Lesson 5 NYS COMMON CORE MATHEMATICS CURRICULUM 8β’4 T Exit Ticket Sample Solutions For each of the following problems, write an equation and solve. 1. Given a right triangle, find the measures of all of the angles, in degrees, if one angle is a right angle and the measure of the second angle is six less than seven times the measure of the third angle. Let π represent the measure of the third angle. Then, ππ β π can represent the measure of the second angle. The sum of the two angles in the right triangle will be ππ°. ππ β π + π = ππ ππ β π = ππ ππ β π + π = ππ + π ππ = ππ π ππ π= π π π = ππ The third angle is π and therefore measures ππ°. Replacing π with ππ in ππ β π gives π(ππ) β π = ππ β π = ππ. Therefore, the measure of the second angle is ππ°. The measure of the third angle is ππ°. 2. In a triangle, the measure of the first angle is six times a number. The measure of the second angle is nine less than the first angle. The measure of the third angle is three times the number more than the measure of the first angle. Determine the measure of each angle in degrees. Let π be the number. Then, the measure of the first angle is ππ, the measure of the second angle is ππ β π, and the measure of the third angle is ππ + ππ. The sum of the measures of the angles in a triangle is πππ°. ππ + ππ β π + ππ + ππ = πππ πππ β π = πππ πππ β π + π = πππ + π πππ = πππ ππ πππ π= ππ ππ π=π Replacing π with π in ππ gives π(π) = ππ. Replacing π with π in ππ β π gives π(π) β π = ππ β π = ππ. Replacing π with π in ππ + ππ gives ππ + π(π) = ππ + ππ = ππ. Therefore, the angle measures are ππ°, ππ°, and ππ°. Note to teacher: There are several ways to solve problems like these. For example, a student may let π₯ be the measure of the first angle and write the measure of the other angles accordingly. Either way, make sure that students are defining their symbols and correctly using the properties of equality to solve. Lesson 5: Writing and Solving Linear Equations This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G8-M4-TE-1.3.0-09.2015 53 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 5 NYS COMMON CORE MATHEMATICS CURRICULUM 8β’4 T Problem Set Sample Solutions Students practice writing and solving linear equations. For each of the following problems, write an equation and solve. 1. The measure of one angle is thirteen less than five times the measure of another angle. The sum of the measures of the two angles is πππ°. Determine the measure of each angle in degrees. Let π be the measure of the one angle. Then, the measure of the other angle is ππ β ππ. π + ππ β ππ = πππ ππ β ππ = πππ ππ β ππ + ππ = πππ + ππ ππ = πππ π = ππ. π Since one angle measure is π, it is ππ. π°. Replacing π with ππ. π in ππ β ππ gives π(ππ. π) β ππ = πππ β ππ. π = πππ. π. Therefore, the other angle measures πππ. π°. 2. An angle measures seventeen more than three times a number. Its supplement is three more than seven times the number. What is the measure of each angle in degrees? Let π be the number. Then, the measure of one angle is ππ + ππ. The measure of the other angle is ππ + π. Since the angles are supplementary, the sum of their measures will be πππ. ππ + ππ + ππ + π = πππ πππ + ππ = πππ πππ + ππ β ππ = πππ β ππ πππ = πππ π = ππ Replacing π with ππ in ππ + ππ gives π(ππ) + ππ = ππ. Replacing π with ππ in ππ + π gives (ππ) + π = πππ + π = πππ. Therefore, the angle measures are ππ° and πππ°. 3. The angles of a triangle are described as follows: β π¨ is the largest angle; its measure is twice the measure of β π©. The measure of β πͺ is π less than half the measure of β π©. Find the measures of the three angles in degrees. Let π be the measure of β π©. Then, the measure of β π¨ is ππ, and the measure of β πͺ is measures of the angles must be πππ°. π + ππ + ππ + Lesson 5: π π β π. The sum of the π β π = πππ π π β π + π = πππ + π π π ππ + = πππ π ππ π + = πππ π π ππ = πππ π Writing and Solving Linear Equations This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G8-M4-TE-1.3.0-09.2015 54 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Lesson 5 NYS COMMON CORE MATHEMATICS CURRICULUM 8β’4 T ππ = πππ π = ππ Since π is the measure of β π©, then β π© is ππ°. Replacing π with ππ in ππ gives π(ππ) = πππ. Therefore, the measure of β π¨ is πππ°. Replacing π with ππ in is ππ°. 4. π π β π gives ππ π β π = ππ β π = ππ. Therefore, the measure of β πͺ A pair of corresponding angles are described as follows: The measure of one angle is five less than seven times a number, and the measure of the other angle is eight more than seven times the number. Are the angles congruent? Why or why not? Let π be the number. Then, the measure of one angle is ππ β π, and the measure of the other angle is ππ + π. Assume they are congruent, which means their measures are equal. ππ β π = ππ + π ππ β ππ β π = ππ β ππ + π βπ β π Since βπ β π, the angles are not congruent. 5. The measure of one angle is eleven more than four times a number. Another angle is twice the first angleβs measure. The sum of the measures of the angles is πππ°. What is the measure of each angle in degrees? Let π be the number. The measure of one angle can be represented with ππ + ππ, and the other angleβs measure can be represented as π(ππ + ππ) = ππ + ππ. ππ + ππ + ππ + ππ = πππ πππ + ππ = πππ πππ + ππ β ππ = πππ β ππ πππ = πππ π = ππ. π Replacing π with ππ. π in ππ + ππ gives π(ππ. π) + ππ = ππ + ππ = ππ. Replacing π with ππ. π in π(ππ + ππ) gives π(π(ππ. π) + ππ) = π(ππ + ππ) = π(ππ) = πππ. Therefore, the measures of the angles are ππ° and πππ°. 6. Three angles are described as follows: β π© is half the size of β π¨. The measure of β πͺ is equal to one less than two times the measure of β π©. The sum of β π¨ and β π© is πππ°. Can the three angles form a triangle? Why or why not? π π π Let π represent the measure of β π¨. Then, the measure of β π© is , and the measure of β πͺ is π ( ) β π = π β π. π The sum of the measures of β π¨ and β π© is πππ. π = πππ π ππ = πππ π ππ = πππ π+ π = ππ Since π is the measure of β π¨, then β π¨ is ππ°. Replacing π with ππ in π π gives ππ π = ππ; therefore, the measure of β π© is ππ°. Replacing π with ππ in π β π gives ππ β π = ππ, therefore the measure of β πͺ is ππ°. The sum of the three angle measures is ππ° + ππ° + ππ° = πππ°. Since the sum of the measures of the interior angles of a triangle must equal πππ°, these angles do not form a triangle. The sum is too large. Lesson 5: Writing and Solving Linear Equations This work is derived from Eureka Math β’ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from G8-M4-TE-1.3.0-09.2015 55 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
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