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Transcript
Triangles Part 1 The sum of the angles in a triangle is always equal to: 180° Classification By Angle Acute A triangle that has all 3 acute angles Obtuse A triangle with one obtuse angle and 2 acute angles Right A triangle with 1 right angle and 2 acute angles The two acute angles must = 90° therefore they are complimentary Equiangular A triangle with all 3 angles congruent They must each = 60 ° Classification by Sides Scalene All three sides have different lengths Isosceles Two sides have the same length Equilateral All three sides have the same length Isosceles Triangles B A C Equilateral Triangle An equilateral triangle is also equiangular. An equiangular triangle is also equilateral B A C Classify each triangle by its angles and sides. Using the Distance Formula to classify triangles by their sides Find the measure of the sides of triangle DCE, then classify the triangle by sides. D(3, 9); E(5, 3);C(2, 2) Step 1: Find the distance of all three sides using the distacne formula. D = (x2- x 1)2 + ( y2- y1)2 DE = (-5 - 3) 2 + (3-9) 2 = (-8) 2+(-6) 2 = 64+36 = 100 = 10 EC = DC = Step 2: Classify the traingle Since _____ sides are congruent the traingle is called ______ You Try Find the measure of the sides of RST. Classify the triangle by sides. R(1. 3);S(4, 4);T (8,1) RS 74;ST 41; RT 85 RST is Scalene Find the missing Values Find x and the measure of each side of an equilateral triangle RST if: RS x 9;ST 2x; RT 3x 9 Step 1: Draw and equilateral traingle and label the given information. Step 1: 1: Draw Draw and and equilateral equilateral Step SSS traingle and label the givento Step 2: Setand anylabel 2 sides traingle the equal given information. eachother and solve for x. (It does information. not matter which two sides you x+9 2x x+9 2x x+9 2x choose since are equal) Step 2: Set all anythree 2 sides equal to eachother and x=9 Step 3: Plug into onenot side to get solve for x.x(It does TT RR T R allmatter three side lengths. (To check which two sides you your answer plug x into are the other 3x-9 choose since all three 3x-9 3x-9 two sides and make sure all three equal) sides are equal.) x+9 = 3x - 9 3x-9 = 2x x+9 = 2x RT 9 ==2x -9 ==-x3x-9 RS x +- 9 ST 9= =2xx RT 18 = 2x 9 ==x3(9)-9 RS 9+9 ST = 2(9) RT = 27-9 9 ==x18 RS ST = 18 RT = 18 You Try Find d and the measure of each side of an equilateral triangle KLM if: KL d 2; LM 12 d; KM 4d 13 d 5; KL LM KM 7 One more! (This one is a little different) Find x and the measure of all sides if COW is isosceles, with CO=CW, and CO x 7;CW 3x 5;OW x 1 x 6;CO 13,CW 13,OW 5 Finding the Measure of Missing Angles The sum of the angles in a triangle is always equal to: 180° Examples Find X: 40° 2.) x 1.) 65° 39° x 4.) 3.) 2x 30° x x Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the remote interior angles. Exterior Angle: An angle formed when one side of a triangle is extended Remote Interior Angles: The interior angles of the triangle that are not adjacent to the exterior angle 4 is an exterior angle 2 3 4 1 m1 + m 2 = m 4 1 & 2 are remote interior angles to 4 Proof of Exterior Angle Theorem 2 1 1.) m1 + m 2 + m 3 = 180 2.) m3 + m 4 = 180 3 4 By Def of a Triangle By Def of Liner Pair 3.) m1 + m 2 + m 3 = m 3 + m 4 By Substitution 4.) m1 + m 2 = m 4 By SPOE Bigger Picture Find all missing angles 38° 4 5 32° 3 1 2 41° 64°