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Transcript
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 o all 3 angles less than 90 • Acute- • Obtuse- one angle greater than 90o, less than 180o • Right- One angle = 90o • Equiangular- All 3 angles are congruent 2) Sides • Scalene - No sides congruent • Isosceles -2 sides congruent • Equilateral - All sides are congruent Parts of 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. 52 Ans: DA = 116 116 AR = 80 DR = 52 80 So……. Its SCALENE Name the triangle by its angles and sides Isosceles Triangle Vertex- Angle where the 2 congruent sides meet A Legs – the congruent sides Leg Base Angles: •Congruent •Formed where the base meets the leg B C 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 T 6x + 11 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 C CD = 8x+6 12x-10 12(4)-10 8x+6 8(4)+6 38 38 base B D X+5 (4)+5 9 Ans: 12x-10 = 8x+6 X=4 Re-read the question, you need to find the perimeter Perimeter =38 + 38 + 9 = 85 Final answer Triangle Sum Thm • The sum of the measures of the interior angles of a triangle is 180o. A • mA + mB+ mC=180o + + = 180 B C Example 1 • Name Triangle AWE by its angles mA + mW+ mE=180o (3x+5) + ( 8x+22) + (4x-12) = 180 A 15x + 15 = 180 3x +5 15x = 165 x = 11 mA = 3(11) +5 = 38 8x + 22 o mW = 8(11)+22 = 110o mE = 4(11)-12 = 32o W E Triangle AWE is obtuse Example 2 Solve for x . Ans: (5x+24) + (5x+24) + (4x+6) = 180 5x +24 5x +24 + 5x+ 24 + 4x+6 = 180 14x + 54 = 180 14x = 126 x=9 The end