Transcript Slide 1
Congruent Triangles Triangle Inequality 4.7 Holt Geometry Congruent Triangles • Find the range of values for the third side of a triangle, with 2 sides known. • Order the lengths of the sides of a triangle, given the angles • Order of the angle measures of a triangle, given the lengths of the sides homework Holt Geometry Congruent Triangles Finding the range of values for the length of the third side of a triangle: Since the third side cannot be larger than the other two added together, we find the maximum value by adding the two sides. Since the third side and the smallest side cannot be larger than the other side, we find the minimum value by subtracting the two sides. Example: Given a triangle with sides of length 3 and 8, find the range of possible values for the third side. Difference < Third Side < Sum 8 – 3 < Third Side < 8 + 3 Range of the third side is 5 < x < 11. Holt Geometry homework Congruent Triangles Inequalities Within a Triangle If the measures of three sides of a triangle are unequal, then the measures of the angles opposite those sides are unequal in the same order. P 11 M 8 13 L LP < PM < ML mM < mL < mP Holt Geometry homework Congruent Triangles Inequalities Within a Triangle If the measures of three angles of a triangle are unequal, then the measures of the sides opposite those angles are unequal in the same order. W 45° 75° K 60° J mW < mJ < mK JK < KW < WJ homework Holt Geometry Congruent Triangles Inequalities Within a Triangle The longest side is BC So, the largest angle is The largest angle is A L So, the longest side is MN homework Holt Geometry Congruent Triangles Inequalities Within a Triangle Can 16, 10, and 5 be the measures of the sides of a triangle? Select the third side. Difference < Third Side < Sum 16 – 10 < Third Side < 16 + 10 6 < Third Side < 26 No! Because 5 is not greater than 6 homework Holt Geometry Congruent Triangles Determine if the three numbers can be measures of the sides of a triangle. If no, explain. a. 13, 28, 19 Yes, 15 < 19 < 41 28 – 13 < Third Side < 13 + 28 b. 9, 4, 4 NO, 5 < 4 < 13 9 – 4 < Third Side < 9 + 4 c. 9, 7, 18 NO, 2 < 18 < 16 9 – 7 < Third Side < 9 + 7 Holt Geometry homework Congruent Triangles If two sides of a triangle have the following measures, find the range of possible measures of the third side. a. 10, 7 10 – 7 < x < 10 + 7 3 < x < 17 b. 18 , 11 18 – 11 < x < 18 + 11 7 < x < 29 homework Holt Geometry Congruent Triangles Write the angles in order from least to greatest. A 12 B 52 43 C C, A, B homework Holt Geometry Congruent Triangles Write the sides in order from greatest to least. Y 128º 22º 30º Z X XZ, XY, YZ homework Holt Geometry Congruent Triangles Write the angles in order from least to greatest. a. S, R, T b. W, Y, Z c. S, T, R homework Holt Geometry Congruent Triangles Is it possible to form a triangle with the given side lengths? If not, explain why not. a. NO b. YES c. e. NOd. NO f. NO YES Find the range for the measure of the third side of a triangle given the measures of two sides. g. 4 < x < 12 h. 6 < x < 16 i. k. Holt Geometry j. 1.5 < x < 6.9 l. 2¾ < x < 3¾ 5.4 < x < 13 5¼ < x < 10 homework Congruent Triangles Write the sides in order from least to greatest. 58 44 60 a. DE, CE, CD c. BC, AB, AC b. BC, AC, AB 109 39 46 32 d. MP, LM, LP 70 44 e. MN, LM, LN f. QW, MQ, MW 55 41 42 55 g. MN = LM, LN Holt Geometry 89 49 h. YZ, XZ, XY 49 i. MQ, MP, PQ homework Congruent Triangles Assignment Geometry: Inequalities in One Triangle and Triangle Inequality Holt Geometry Congruent Triangles Midsegment 4.8 Holt Geometry Congruent Triangles • To review midsegments in Trapezoids. homework Holt Geometry Congruent Triangles If a quadrilateral is a Trapezoid homework Holt Geometry Congruent Triangles MN is the midsegment of trapezoid PQRS. What is x? What is MN? M = ½(b1 + b2) 2x + 11 = ½(8x – 12 + 10) 2x + 11 = ½(8x – 2) 2•6 + 11 = 23 2x + 11 2x + 11 = 4x – 1 N M -2x -2x 8•6 -12 = 36 8x – 12 11 = 2x – 1 S P +1 +1 •How can you check your answer? 12 = 2x 10 + 36 = 46 2 2 homework = 23 2 2 6=x Q Holt Geometry 10 R Congruent Triangles Find the length of the midsegment and the bases of the trapezoid. 3•5+5 = 20 M = ½(b1 + b2) 2x + 16 = ½(3x + 5 + 5x + 7) 2x + 16 = ½(8x + 12) = 26 2•5+16 2x + 16 2x + 16 = 4x + 6 -2x -2x 5•5+7 = 32 5x + 7 16 = 2x + 6 -6 -6 10 = 2x homework 2 2 5=x 3x + 5 Holt Geometry Congruent Triangles Find x in the trapezoids with the indicated midsegments. a. x = 7 b. x = 2 homework Holt Geometry Congruent Triangles Assignment Geometry: Midsegments Holt Geometry