2.1:a Prove Theorems about Triangles CCSS G-CO.10 Prove 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. GSE’s 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

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

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

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

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

Converse of the Pythagorean Theorem Where c is chosen to be the longest of the three sides: If a2 + b2 = c2, then the triangle is right. If a2 + b2 > c2 , then the triangle is acute. If a2 + b2 < c2, 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

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

Example Triangle TAP is isosceles with angle P as the Vertex. TP = 14x -5 , TA = 6x + 11 , PA = 10x + 43. Is this triangle also equilateral? P TP PA 14x – 5 = 10x + 43 14x-5 10x + 43 4x = 48 X = 12 TP = 14(12) -5 = 163 6x + 11 T A 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 C 12x-10 8x+6 Ans: 12x-10 = 8x+6 12(4)-10 38 8(4)+6 38 X = 4 base Re-read the question, you need to find the perimeter D B X+5 Perimeter =38 + 38 + 9 = 85 (4)+5 9 Final answer

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

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

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

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