Download Finding Unknown Measures of Angles Use the Triangle Sum

Finding Unknown Measures of Angles Use the Triangle Sum Conjecture in Euclidean Geometry to determine the missing measures of angles. Justify your answers and explain your reasoning 1. 2. 3. 4. 5. 6. (Short response) In triangle ABC, the measure of angle A is 2x + 3, the measure of angle B is 4x + 2, and the measure of angle C is 2x – 1. What are the measures of the angles? 7. (Short response) The measures of the angles in a triangle are in the ratio 1 : 3 : 5. Find the measure of each angle. Answer Key 1. The sum of the triangle angles are supplementary it is given that when you add the three angles the answer will be 180. A + B + C = 180 Angles of triangle are supplementary 52 + 55 + C = 180 Substitution 107 + C = 180 Combine like terms 107 -­‐ 107 + C = 180 -­‐ 107 Inverse operation of addition – subtraction C = 73 Solve 52 + 55 + 73 = 180 Check 2. = 110 70 130 60 50 100 80 120 50 80 In this problem the students should analyze the picture for details of the various concepts that are going to help them solve the most obvious measurements. For instance, there are two vertical angle situations and three supplementary. Once those are complete then the student should realize that they can use the sum of the angles in a triangle as well as, the sum of the angles of a quadrilateral, which will then move them back to using supplementary angles to solve for x. Students may notice the supplementary relationship or the vertical relationship either way the students are going in the right direction. 120 & 100 are situated in a figure that is formed by two line segments that form vertical angles therefore giving you congruent angles across from one another. They are also form supplementary angle relationship along with the m130 . 180 -­‐ 120 = 60 Supplementary Relationship 180 -­‐ 100 = 80 Supplementary Relationship 180 -­‐ 130 = 50 Supplementary Relationship Now they should notice they have measurements for two angles in the triangle. They can use the sum of the angles in the triangle are supplementary. A + B + C = 180 Sum of the angles in a triangle 50 + 80 + C = 180 Substitution 130 + C = 180 Combine like terms 130 -­‐ 130 + C = 180 -­‐ 130 Inverse operations of addition -­‐ subtraction C = 50 Solve Now there is only one missing angle for the quadrilateral, which means you can solve for the angle by doing the sum of the angles of the quadrilateral then using supplementary relationships to solve for x. D + E + F + G = 360 Sum of the Angles of Quadrilateral 60 + 100 + 130 + G = 360 Substitution 290 + G = 360 Combine like terms 290 -­‐ 290 + G = 360 -­‐ 290 Inverse operation of addition – subtraction G = 70 Solve Since x is sitting next to this angle, we can now use the supplementary relationship to solve for x. x + 70 = 180 Supplementary Relationship x + 70 -­‐ 70 = 180 -­‐ 70 Inverse operation of addition – subtraction x = 110 Solve 3. a is a part of the a bigger triangle with the angles 71 & 40. Can use the sum of the angles of a triangle is supplementary. a + b + b = 180 Sum of the angles in a triangle a + 71 + 40 = 180 Substitution a + 111 = 180 Combine like terms a + 111 -­‐ 111 = 180 -­‐ 111 Inverse operations of addition -­‐ subtraction a = 69 Solve b is supplementary to 133 180 -­‐ 133 = b Solve 47 = b The student may use two concepts to solve for c. Either the sum of the angle of triangle is supplementary and then use that angle in relationship to c. Or they can use the exterior angle of a triangle is the sum of the angles opposite it in the triangle. c = 69 + 47 c = 116 or 180 = f + 69 + 47 180 = f + 116 180 -­‐ 116 = f + 116 -­‐ 116 64 = f 180 -­‐ 64 = c 116 = c d can be solved by using the idea exterior 140 is the sum of the two interior angles of the triangle. Or a student may use the supplementary relationship and solve for the missing angles. d = 140 -­‐ 47 d = 93 or 140 + g = 180 Supplementary -­‐140 -­‐140 Inverse operation of addition – subtraction g = 40 Solve 180 -­‐ 40 -­‐ 47 = d Sum of the angles in triangle 93 = d Solve d has a vertical relationship of a missing angle in the triangle above it making that angle equal to 93. Using sum of angles in triangle the 3rd angle will be 53. There is a vertical relationship with this angle in the adjacent triangle. Then using the supplementary relationship between 133 and the 2nd missing angle will give you 47. Since we have two measurements in the triangle we can use the sum of the angles in a triangle are supplementary to solve for e. 53 + 47 + e = 180 100 + e = 180 -­‐100 -­‐100 e = 80 4. In this figure students should know that they are using sum of the angles in a triangle. The perpendicular line that is forming the 90 angle form two 90. Therefore, you have two known angles in the triangle and you can solve for x and y. x = 180 – (90 + 65) x = 180 – (155) x = 25 y = 180 – (90 + 40) y = 180 – (130) y = 50 5. Use the fact that the angles of a triangle are supplementary. x + 72 +  (x + 2)  = 180 x + 72 + x + 2 = 180 2x + 74 = 180 2x + 74 – 74 = 180 – 74 2x = 106 2x/2 = 106/2 x = 53 x + 2 53 + 2 = 55 Check: 55 + 53 + 72 = 180 6. In triangle ABC, the measure of angle A is 2x + 3, the measure of angle B is 4x + 2, and the measure of angle C is 2x – 1. What are the measures of the angles? ∆ABC: m∠A + m∠B + m∠C = 180° m∠(2x + 3) + m∠(4x + 2) + m∠(2x – 1) = 180° 2x + 3 + 4x + 2 + 2x – 1 = 180 2x + 4x + 2x + 3 + 2 – 1 = 180 8x + 4 = 180 8x = 180 – 4 8x = 176 x = 176/8 x = 22 m∠A = m∠(2x + 3) m∠A = m∠(2(22) + 3) m∠A = m∠(44 + 3) m∠A = m∠47° m∠B = m∠(4x + 2) m∠B = m∠(4(22) + 2) m∠B = m∠(88 + 2) m∠B = m∠90° m∠C = m∠(2x -­‐ 1) m∠C = m∠(2(22) -­‐ 1) m∠C = m∠(44 -­‐ 1) m∠C = m∠43° Check: ∠47° + ∠90° + ∠43° = 180° 7. (Short response) The measures of the angles in a triangle are in the ratio 1 : 3 : 5. Find the measure of each angle. We have 3 unknown angles: a, b, and c. However, we are told they have a specific relationship: 1 to 3 to 5. So the second angle is 3 times the first, and the third angle is 5 times the first. So we can let: 𝑎 = 𝑎 𝑏 = 3𝑎 𝑐 = 5𝑎 Since we know the sum of the angles of a triangle is 180°, we can say 𝑎 + 𝑏 + 𝑐 = 180°. Using substitution, 𝑎 + 3𝑎 + 5𝑎 = 180°. So, 9𝑎 = 180° Divide both sides by 9 𝑎 = 20 Substituting back into the formulas we initially defined. 𝑎 = 20° 𝑏 = 3𝑎 = 3(20) = 60° 𝑐 = 5𝑎 = 5(20) = 100° Let’s check our work: 20° + 60° + 100° = 180°