MA.8.GR.1.5

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Solve problems involving the relationships of interior and exterior angles of a triangle.

Clarifications

Clarification 1: Problems include using the Triangle Sum Theorem and representing angle measures as algebraic expressions.
General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 8
Strand: Geometric Reasoning
Standard: Develop an understanding of the Pythagorean Theorem and angle relationships involving triangles.
Date Adopted or Revised: 08/20
Status: State Board Approved

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

 

Terms from the K-12 Glossary

  • Angle (∠)
  • Supplementary Angles

 

Vertical Alignment

Previous Benchmarks

Next Benchmarks

 

Purpose and Instructional Strategies

In grades 4 and 5, students used angle measures as an attribute of two-dimensional figures and identified and classified angles as acute, right, obtuse, straight and reflex. In grade 8, students solve problems involving the interior and exterior angles of a triangle. In Geometry, the work in this benchmark will be extended to proving relationships and theorems about triangles.
  • Students should explore the concept before being provided the theorem. Once conceptual understanding is developed, students can use numerical equations to solve problems involving finding missing interior or exterior angles. From there, students should develop algebraic equations to solve for both missing angle measurements as well as variables.
  • Instruction includes students exploring and using deductive reasoning to determine relationships that exist between angle sums and exterior angle sums of triangles. Students should construct various triangles and find the measures of both the interior and exterior angles. Applying knowledge of these relationships, students can use deductive reasoning to find the measure of any missing angles.
  • Using an investigation, such as the one below, for the Triangle Sum Theorem will help students conceptually understand the total degree measures of a triangle related to a straight line.
    • Give each student a triangle (a variety of triangle sizes will allow for discussion).
    • Have students use a lined sheet of paper or draw a straight line using a ruler to represent the straight angle of 180°.
    • Next, have the students tear off each angle of the triangle.
    • Then, putting the angle side towards the line, the students will be able to model the 180° of the triangle measures to a straight line. This is illustrated in the top part of the figure below.
      180° of the triangle measures to a straight line.
    • A similar investigation can be used to connect the measure of an exterior angle of a triangle and the two opposite interior angles in the triangle.
  • Students can also use patty paper to trace the concepts to make connections.

 

Common Misconceptions or Errors

  • Students may incorrectly think that the exterior angle is the whole reflex angle of the interior angle.
  • Students may not recognize that there are two exterior angles for each interior angle and that the exterior angles are congruent.

 

Strategies to Support Tiered Instruction

  • Teacher provides a visual with one angle of the triangle on a straight line of 180°, similar to the illustration below. Teacher provides instruction on how to measure each of the angles with a protractor and color code congruent angles.
  • Teacher provides tracing paper or patty paper to trace the related interior and exterior angles and have discussion about the angle relationships.
    interior and exterior angles
  • Instruction includes providing a visual with one angle of the triangle on a straight line of 180° and measuring each angle in the figure. Teacher co-constructs a graphic organizer with students to document the angle measurements, color-code congruent angles in the figure and in the graphic organizer, and identifies the relationships between interior and exterior angles of a triangle and the Triangle Sum Theorem.
    interior and exterior angles
    Table with columns 'Interior Angles', 'Exterior Angles', 'Relationship'.

 

Instructional Tasks

Instructional Task 1 (MTR.1.1, MTR.2.1)
ΔABC and ΔBCD share a common side of BC. Angle BAC is 30° and angle ABC is 60°.
  • Part A. Create a diagram to represent this description.
  • Part B. What will be the measure of angle BCD? Provide an explanation to support how to find the measurement of angle BCD.
  • Part C. Are there questions that still need to be answered to approach Part B?

 

Instructional Items

Instructional Item 1
In triangle LMN, the measure of angle L is 50° and the measure of angle M 70°. What is the measure of the exterior angle to angle N?

Instructional Item 2
One measure of an angle in a triangle is 96°. The other two angle measures are represented by 2x and x + 12. Determine the other two degree measures for the missing angles.

 

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

Related Courses

This benchmark is part of these courses.
1205050: M/J Accelerated Mathematics Grade 7 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 - 2024, 2024 and beyond (current))
1205070: M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 - 2024, 2024 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 - 2024, 2024 and beyond (current))
7812030: Access M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.8.GR.1.AP.5: Given an image, solve simple problems involving the relationships of interior and exterior angles of a triangle.

Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

Formative Assessment

Justifying the Exterior Angle of a Triangle Theorem:

Students are asked to apply the Exterior Angle of a Triangle Theorem and provide an informal justification.

Type: Formative Assessment

Lesson Plans

The Laws of Sine and Cosine:

In this lesson, students determine when to use the Law of Cosine and/or the Law of Sine to calculate the measures of sides and angles of non-right triangles.

Type: Lesson Plan

Sine and Cosine Relationship between Complementary Angles:

This is a lesson on the relationship between the Sine and Cosine values of Complementary Angles.

Type: Lesson Plan

Sine, Sine, Everywhere a Sine:

Students discover the complementary relationship between sine and cosine in a right triangle.

Type: Lesson Plan

Will You Survive?:

Students are stranded on a desert island and will need to use the law of sines in order to find the quickest path to a rescue vessel.

Note: This is not an introductory lesson for the standard.

Type: Lesson Plan

The Copernicus' Travel:

This lesson uses Inverse Trigonometric Ratios to find acute angle measures in right triangles. Students will analyze the given information and determine the best method to use when solving right triangles. The choices reviewed are Trigonometric Ratios, The Pythagorean Theorem, and Special Right Triangles.

Type: Lesson Plan

Discovering Triangle Sum:

This lesson is designed to address all levels and types of learners to improve understanding of the triangle sum theorem from the simplest perspective and progress steadily by teacher led activities to a more complex level. It is intended to create a solid foundation in geometric reasoning to help students advance to higher levels in confidence.

Type: Lesson Plan

The Trig Song:

This lesson is a group project activity designed to reinforce the concepts of sine and cosine. The lesson begins with a spiral review of the concepts, which will move into the group project - writing an original song to demonstrate understanding and application of sine and cosine ratios.

Type: Lesson Plan

Following the Law of Sine:

This lesson introduces the law of sine. It is designed to give students practice in using the law to guide understanding. The summative assessment requires students to use the law of sine to plan a city project.

Type: Lesson Plan

Geometer Sherlock: Triangle Investigations:

The students will investigate and discover relationships within triangles; such as, the triangle angle sum theorem, and the triangle inequality theorem.

Type: Lesson Plan

How Tall am I?:

Students will determine the height of tall objects using three different calculation methods. They will work in groups to gather their data and perform calculations. A whole-class discussion is conducted at the end to compare results and discuss some of the possible errors.

Type: Lesson Plan

Triangles: Finding Interior Angle Measures:

The lesson begins with a hands-on activity and then an experiment with a GeoGebra-based computer model to discover the Triangle Angle Sum Theorem. The students write and solve equations to find missing angle measures in a variety of examples.

Type: Lesson Plan

Problem-Solving Tasks

Tile Patterns I: octagons and squares:

Students use interior and exterior angles to to verify attributes of an octagon and square. Students are given a tile pattern involving congruent regular octagons and squares.

Type: Problem-Solving Task

Find the Angle:

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Type: Problem-Solving Task

Tile Patterns II: hexagons:

This task is ideally suited for instruction purposes where students can take their time and develop several of the standards, as the mathematical content is directly related to, but somewhat exceeds, the content of the standard on sums of angles in triangles. Careful analysis of the angles requires students to construct valid arguments using abstract and quantitative reasoning. Producing the picture in part (c) helps students identify a common mathematical argument repeated multiple times. Students may use pattern blocks to develop the intuition for decomposing the hexagon into triangles.

Type: Problem-Solving Task

Tutorial

Finding Missing Angle Measures:

In this video, we find missing angle measures from a variety of examples.

 

Type: Tutorial

MFAS Formative Assessments

Justifying the Exterior Angle of a Triangle Theorem:

Students are asked to apply the Exterior Angle of a Triangle Theorem and provide an informal justification.

Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

Problem-Solving Tasks

Tile Patterns I: octagons and squares:

Students use interior and exterior angles to to verify attributes of an octagon and square. Students are given a tile pattern involving congruent regular octagons and squares.

Type: Problem-Solving Task

Find the Angle:

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Type: Problem-Solving Task

Tile Patterns II: hexagons:

This task is ideally suited for instruction purposes where students can take their time and develop several of the standards, as the mathematical content is directly related to, but somewhat exceeds, the content of the standard on sums of angles in triangles. Careful analysis of the angles requires students to construct valid arguments using abstract and quantitative reasoning. Producing the picture in part (c) helps students identify a common mathematical argument repeated multiple times. Students may use pattern blocks to develop the intuition for decomposing the hexagon into triangles.

Type: Problem-Solving Task

Tutorial

Finding Missing Angle Measures:

In this video, we find missing angle measures from a variety of examples.

 

Type: Tutorial

Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

Problem-Solving Tasks

Tile Patterns I: octagons and squares:

Students use interior and exterior angles to to verify attributes of an octagon and square. Students are given a tile pattern involving congruent regular octagons and squares.

Type: Problem-Solving Task

Find the Angle:

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Type: Problem-Solving Task

Tile Patterns II: hexagons:

This task is ideally suited for instruction purposes where students can take their time and develop several of the standards, as the mathematical content is directly related to, but somewhat exceeds, the content of the standard on sums of angles in triangles. Careful analysis of the angles requires students to construct valid arguments using abstract and quantitative reasoning. Producing the picture in part (c) helps students identify a common mathematical argument repeated multiple times. Students may use pattern blocks to develop the intuition for decomposing the hexagon into triangles.

Type: Problem-Solving Task