# MA.8.GR.1.4

Solve mathematical problems involving the relationships between supplementary, complementary, vertical or adjacent angles.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning
Date Adopted or Revised: 08/20
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Angle (∠)
• Complementary (∠)
• Supplementary (∠)
• Vertical

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In the elementary grades, students were introduced to acute, right, obtuse, straight reflex angles and solved real-world and mathematical problems involving angle measures. They also used angles to classify triangles and quadrilaterals. In grade 8, students solve problems involving supplementary, complementary, vertical and adjacent angles. In Geometry, students will extend the learning from this benchmark to prove relationships and theorems involving lines and angles.
• This benchmark is foundational to help develop the understanding of angles and connections related to parallel lines cut by a transversal.
• In order for students to learn relationships between angles, it is important to provide an opportunity to connect complementary and supplementary angles to work with triangles. Students should draw or be given a right triangle to explore rearranging the angles to show both the 90 and 180 degrees that can be created for a right angle and a straight line, respectively.
• To support the concept of adjacent angles, students should have examples and non-examples to write their own definition and revise it based on critiques from others (MTR.4.1). Students should trace each angle with different colors to ensure that there isn’t overlap, but has a common side.

• When discussing vertical angles, use a model of two strips of paper with a small brad at the center where they cross. Then, moving the paper to create different sized angles, measure each angle to show the vertical angle measures to lead to understanding that the vertical angles will have the same measure.
• Vertical angles can be explored using the same activity as adjacent angles with examples and non-examples. The criteria could include the following:
• Formed from exactly 2 straight intersecting lines
• Pair of angles
• Common vertex
• It is important to have students’ reasoning supported. This can be done by making statements with reasoning such as “always true, sometimes true, never true.”
• For example, a linear pair of angles (a type of adjacent angles) are always supplementary because they form a straight line.
• Once conceptual understanding and definitions are built, introduce algebraic concepts for students to write and solve equations using facts about the angle relationships. Students should be able to generate equations written in different forms.
• For example, if students are provided the figure below, they can generate multiple equivalent equations to represent their thinking. For this figure, three possible equations are:
180 = 147 + 2$x$ + 3
180 − 147 = 2$x$ +3
2$x$ + 150 = 180

### Common Misconceptions or Errors

• Students may invert the definition of complementary and supplementary.

### Strategies to Support Tiered Instruction

• Instruction includes co-constructing a graphic organizer with students to measure, label and record the angle measurement of two intersecting lines. The teacher labels the angles, and measures and record the angle measurements. The teacher then leads a discussion and documents the relationships between different angle pairs.
• Instruction includes erasing or covering part of a line for students to visually see the supplementary angles within two intersecting lines.

• Instruction includes co-creating a graphic organizer identifying the relationships between supplementary, complementary, vertical, and adjacent angles. Include a strategy for solving problems involving each type of angle pair, such as setting vertical angle measures equal.

Instructional Task 1 (MTR.1.1, MTR.2.1)
Complete the table below that includes the following types of angles:

Instructional Task 2 (MTR.4.1)
Determine if each of the following statements is always true, sometimes true or never true. For each statement that you chose as “sometimes true”, provide an example and non-example.
a. The sum of the measures of two supplementary angles is 180°.
b. Vertical angles are also adjacent angles.
c. Two adjacent angles are complementary.
d. If two lines intersect, each pair of vertical angles are complementary.

### Instructional Items

Instructional Item 1
The measure of angle 1 is 12 more than the measure of angle 2. What is the degree measure of angle 3?

*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 and beyond (current))
1205070: M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 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.4: Identify supplementary, complementary, vertical or adjacent angle relationships.

## Related Resources

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

## Formative Assessments

What Is Your Angle?:

Students are asked use knowledge of angle relationships to write and solve an equation to determine an unknown angle measure.

Type: Formative Assessment

Straight Angles:

Students are asked to write and solve equations to determine unknown angle measures in supplementary angle relationships.

Type: Formative Assessment

Solve for the Angle:

Students are asked to write and solve equations to determine unknown angle measures in supplementary and complementary angle pairs.

Type: Formative Assessment

Find the Angle Measure:

Students are asked to use knowledge of angle relationships to write and solve equations to determine unknown angle measures.

Type: Formative Assessment

## Lesson Plans

Parallel Thinking Debate:

Students prove theorems related to parallel lines using vertical, corresponding, and alternate interior angles.

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

What's the Problem:

Students solve problems using triangle congruence postulates and theorems.

Type: Lesson Plan

Proving and Using Congruence with Corresponding Angles:

Students, will prove that corresponding angles are congruent. Directions for using GeoGebra software to discover this relationship is provided.

Type: Lesson Plan

Sine, Sine, Everywhere a Sine:

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

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

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

Angles, angles, everywhere!:

This is an introductory lesson that allows students to explore complementary and supplementary angles and their relationships. Students will measure the degrees for sets of angles. They will then use their knowledge of straight, right, obtuse, and acute angles to make connections to complementary, supplementary, adjacent, and vertical angles. This lesson provides opportunities for kinesthetic, auditory, and visual learning.

Type: Lesson Plan

Special Angle Pairs Discovery Activity:

This lesson uses a discovery approach to identify the special angles formed when a set of parallel lines is cut by a transversal. During this lesson, students identify the angle pair and the relationship between the angles. Students use this relationship and special angle pairs to make conjectures about which angle pairs are considered special angles.

Type: Lesson Plan

Kissing “V”s:

This lesson uses a paper-cutting activity to teach vertical angles. The lesson provides examples in which students must solve equations to find missing measures.

Type: Lesson Plan

Through a hands-on-activity and guided practice, students will explore parallel lines intersected by a transversal and the measurements and relationships of the angles created. They will solve for missing measurements when given a single angle's measurement. They will also use the relationships between angles to set up equations and solve for a variable.

Type: Lesson Plan

## Original Student Tutorials

Playground Angles Part 1:

Explore complementary and supplementary angles around the playground with Jacob in this interactive tutorial.

This is Part 1 in a two-part series. Click HERE to open Playground Angles: Part 2.

Type: Original Student Tutorial

Playground Angles: Part 2:

Help Jacob write and solve equations to find missing angle measures based on the relationship between angles that sum to 90 degrees and 180 degrees in this playground-themed, interactive tutorial.

This is Part 2 in a two-part series. Click HERE to open Playground Angles: Part 1.

Type: Original Student Tutorial

## Tutorials

Finding Missing Angle Measures:

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

Type: Tutorial

Finding the Measure of Complementary Angles:

The video will use algebra to find the measure of two angles whose sum equals 90 degrees, better known as complementary angles.

Type: Tutorial

Find Measure of Complementary Angles:

Watch as we use algebra to find the measure of two complementary angles.

Type: Tutorial

Find Measure of Supplementary Angles:

Watch as we use algebra to find the measure of supplementary angles, whose sum is 180 degrees.

Type: Tutorial

Find Measure of Vertical Angles:

This video uses knowledge of vertical angles to solve for the variable and the angle measures.

Type: Tutorial

Introduction to Vertical Angles:

This video uses facts about supplementary and adjacent angles to introduce vertical angles.

Type: Tutorial

Find Measure of Angles in a Word Problem:

This video demonstrates solving a word problem involving angle measures.

Type: Tutorial

Complementary and Supplementary Angles:

The video will demonstrate the difference between supplementary angles and complementary angles, by using the given measurements of angles.

Type: Tutorial

## MFAS Formative Assessments

Find the Angle Measure:

Students are asked to use knowledge of angle relationships to write and solve equations to determine unknown angle measures.

Solve for the Angle:

Students are asked to write and solve equations to determine unknown angle measures in supplementary and complementary angle pairs.

Straight Angles:

Students are asked to write and solve equations to determine unknown angle measures in supplementary angle relationships.

What Is Your Angle?:

Students are asked use knowledge of angle relationships to write and solve an equation to determine an unknown angle measure.

## Original Student Tutorials Mathematics - Grades 6-8

Playground Angles Part 1:

Explore complementary and supplementary angles around the playground with Jacob in this interactive tutorial.

This is Part 1 in a two-part series. Click HERE to open Playground Angles: Part 2.

Playground Angles: Part 2:

Help Jacob write and solve equations to find missing angle measures based on the relationship between angles that sum to 90 degrees and 180 degrees in this playground-themed, interactive tutorial.

This is Part 2 in a two-part series. Click HERE to open Playground Angles: Part 1.

## Student Resources

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

## Original Student Tutorials

Playground Angles Part 1:

Explore complementary and supplementary angles around the playground with Jacob in this interactive tutorial.

This is Part 1 in a two-part series. Click HERE to open Playground Angles: Part 2.

Type: Original Student Tutorial

Playground Angles: Part 2:

Help Jacob write and solve equations to find missing angle measures based on the relationship between angles that sum to 90 degrees and 180 degrees in this playground-themed, interactive tutorial.

This is Part 2 in a two-part series. Click HERE to open Playground Angles: Part 1.

Type: Original Student Tutorial

## Tutorials

Finding Missing Angle Measures:

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

Type: Tutorial

Finding the Measure of Complementary Angles:

The video will use algebra to find the measure of two angles whose sum equals 90 degrees, better known as complementary angles.

Type: Tutorial

Find Measure of Complementary Angles:

Watch as we use algebra to find the measure of two complementary angles.

Type: Tutorial

Find Measure of Supplementary Angles:

Watch as we use algebra to find the measure of supplementary angles, whose sum is 180 degrees.

Type: Tutorial

Find Measure of Vertical Angles:

This video uses knowledge of vertical angles to solve for the variable and the angle measures.

Type: Tutorial

Introduction to Vertical Angles:

This video uses facts about supplementary and adjacent angles to introduce vertical angles.

Type: Tutorial

Find Measure of Angles in a Word Problem:

This video demonstrates solving a word problem involving angle measures.

Type: Tutorial

Complementary and Supplementary Angles:

The video will demonstrate the difference between supplementary angles and complementary angles, by using the given measurements of angles.

Type: Tutorial

## Parent Resources

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