4 - 17 - 18 unit 11 lesson 2 triangle inequalities.notebook file4 17 18 unit 11 lesson 2 triangle...
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4 17 18 unit 11 lesson 2 triangle inequalities.notebook
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4/17 Aim: Triangle inequality theoremDo now: Homework: TBA
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What makes a triangle?
http://www.mathwarehouse.com/geometry/triangles/triangleinequalitytheoremruleexplained.php
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Theorem 1: The sum of the lengths of two sides of a triangle is greater than the length of the third side.
AC + CB > ABCB + AB > ACAB + AC > CB
5 + 3 > 73 + 7 > 57 + 5 > 3
A
C
B
5 3
7
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In #’s 1‐6 tell whether the given lengths may be the measures of the sides of a triangle?1) 3, 4, 5 2) 5, 8, 13
3) 6, 7, 10 4) 3, 9, 10
5) 2, 2, 3 6) 2, 4, 6
WRite these values next to the values there.
Use these to values instead
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To calculate the range of the third side of a triangleyou subtract the values and add the values, then write an inequality.
9, 5
9 ‐ 5 = 9 + 5 =
< x<
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To calculate the range of the third side of a triangleyou subtract the values and add the values, then write an inequality. < x<14) 5, 8
15) 6, 10
16) 6, 9
17) 11, 8
18) 14, 11
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Theorem 2: In a triangle, the longest side is across from the largest angle.
Since 7 is the longest side in the triangle, <C, across from it, is the largest angle.
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Theorem 3: In a triangle, the largest angle is across from the longest side.
Since 100 degrees is the largest angle in this triangle, AB, across from it, is the longest side.
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Theorem 4: The measure of the exterior angle is equal to the sum of the two nonadjacent interior angles.
m<2+ m<3 = m<1
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Theorem 5: The measure of the exterior angle of a triangle is greater than the measure of either nonadjacent interior angle.
<1 is the exterior angle<2 and <3 are its nonadjacent interior angles
A B D
C
3
2
1
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Theorem 6: The whole is greater than any of its parts.
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1) Which of the following may be lengths of the sides of a triangle?
a) 4,6,10 b) 8,8,16 c) 6,8,16 d) 10,12,14
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2) Two sides of a triangle have lengths of 3 and 7. Find the range of possible lengths of the third side.
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3) Given the lengths of two sides of a triangle are a and b, can you use the Triangle Inequality Theorem to find a pattern for finding a possible length of the third side x.
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4) The length of the base of an isosceles triangle is 12. What can be said about the lengths of the legs of the triangle?
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5) If the lengths of two sides of a triangle are 4 and 13, the length of the third side can be what values?
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6) The lengths of two sides of an isosceles triangle are 9 and 15. Use an inequality to express the range of the length of the third side.
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8) The lengths of two sides of a triangle are 9 and 6. What is the length of the third side?
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9) What do you know about the length of the third side of a triangle if two side lengths are 12 and 5?
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10) The length of one of the legs of an isosceles triangle is 8. Write an inequality to express the range of the length of the base side.
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11) Two side lengths of a triangle are 6 and 8. The third side can't be what?
a) 4 b) 6 c) 9 c) 15
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12) If the lengths of two sides of a triangle are 7 and 12, the length of the third side cannot be what?
a) 4 b) 8 c) 10 c) 16
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13) The length of the base of an isosceles triangle is 10. What can be the lengths of the legs of the triangle?
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#18: Determine in each case whether the given lengths can be measures of sides of a triangle.
1) 3,4,5 2) 5,8,13 3) 6,7,10
4) 3,9,15 5) 2,2,3 6) 1,1,2
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#710: Write an inequality to display the possible length of the third side of a triangle, given the two lengths.
7) 2 and 4 8) 10 and 16
9) 9.6 and 12.5 10) 2 and 17
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11) If the sides of a triangle have lengths 3 and 10, the lengths of the third side may be any number:
(A) greater than 3, but less than 10 (B) less than 13(C) greater than 7 (D) greater than 7, but less than 13
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12) Two side lengths of a triangle are 5 and 7. The third side length cannot be:
(A) 3 (B) 7 (C) 10 (D) 12
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13) The lengths of two sides of an isosceles triangle are 8 and 10. The length of the third side can be:
(A) 6, only (B) 8, only (C) 10, only (D) 8 or 10
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14) The lengths of two sides of a triangle are 7 and 10. A possible third side length is:
(A) 17 (B) 20 (C) 3 (D) 8
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2. If two sides of a triangle measure 11 and 20, then the length of third side could NOT be:10 18 27 32
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5. If two sides of a triangle measure 12 and 14, the third side could be:12 2 20 26
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8. The vertex angle of an isosceles triangle is 30° less than the measure of a base angle. What is the measure of a base angle?
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How do Angles and Sides Relate?
When the sides of triangle are not congruent, there is a relationship between the sides and the angles of the triangle.
• If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side.• CONVERSE: If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.
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Solve for x.
D
E
F4x + 5
3x + 2
8x + 15
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Find the measure of an exterior angle at the baseof an isosceles triangle whose vertex angle measures400
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The vertex of an isosceles triangle measures eight timesthe measure of a base angle. Find the measure of the base angle. Find the measure of an exterior base angle.
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In triangle ABC, m<A = 50 and m<B = 20. Which is the longest side of the triangle?
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2. In triangle CDE, m<C = 70 and m<D = 60. Which is the shortest side of the triangle?
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3. In the accompanying diagram of triangle ABC, AC is extended to D. If m<BCD = 120 and m<B = 55,which is the longest side of the triangle?
DCA
B
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4. In triangle RST, the measure of angle R is 58 an exterior angle at S measures 118. Which is the longest side of the triangle?
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5. In triangle ABC, AC= 3 CB = 4 and AB = 5. What is the smallest angle of the triangle?
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6. Triangle PQR is a right triangle with the right angle at Q. What is the longest side of triangle PQR?
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7. In triangle ABC, m<A = 80 and AB >AC. What is the smallest angle of triangle ABC?
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8. In the accompanying diagram AB = BD and ADC. If m<BDC = 130, what is the longest side of triangle ABC?
A C
B
D
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1. In any triangle ABC, which statement is always true?1) AB = BC = CA2) AB +BC < CA3) AB + BA > BC4) AB +AC > BC
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2. which of the following may be the lengths of the sides of a triangle?1) 2, 2, 42) 2, 3, 43) 2, 3, 54) 3, 4, 8
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3. If the lengths of two sides of a triangle are 4 and 7, the length of the third side may not be1) 52) 73) 114) 4
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4. Which set of numbers cannot represent the lengths of the sides of a triangle?1) 3, 4, 52) 1, 2, √33) 1, 1, 14) 4, 1, 2
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5. Which set of numbers represents the lengths of the sides of an isosceles triangle?1) 4, 4, 92) 4, 9, 93) 5, 5, 104) 3, 4, 5
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6. If the lengths of two sides of a triangle are 5 and 11, which could be the length of the third side?
1) 62) 103) 164)22
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7. If the sum of the lengths of two sides of a triangle is 10 cm, the length of the third side must always be
1) equal to 10 cm2) less than 10 cm3) greater than 10cm40 none of the above
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14. In isosceles right triangle ABC, angle C is the right angle. Find the measure of the exterior angle at A.
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15. In ΔABC, AB = 5 feet and BC = 3 feet. Which inequality represents all possible values for the length of AC, in feet?1) 2 ≤ AC ≤ 82) 2 < AC < 83) 3 ≤ AC ≤ 74) 3 < AC < 7
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A
B
C D
12550
16. In the accompanying diagram of ΔABC, find in degrees, m<ABC.
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17. In ΔABC, m<C = 118 and m<B = 44. Which is the shortest side of the triangle?
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A B D
C
2x + 10
x + 30
4x + 30
18. In the accompanying diagram of ΔABC, side AB is extended to D. What is the value of x?
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19. In isosceles triangle ABC, AB = 10 and BC = 5. Which is the smallest angle of the triangle?
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3x
x+ 20 120
B
CA D
20. In the accompanying diagram, <BCD is an exterior angle formed by extending AC to point D. Find the value of x.
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21. For which measures of the sides of ΔABC is angle B the largest angle of the triangle?
1) AB = 2, BC = 6, AC = 72) AB = 6, BC = 12, AC = 83) AB = 16, BC = 9, AC = 104) AB = 18, BC = 14, AC = 5
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22. In triangle PQR, PR is extended through R to S. If m<SRQ = 28x and m<Q = 125, and m<P =3x, find x.
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3y2y 6y
A
B C D
23. In the accompany diagram of ΔABC, BC is extended to D. What is m<A?1) 152) 1730 204) 24
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24. In ΔABC, an exterior angle at C measures 85. What is the longest side of ΔABC?
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4x
110 x2 +5x
Q
P R S
25. In the diagram of ΔPQR shown below, PR is extended to S.What is m<Q?1) 442) 403) 114) 10
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2x + 1
x + 15 x
26. What is the measure of the largest angle in the accompanying triangle?1) 412) 46.53) 564) 83
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27. If ΔJSO is an equilateral triangle, find the measure of an exterior angle at S.
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B
A
C D50 5x
3x
28. In the accompanying diagram, <ACD is an exterior angle of ΔABC, m<A = 3x, m<ACD = 5x and m<5 = 50. What is the value of x?1) 252) 303) 604) 100
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29. ΔABC, m<A = x2 + 12, m<B = 11x + 5, and m<C = 13x 17. Determine the longest side of ΔABC.
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30. In triangle ABC, m<A = 80 and AB > AC what is the smallest angle of triangle ABC?
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