Lesson 10: Angle Problems And Solving Equations
Lesson 10NYS COMMON CORE MATHEMATICS CURRICULUM7 3Lesson 10: Angle Problems and Solving EquationsStudent Outcomes Students use vertical and adjacent angles and angles on a line and angles at a point in a multi-step problem towrite and solve simple equations for an unknown angle in a figure.Related Topics: More Lesson Plans for Grade 7 Common Core MathLesson NotesIn Lessons 10 and 11, students apply their understanding of equations to unknown angle problems. The geometry topicis a natural context within which they apply algebraic skills. Students understand that the unknown angle is an actual,measureable angle; they simply need to find the value that makes each equation true. They set up the equations basedon the angle facts they have learned in Grade 4. The problems presented are not as simple as in Grade 4 becausediagrams incorporate angle facts in combination, rather than in isolation. Encourage students to verify their answers bymeasuring relevant angles in each diagram―all diagrams are drawn to scale.ClassworkOpening (5 minutes)Discuss the ways in which angles are named and notated. What do you notice about the three figures below? What is the same about all three figures; what isdifferent? There are three angles that appear to be the same measurement but are notated differently.What is a likely implication of the three different kinds of notation? They indicate the different ways of labeling or identifying the angle. Students are familiar with addressing Figure 1 as and having a measurement of and addressing Figure 2 asangle . Elicit this from students and say that in a case like Figure 1, the angle is named by the arc, and in acase like Figure 2, the angle is named by the single letter. In a case like Figure 3, we use three letters when we name the angle. Why use three points to name an angle? In a figure where several angles share the same vertex, naming a particular angle by the vertex point isnot sufficient information to distinguish that angle. Two points, one belonging to each side of theintended angle, are necessary to identify it.Encourage students to use both multiple forms of angle notation in the table to demonstrate each anglerelationship.Lesson 10:Date: 2013 Common Core, Inc. Some rights reserved. commoncore.orgAngle Problems and Solving Equations3/19/14146This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
a.b.b c.CAANaming by the arc.7 3Lesson 10NYS COMMON CORE MATHEMATICS CURRICULUMNaming by the vertex.DNaming by three points.orRecall the definitions of adjacent and vertical and the facts regarding angles on a line and angles at a point. If anabbreviation exists, students should include the abbreviation of the angle fact under the name of each relationship. Inthe Angle Fact column, students should write the definitions and practice the different angle notations when describingthe relationship in the angle fact.Note: The angles on a line fact applies to two or more angles.Angle Facts and DefinitionsName of AngleRelationshipAdjacent AnglesAngle FactDiagramBTwo anglesandwith acommon side ⃗⃗⃗⃗⃗ , are adjacent angles ifbelongs to the interior of.AnglesandandCa are adjacent angles;are adjacent angles.b ATwo angles are vertical angles (orvertically opposite angles) if their sidesform two pairs of opposite rays.DDGa Vertical Angles(vert. )b CFEThe sum of the measures of two anglesthat share a ray form a line of.CAngles on a Line( on a line)a b BADBThe measure of all angles formed by threeor more rays with the same vertex is.Angles at a Point( at a point)a b ADc CLesson 10:Date: 2013 Common Core, Inc. Some rights reserved. commoncore.orgAngle Problems and Solving Equations3/19/14147This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 10NYS COMMON CORE MATHEMATICS CURRICULUM7 3Opening Exercise (4 minutes)Opening ExerciseUse a protractor, measure all the angles and complete the chart to follow.Name the Angles that areVerticalandAdjacentAnswers include:andandAngles on a Line,andAnswers include:,, and,, andAngles at a Point,,,,,Example 1 (4 minutes)Students describe the angle relationship in the diagram and set up and solve an equation that models it. Have studentsverify their answers by measuring the unknown angle with a protractor.Example 1CEstimate the measurement of .In a complete sentence, describe the angle relationship in thediagram.andare angles on a line and sum tox B132 AWrite an equation for the angle relationship shown in the figure and solve for . Then find the measures ofconfirm your answers by measuring the angle with a protractor.(Dand)Lesson 10:Date: 2013 Common Core, Inc. Some rights reserved. commoncore.orgAngle Problems and Solving Equations3/19/14148This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 10NYS COMMON CORE MATHEMATICS CURRICULUM7 3Exercise 1 (4 minutes)Students describe the angle relationship in the diagram and set up and solve an equation that models it. Have studentsverify their answers by measuring the unknown angle with a protractor.Exercise 1In a complete sentence, describe the angle relationship in thediagram.,, andCDare angles on a line and sum to.3x Find the measurements of( )()( )(()()and.2x ABE)Example 2 (4 minutes)Students describe the angle relationship in the diagram and set up and solve an equation that models it. Have studentsverify their answers by measuring the unknown angle with a protractor.Example 2In a complete sentence, describe the angle relationship in thediagram.andare supplementary and sum to.are vertical angles and are of equal measurement.Land144 AE(or vert. s are )Lesson 10:Date: 2013 Common Core, Inc. Some rights reserved. commoncore.orgBy Write an equation for the angle relationship shown in the figure andsolve for and . Find the measurements ofand.;x KAngle Problems and Solving Equations3/19/14149This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 10NYS COMMON CORE MATHEMATICS CURRICULUM7 3Exercise 2 (4 minutes)Students describe the angle relationship in the diagram and set up and solve an equation that models it. Have studentsverify their answers by measuring the unknown angle with a protractor.Exercise 2NIn a complete sentence, describe the angle relationships in the diagram.andmeasure ofmeasurement.are adjacent angles and when added together are the;andare vertical angles and are of equal16 MJ3x EWrite an equation for the angle relationship shown in the figure and solvefor .( )85 LK( )Example 3 (4 minutes)Students describe the angle relationship in the diagram and set up and solve an equation that models it. Have studentsverify their answers by measuring the unknown angle with a protractor.Example 3In a complete sentence, describe the angle relationships in the diagram.,, andare angles at a point and sum toE.Write an equation for the angle relationship shown in the figure and solvefor . Find the measurement ofand confirm your answers bymeasuring the angle with a protractor.x GK135 FLesson 10:Date: 2013 Common Core, Inc. Some rights reserved. commoncore.orgAngle Problems and Solving Equations3/19/14150This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 10NYS COMMON CORE MATHEMATICS CURRICULUM7 3Exercise 3 (4 minutes)Students describe the angle relationship in the diagram and set up and solve an equation that models it. Have studentsverify their answers by measuring the unknown angle with a protractor.Exercise 3FIn a complete sentence, describe the angle relationships in thediagram.,,, andare angles at a point and sum toE.103 AFind the measurement of(.59 G(x 1) 167 )H(())Example 4 (5 minutes) MP.8List pairs of angles whose measurements are in a ratio ofScaffolding:.Students may find it helpful tohighlight the pairs of equalvertical angles. Examples include: What does it mean for the ratio of the measurements of two angles tobe? The measurement of one angle is two times the measure of the other angle. If the smaller angle isdefined as , then the larger angle is. If the larger angle is defined as , then the smaller angle isand,and,and. Based on the following figure, which angle relationship(s) can be utilized to find the measure of an obtuse andacute angle? Any adjacent angle pair are on a line and sum to.Students describe the angle relationship in the question and set up and solve an equation that models it. Have studentsverify their answers by measuring the unknown angle with a protractor.Example 4Two lines intersect in the following figure. In the figure, the ratio of themeasurements of the obtuse angle to the acute angle in any adjacentangle pair is.In a complete sentence, describe the angle relationships in the diagram.The measurement of an obtuse angle is twice the measurement of anacute angle in the diagram.Lesson 10:Date: 2013 Common Core, Inc. Some rights reserved. commoncore.org2x x Angle Problems and Solving Equations3/19/14151This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 10NYS COMMON CORE MATHEMATICS CURRICULUM7 3Label the diagram with expressions that describe this relationship. Write an equation that models the angle relationshipand solve for . Find the measurements of the acute and obtuse angles.( )()( )()Acute angle(Obtuse angle)Exercise 4 (4 minutes)Students describe the angle relationship in the diagram and set up and solve an equation that models it. Have studentsverify their answers by measuring the unknown angle with a protractor.Exercise 4The ratio oftoisrelationships in the diagram.The measurement of. In a complete sentence, describe the angleis the measurement ofcomplementary and sum to;andEareH.3x Find the measures ofand( )(()()2x .)( )(FG)Relevant VocabularyAdjacent Angles: Two angles.andwith a common side ⃗⃗⃗⃗⃗ are adjacent angles ifbelongs to the interior ofVertical Angles: Two angles are vertical angles (or vertically opposite angles) if their sides form two pairs of oppositerays.Angles on a Line: The sum of the measures of adjacent angles on a line isAngles at a Point: The sum of the measures of adjacent angles at a point is.Exit Ticket (3 minutes)Lesson 10:Date: 2013 Common Core, Inc. Some rights reserved. commoncore.orgAngle Problems and Solving Equations3/19/14152This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 10NYS COMMON CORE MATHEMATICS CURRICULUMName7 3DateLesson 10: Angle Problems and Solving EquationsExit TicketEKIn a complete sentence, describe the relevant angle relationships in thefollowing diagram. That is, describe the angle relationships you could use todetermine the value of .30 J3x HA5x FGUse the angle relationships described above to write an equation to solve for .Lesson 10:Date: 2013 Common Core, Inc. Some rights reserved. commoncore.orgAngle Problems and Solving Equations3/19/14153This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 10NYS COMMON CORE MATHEMATICS CURRICULUM7 3Exit Ticket Sample SolutionsIn a complete sentence, describe the relevant angle relationships in thefollowing diagram. That is, describe the angle relationships you could use todetermine the value of .and;andEKare adjacent angles whose measurements are equal toare vertical angles and are of equal measurement.Use the angle relationships described above to write an equation to solve for. Then determine the measurements ofand.( )(()())( )()30 J3x A5x FGHProblem Set Sample SolutionsFor each question, use angle relationships to write an equation in order to solve for each variable. Determine theindicated angles. You can check your answers by measuring each angle with a protractor.1.In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measurementof.One possible response:,are angles on a line and sum to, andD.Ex 65 C2.AFIn a complete sentence, describe the relevant angle relationships in the following diagram. Find the measurementof.,sum to, andare angles on a line and.( )Lesson 10:Date: 2013 Common Core, Inc. Some rights reserved. commoncore.org( )Angle Problems and Solving Equations3/19/14154This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 10NYS COMMON CORE MATHEMATICS CURRICULUM3.In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measurementsofand.,,,on a line and sum to, andare anglesE.D( )()()( )103 2x C4.7 310 3x F12 QBGIn a complete sentence, describe the relevant angle relationships in the following diagram. Find the measure of .All the angles in the diagram are angles at a point and sum to.()x 71 ( )x 71 ( )71 x 71 5.x In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measure ofand .B,, andare angles on a line and sum to. Sinceandare supplementary and themeasurement ofis,is, makingandcomplementary angles that sum to.D25 KE(Fx )y CLesson 10:Date: 2013 Common Core, Inc. Some rights reserved. commoncore.orgAAngle Problems and Solving Equations3/19/14155This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 10NYS COMMON CORE MATHEMATICS CURRICULUM6.In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measure ofand .andare vertical angles and are of equalmeasurement.andare complementaryangles and sum to.EFx Gx C( )y y y H( )FIn a complete sentence, describe the relevant angle relationships in the following diagram. Find the measureofand.and(are complementary angles and sum to.)( )8.D( )J( )24 AK7.7 3( )()()In a complete sentence, describe the relevant angle relationships in the following diagram. Find the measure of.andare vertical angles and are of equalmeasurement.andare adjacent and sum to themeasurement of.G3x ( )(( ) 2013 Common Core, Inc. Some rights reserved. commoncore.orgQ155 CFDE)Lesson 10:Date:56 Angle Problems and Solving Equations3/19/14156This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 10NYS COMMON CORE MATHEMATICS CURRICULUM9.The ratio of the measures of a pair of adjacent angles on a line isa.Find the measures of the two angles.Angle, Angle( )b.7 3Angle()Angle()( )Draw a diagram to scale of these adjacent angles. Indicate the measurements of each angle.50 40 10. The ratio of the measures of three adjacent angles on a line isa.Angle, Angle, Angle(b. Find the measures of the three angles.Find the measures of the three angles.Angle()Angle()Angle())()Draw a diagram to scale of these adjacent angles. Indicate the measurements of each angle.75 Lesson 10:Date: 2013 Common Core, Inc. Some rights reserved. commoncore.org60 45 Angle Problems and Solving Equations3/19/14157This work is licensed under aCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Adjacent Angles: Two angles and with a common side ⃗⃗⃗⃗⃗ are adjacent angles if belongs to the interior of . Vertical Angles: Two angles are vertical angles (or vertically opposite angles) if their sides form two pairs of opposite rays. Angles on a Line: The sum of the me
4 Step Phonics Quiz Scores Step 1 Step 2 Step 3 Step 4 Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5 Lesson 6 Lesson 7 Lesson 8 Lesson 9 Lesson 10 Lesson 11 Lesson 12 Lesson 13 Lesson 14 Lesson 15 . Zoo zoo Zoo zoo Yoyo yoyo Yoyo yoyo You you You you
Before measuring an angle, it is helpful to estimate the measure by using benchmarks, such as a right angle and a straight angle. For example, to estimate the measure of the blue angle below, compare it to a right angle and to a straight angle. 90 angle 180 angle A right angle has a measure of 90
An angle is said to be an obtuse angle if it is greater than 90 but is less than 120 . Straight Angle An angle whose measure is 180 is a straight angle. This angle got its name as it forms a straight line. AOB is a straight angle. Reflex Angle A reflex angle
The Angle Bisector Theorem 781 Lesson 13-2 The Angle Bisector Theorem Lesson 13-2 BIG IDEA The bisector of any angle of a triangle splits the opposite side into two segments whose lengths are in the same ratio as the sides including the original angle. In Lesson 6-2, you learned that the bisector of the vertex angle of
Participant's Workbook Financial Management for Managers Institute of Child Nutrition iii Table of Contents Introduction Intro—1 Lesson 1: Financial Management Lesson 1—1 Lesson 2: Production Records Lesson 2—1 Lesson 3: Forecasting Lesson 3—1 Lesson 4: Menu Item Costs Lesson 4—1 Lesson 5: Product Screening Lesson 5—1 Lesson 6: Inventory Control Lesson 6—1
Once the measure of an angle is known, the angle can be classified as one of three types of angles. These types are defined in relation to a right angle. Types of Angles A right angle m A 90 acute angle 0 m A 90 A obtuse angle 90 m A 180 A Angle Measure
Vertical Angles: Two angles whose sides form two pairs of opposite rays. Acute Angle: An angle less than 90 degrees. Obtuse Angle: An angle greater than 90 degrees. Right Angle: An angle equal to 90 degrees. Straight Angle: An angle equal to 180 degrees. Midpoint: The point that divide
Lesson 41 Day 1 - Draft LESSON 42 - DESCRIPTIVE PARAGRAPH Lesson 42 Day 1 - Revise Lesson 42 Day 1 - Final Draft Lesson 42 - Extra Practice LESSON 43 - EXPOSITORY PARAGRAPH Lesson 43 Day 1 - Brainstorm Lesson 43 Day 1 - Organize Lesson 43 Day 1 - Draft LESSON 44 - EXPOSITORY PARAGRAPH Lesson 44 Day 1 - Revise
