2.1:a Prove Theorems about Triangles 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 G-CO.10Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. CCSS

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.

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

Triangle Sum Thm The sum of the measures of the interior angles of a triangle is 180 o. m  A + m  B+ m  C=180 o + + = 180 A BC

Example 1 Name Triangle AWE by its angles 4x x x +5 A W E (3x+5) + ( 8x+22) + (4x-12) = 180 m  A + m  W+ m  E=180 o 15x + 15 = x = 165 x = 11 m  A = 3(11) +5 = 38 o m  W = 8(11)+22 = 110 o m  E = 4(11)-12 = 32 o Triangle AWE is obtuse

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

The end