2.1:Triangles Properties - properties M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems involving angles, lines, polygons, circles, or right triangle ratios (sine, cosine, tangent) within mathematics or across disciplines or contexts GSE’s

Triangles Triangle-figure formed by 3 segments joining 3 noncollinear pts. Triangles are named by these three pts (ΔQRS) Would it matter if you named in a different order? Nope! ΔRQS, ΔSRQ all mean the same thing Q R S

Parts of a Triangle Sides A B C Segment AB, AC, BC Points A, B, C Angles A, B, C Angles Vertices

2 Ways to classify triangles 1) by their Angles 2) by their Sides

1)Angles Acute- Obtuse- Right- Equiangular- all 3 angles less than 90 o one angle greater than 90 o, less than 180 o One angle = 90 o All 3 angles are congruent

2) Sides Scalene Isosceles Equilateral - No sides congruent -2 sides congruent - All sides are congruent

Parts of a Right Triangle Leg Hypotenuse Sides touching the 90 o angle Side across the 90 o angle. Always the largest in a right triangle

Converse of the Pythagorean Theorem Where c is chosen to be the longest of the three sides: If a 2 + b 2 = c 2, then the triangle is right. If a 2 + b 2 > c 2, then the triangle is acute. If a 2 + b 2 < c 2, then the triangle is obtuse.

Example of the converse Name the following triangles according to their angles 1) 4in, 8in, 9 in 2) 5 in, 12 in, 13 in 4) 10 in, 11in, 12 in

Example on the coordinate plane Given DAR with vertices D(1,6) A (5,-4) R (-3, 0) Classify the triangle based on its sides and angles. Ans: DA = AR = DR = So……. Its SCALENE

Name the triangle by its angles and sides

Legs – the congruent sides Isosceles Triangle A B C Leg Base-Non congruent side Across from the vertex Vertex- Angle where the 2 congruent sides meet Base Angles: Congruent Formed where the base meets the leg

Example Triangle TAP is isosceles with angle P as the Vertex. TP = 14x -5, TA = 6x + 11, PA = 10x Is this triangle also equilateral? TA P 14x-5 6x x + 43 TP PA 14x – 5 = 10x x = 48 X = 12 TP = 14(12) -5 = 163 PA= 10(12) + 43 = 163 TA = 6(12) + 11 = 83

1. 2.

How can ΔABC be right and isosceles? A B C ____ _____

Example BCD is isosceles with BD as the base. Find the perimeter if BC = 12x-10, BD = x+5 CD = 8x+6 B C D base 12x-108x+6 X+5 Ans: 12x-10 = 8x+6 X = 4 Re-read the question, you need to find the perimeter 12(4) (4)+6 38 (4)+5 9 Perimeter = = 85 Final answer

Example 2 Solve for x. 5x +24 Ans: (5x+24) + (5x+24) + (4x+6) = 180 5x x x+6 = x + 54 = x = 126 x = 9

Exterior Angles Thm The measure of an exterior angle of a triangle is equal to the sum of the measures of the 2 nonadjacent interior angles. m  1=m  A+ m  B m  1= + 1 A B C

Example Exterior angle = 2 nonadjacent angles Find every angle measure x-10 = (25) + (x+15) 3x-10 = x +40 2x= 50 x = 25

Exterior angle example Solve for q

Example Find the missing angles

Assignment