAS/ASA/AAS/ HL ) Step 4: List the Parts in the order of the method Step 5: Fill in the reasons Step 6: Is there more? Given Vertical Angle Thm Given AAS Postulate Lesson 4-4: AAS & HL Postulate

  33. Problem 2  Step 1: Mark the Given Step 2: Mark reflexive sides HL Step 3: Choose a Method (SSS /SAS/ASA/AAS/ HL ) Step 4: List the Parts in the order of the method Step 5: Fill in the reasons Step 6: Is there more? Given Given Reflexive Property HL Postulate Lesson 4-4: AAS & HL Postulate

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