Triangles and Angles

1. Triangles and Angles

2. Standard/Objectives: Standard 3: Students will learn and apply geometric concepts. Objectives: • Classify triangles by their sides and angles. • Find angle measures in triangles DEFINITION: A triangle is a figure formed by three segments joining three non-collinear points.

3. Names of triangles Triangles can be classified by the sides or by the angle Equilateral—3 congruent sides Isosceles Triangle—2 congruent sides Scalene—no congruent sides

4. Acute Triangle 3 acute angles

5. Equiangular triangle • 3 congruent angles. An equiangular triangle is also acute.

6. 1 right angle Right Triangle Obtuse Triangle

8. Each of the three points joining the sides of a triangle is a vertex.(plural: vertices). A, B and C are vertices. Two sides sharing a common vertext are adjacent sides. The third is the side opposite an angle Parts of a triangle adjacent Side opposite A adjacent

9. Red represents the hypotenuse of a right triangle. The sides that form the right angle are the legs. Right Triangle hypotenuse leg leg

10. An isosceles triangle can have 3 congruent sides in which case it is equilateral. When an isosceles triangle has only two congruent sides, then these two sides are the legs of the isosceles triangle. The third is thebase. Isosceles Triangles leg base leg

11. Explain why ∆ABC is an isosceles right triangle. In the diagram you are given that C is a right angle. By definition, then ∆ABC is a right triangle. Because AC = 5 ft and BC = 5 ft; AC BC. By definition, ∆ABC is also an isosceles triangle. Identifying the parts of an isosceles triangle About 7 ft. 5 ft 5 ft

12. Identify the legs and the hypotenuse of ∆ABC. Which side is the base of the triangle? Sides AC and BC are adjacent to the right angle, so they are the legs. Side AB is opposite the right angle, so it is t he hypotenuse. Because AC BC, side AB is also the base. Identifying the parts of an isosceles triangle Hypotenuse & Base About 7 ft. 5 ft 5 ft leg leg

14. Using Angle Measures of Triangles Smiley faces are interior angles and hearts represent the exterior angles Each vertex has a pair of congruent exterior angles; however it is common to show only one exterior angle at each vertex.

15. Ex. 3 Finding an Angle Measure. Exterior Angle theorem: m1 = m A +m 1 x + 65 = (2x + 10) 65 = x +10 55 = x 65 (2x+10) x

17. Corollary to the triangle sum theorem The acute angles of a right triangle are complementary. m A + m B = 90 Finding angle measures 2x x

18. X + 2x = 90 3x = 90 X = 30 So m A = 30 and the m B=60 Finding angle measures B 2x x A C

21. Congruence and Triangles

22. Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives: • Identify congruent figures and corresponding parts • Prove that two triangles are congruent

24. Identifying congruent figures • Two geometric figures are congruent if they have exactly the same size and shape. NOT CONGRUENT CONGRUENT

25. Corresponding angles A ≅ P B ≅ Q C ≅ R Corresponding Sides AB ≅ PQ BC ≅ QR CA ≅ RP Triangles B Q R A C P

26. Z • If Δ ABC is  to Δ XYZ, which angle is  to C?

28. Thm 4.33rd angles thm • If 2 s of one Δ are  to 2 s of another Δ, then the 3rd s are also .

29. Ex: find x ) ) 22o )) 87o )) (4x+15)o

30. Ex: continued 22+87+4x+15=180 4x+15=71 4x=56 x=14

32. 9 cm 91° 86° 113° Ex: ABCD is  to HGFE, find x and y. F E (5y-12)° G 4x – 3 cm H 4x-3=9 5y-12=113 4x=12 5y=125 x=3 y=25

33. Thm 4.4Props. of Δs A B • Reflexive prop of Δ - Every Δ is  to itself (ΔABC ΔABC). • Symmetric prop of Δ- If ΔABC ΔPQR, then ΔPQR ΔABC. • Transitive prop of Δ - If ΔABC ΔPQR & ΔPQR ΔXYZ, then ΔABC  ΔXYZ. C P Q R X Y Z

39. Proving Δs are  : SSS and SAS

40. Standards/Benchmarks Standard 2: Students will learn and apply geometric concepts Objectives: • Prove that triangles are congruent using the SSS and SAS Congruence Postulates. • Use congruence postulates in real life problems such as bracing a structure.

41. Remember? • As of yesterday, Δs could only be  if ALL sides AND angles were  • NOT ANY MORE!!!! • There are two short cuts to add.

42. Post. 19Side-Side-Side (SSS)  post • If 3 sides of one Δ are  to 3 sides of another Δ, then the Δs are .

43. A Meaning: ___ ___ ___ ___ If seg AB  seg ED, seg AC  seg EF & seg BC  seg DF, then ΔABC ΔEDF. B C ___ ___ E ___ ___ ___ ___ ___ ___ D F

44. Given: seg QR  seg UT, RS  TS, QS=10, US=10Prove: ΔQRS ΔUTS U Q 10 10 R S T

47. Proof Statements Reasons 1. 1. given 2. QS=US 2. subst. prop. = 3. Seg QS  seg US 3. Def of  segs. 4. Δ QRS Δ UTS 4. SSS post

48. Post. 20Side-Angle-Side post. (SAS) • If 2 sides and the included  of one Δ are  to 2 sides and the included  of another Δ, then the 2 Δs are .

49. If seg BC  seg YX, seg AC  seg ZX, and C X, then ΔABC  ΔZXY. B Y ) ( C A X Z