# MA.912.GR.1.1

Prove relationships and theorems about lines and angles. Solve mathematical and real-world problems involving postulates, relationships and theorems of lines and angles.

### Clarifications

Clarification 1: Postulates, relationships and theorems include vertical angles are congruent; when a transversal crosses parallel lines, the consecutive angles are supplementary and alternate (interior and exterior) angles and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

Clarification 2: Instruction includes constructing two-column proofs, pictorial proofs, paragraph and narrative proofs, flow chart proofs or informal proofs.

Clarification 3: Instruction focuses on helping a student choose a method they can use reliably.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Angle
• Congruent
• Corresponding Angles
• Supplementary Angles
• Transversal
• Vertical

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 3, students described and identified line segments, rays, perpendicular lines and parallel lines. In grade 4, students classified and solved problems involving acute, right, obtuse, straight and obtuse angle measures. In grade 8, students solved problems involving supplementary, complementary, adjacent and vertical angles. In Geometry, students prove relationships and theorems and solve problems involving lines and angle measure. In later courses, students will study lines and angles in two and three dimensions using vectors and they will use radians to measure angles.
• While focus of this benchmark are those postulates, relationships and theorems listed in Clarification 1, instruction includes other definitions, postulates, relationships or theorems such as the midpoint of a segment, angle bisector, segment bisector, perpendicular bisector, the angle addition postulate and the segment addition postulate. Additionally, some postulates and theorems have a converse (i.e., if conclusion, then hypothesis) that can be included.
• Instruction includes the connection to the Logic and Discrete Theory benchmarks when developing proofs. Additionally, with the construction of proofs, instruction reinforces the Properties of Operations, Equality and Inequality. (MTR.5.1
• For example, when proving that vertical angles are congruent, students must be able to understand and use the Substitution Property of Equality and the Subtraction Property of Equality.
• Instruction utilizes different ways students can organize their reasoning by constructing various proofs when proving geometric statements. It is important to explain the terms statements and reasons, their roles in a geometric proof, and how they must correspond to each other. Regardless of the style, a geometric proof is a carefully written argument that begins with known facts, proceeds from there through a series of logical deductions, and ends with the statement you are trying to prove. (MTR.2.1
• For examples of different types of proofs, please see MA.912.LT.4.8
• Instruction includes the connection to compass and straight edge constructions and how the validity of the construction is justified by a proof. (MTR.5.1
• Students should develop an understanding for the difference between a postulate, which is assumed true without a proof, and a theorem, which is a true statement that can be proven. Additionally, students should understand why relationships and theorems can be proven and postulates cannot.
• Instruction includes the use of hatch marks, hash marks, arc marks or tick marks, a form of mathematical notation, to represent segments of equal length or angles of equal measure in diagrams and images.
• Students should understand the difference between congruent and equal. If two segments are congruent  (i.e., PQMN),  then they have equivalent lengths (i.e., PQ = MN) and the converse is true. If two angles are congruent (i.e., ∠ABC ≅ ∠PQR), then they have equivalent angle measure (i.e., $m$ABC = $m$PQR) and the converse is true.
• Instruction includes the use of hands-on manipulatives and geometric software for students to explore relationships, postulates and theorems.
• For example, folding paper (e.g., patty paper) can be used to explore what happens with angle pairs when two parallel lines are cut by a transversal. Students can discuss the possible pairs of corresponding angles, alternate interior angles, alternate exterior angles and consecutive (same-side interior and same-side exterior) angles. (MTR.2.1, MTR.4.1)
• Problem types include mathematical and real-world context where students evaluate the value of a variable that will make two lines parallel; utilize two sets of parallel lines or more than two parallel lines; or write and solve equations to determine an unknown segment length or angle measure.
• Instruction for some relationships or postulates may be necessary in order to prove theorems.
• For example, to prove that consecutive angles are supplementary when a given transversal crosses parallel lines, students will need to know the postulate that states corresponding angles are congruent.
• For example, to prove or use the perpendicular bisector theorem, students will need to have knowledge of the definition of a perpendicular bisector of a segment and midpoint.

### Common Misconceptions or Errors

• Students may misuse the terms corresponding, alternate interior and alternate exterior as synonyms of congruent. To help address this conception, students should develop the understanding that these angles are congruent if and only if two parallel lines are cut by a transversal. Similarly, same-side angles are supplementary if and only if the two lines being cut by a transversal are parallel.

• Runways are identified by their orientation relative to Magnetic North as viewed by an approaching aircraft. Runway directions are always rounded to the nearest ten degrees and the zero in the “ones” column is never depicted (i.e., 170 degrees would be viewed as “17” and 20 degrees would be seen as “2”). The same runway has two names which are dependent on the direction of approach.
• Use the aerial of Northeast Florida Regional Airport in St. Augustine, FL to answer the questions below.

• Part A. Flying to the runway from point F, the runway is Runway 13. This means the heading is 130° off magnetic north. Draw a line through point R that goes to magnetic north, what is true about that line and line NF
• Part B. On the line drawn in Part A, draw and label a point, A
• Part C. Measure angle FRA
• Part E. What is the name of the runway when approaching from point R

• Part A. Given AB, use a compass and straightedge to construct line $l$ such that line $l$ and AB form 90° and the point of intersection, M, is the midpoint of AB
• Part B. Suppose that point P lies on line $l$ as shown below. What conjecture can be made about point P? Which endpoint of AB is closest to point P? Use a compass to test your conjecture.
• Part C. What if a point, Q, was added to line $l$ Which endpoint of AB is closest to point Q How does this compare with your conjecture in Part B?
• Part D. How can the construction from Part A and the conjectures from Part B and Part C be used to prove that AP = BP given that line $l$ is the perpendicular bisector of AB and point P lies on line $l$?

### Instructional Items

Instructional Item 1
• What value of $x$ will make M the midpoint of PQ if  PM = 3$x$ − 1 and  PQ = 5$x$ + 3?

Instructional Item 2
• Two lines intersect at point P. If the measures of a pair of vertical angles are (2$x$ − 7)° and ($x$ + 13)°, determine $x$ and the measures of the other two angles?

Instructional Item 3
• Based on the figure below, complete a proof to prove that ∠1 ≅ ∠2 given that $a$ || $b$ and $c$ || $d$.

*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.
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206310: Geometry (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206315: Geometry for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912065: Access Geometry (Specifically in versions: 2015 - 2022, 2022 and beyond (current))
1209315: Mathematics for ACT and SAT (Specifically in versions: 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.GR.1.AP.1: Use the relationships and theorems about lines and angles to solve mathematical or real-world problems involving postulates, relationships and theorems of lines and angles.

## Related Resources

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

## Formative Assessments

Finding Angle Measures - 1:

Students are asked to find the measures of angles formed by three concurrent lines and to justify their answers.

Type: Formative Assessment

Finding Angle Measures - 3:

Students are asked to find the measures of angles formed by two parallel lines and two transversals.

Type: Formative Assessment

Finding Angle Measures - 2:

Students are asked to find the measures of angles formed by two parallel lines and a transversal.

Type: Formative Assessment

Camping Calculations:

Students are asked to find the measure of an angle formed by the support poles of a tent using the properties of geometric shapes.

Type: Formative Assessment

Same Side Interior Angles:

Students are asked to describe and justify the relationship between same side interior angles.

Type: Formative Assessment

Justifying Angle Relationships:

Students are asked to describe and justify the relationship between corresponding angles and alternate interior angles.

Type: Formative Assessment

Proving the Alternate Interior Angles Theorem:

In a diagram involving two parallel lines and a transversal, students are asked to use rigid motion to prove that alternate interior angles are congruent.

Type: Formative Assessment

Constructions for Parallel Lines:

Students are asked to construct a line parallel to a given line through a given point.

Type: Formative Assessment

Equidistant Points:

Students are asked to prove that a point on the perpendicular bisector of a line segment is equidistant from the endpoints of the segment.

Type: Formative Assessment

Proving the Vertical Angles Theorem:

Students are asked to identify a pair of vertical angles in a diagram and then prove that they are congruent.

Type: Formative Assessment

## Lesson Plans

Triangle Mid-Segment Theorem:

The Triangle Mid-Segment Theorem is used to show the writing of a coordinate proof clearly and concisely.

Type: Lesson Plan

Proof of Quadrilaterals in Coordinate Plane:

This lesson is designed to instruct students on how to identify special quadrilaterals in the coordinate plane using their knowledge of distance formula and the definitions and properties of parallelograms, rectangles, rhombuses, and squares. Task cards, with and without solution-encoded QR codes, are provided for cooperative group practice. The students will need to download a free "QR Code Reader" app onto their SmartPhones if you choose to use the cards with QR codes.

Type: Lesson Plan

To Be or Not to Be a Parallelogram:

Students apply parallelogram properties and theorems to solve real world problems. The acronym, P.I.E.S. is introduced to support a problem solving strategy, which involves drawing a Picture, highlighting important Information, Estimating and/or writing equation, and Solving problem.

Type: Lesson Plan

Parallel Thinking Debate:

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

Type: Lesson Plan

Vertical Angles: Proof and Problem-Solving:

Students will explore the relationship between vertical angles and prove the Vertical Angle Theorem. They will use vertical angle relationships to calculate other angle measurements.

Type: Lesson Plan

Diagonally Half of Me!:

This lesson is an exploration activity assisting students prove that diagonals of parallelograms bisect each other. It allows them to compare other quadrilaterals with parallelograms in order to make conjectures about the diagonals of parallelograms.

Type: Lesson Plan

Students will use formulas they know (distance, midpoint, and slope) to classify quadrilaterals.

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

Proving Parallelograms Algebraically:

This lesson reviews the definition of a parallelogram and related theorems. Students use these conditions to algebraically prove or disprove a given quadrilateral is a parallelogram.

Type: Lesson Plan

How Much Proof Do We Need?:

Students determine the minimum amount of information needed to prove that two triangles are similar.

Type: Lesson Plan

Proving quadrilaterals algebrically using slope and distance formula:

Working in groups, students will prove the shape of various quadrilaterals using slope, distance formula, and polygon properties. They will then justify their proofs to their classmates.

Type: Lesson Plan

In this lesson, students will use coordinates to algebraically prove that quadrilaterals are rectangles, parallelograms, and trapezoids. A through introduction to writing coordinate proofs is provided as well as plenty of practice.

Type: Lesson Plan

Observing the Centroid:

Students will construct the medians of a triangle then investigate the intersections of the medians.

Type: Lesson Plan

Determination of the Optimal Point:

Students will use dynamic geometry software to determine the optimal location for a facility under a variety of scenarios. The experiments will suggest a relation between the optimal point and a common concept in geometry; in some cases, there will be a connection to a statistical concept. Algebra can be used to verify some of the conjectures.

Type: Lesson Plan

This lesson provides a series of assignments for students at the Getting Started, Moving Forward, and Almost There levels of understanding for the Mathematics Formative Assessment System (MFAS) Task Describe the Quadrilateral (CPALMS Resource ID#59180). The assignments are designed to "move" students from a lower level of understanding toward a complete understanding of writing a coordinate proof involving quadrilaterals.

Type: Lesson Plan

The Centroid:

Students will construct the centroid of a triangle using graph paper or GeoGebra in order to develop conjectures. Then students will prove that the medians of a triangle actually intersect using the areas of triangles.

Type: Lesson Plan

What's the Point?:

Students will algebraically find the missing coordinates to create a specified quadrilateral using theorems that represent them, and then algebraically prove their coordinates are correct.

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

Type: Lesson Plan

Polygon...Prove it:

While this is an introductory lesson on the standard, students will enjoy it, as they play "Speed Geo-Dating" during the Independent practice portion. Students will use algebra and coordinates to prove rectangles, rhombus, and squares. Properties of diagonals are not used in this lesson.

Type: Lesson Plan

Parallel Lines:

Students will prove that alternate interior angles and corresponding angles are congruent given two parallel lines and a traversal. Students will use GeoGebra to explore real-world images to prove their line segments are parallel.

Type: Lesson Plan

Let's Prove the Pythagorean Theorem:

Students will use Triangle Similarity to derive the proof of the Pythagorean Theorem and apply this method to develop the idea of the geometric mean with respect to the relationships in right triangles.

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

Help me Find my Relationship!:

In this lesson, students will investigate the relationship between angles when parallel lines are cut by a transversal. Students will identify angles, and find angle measures, and they will use the free application GeoGebra (see download link under Suggested Technology) to provide students with a visual representation of angle relationships.

Type: Lesson Plan

An Investigation of Angle Relationships Formed by Parallel Lines and a Transversal Using GeoGebra:

In this lesson, students will discover angle relationships formed when two parallel lines are cut by a transversal (corresponding, alternate interior, alternate exterior, same-side interior, same-side exterior). They will establish definitions and identify whether these angle pairs are supplementary or congruent.

Type: Lesson Plan

Accurately Acquired Angles:

Students will start the lesson by playing a game to review angle pairs formed by two lines cut by a transversal. Once students are comfortable with the angle pairs the teacher will review the relationships that are created once the pair of lines become parallel. The teacher will give an example of a proof using the angle pairs formed by two parallel lines cut by a transversal. The students are then challenged to prove their own theorem in groups of four. The class will then participate in a Stay and Stray to view the other group's proofs. The lesson is wrapped up through white board questions answered within groups and then as a whole class.

Type: Lesson Plan

What's the Point? Part 1:

This is a patty paper-folding activity where students measure and discover the properties of the point of concurrency of the perpendicular bisectors of the sides of a triangle.

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 Tutorial

Angle UP: Player 1:

Explore the construction processes for constructing an angle bisector, copying an angle and constructing a line parallel to a given line through a point not on the line using a variety of tools in this interactive, retro video game-themed tutorial.

NOTE: This tutorial uses both the angle bisector construction and the construction to copy an angle as an extension opportunity to also construct a line parallel to a given line through a point not on the line. Students also learn to identify corresponding angles created when a transversal crosses parallel lines, and discover using Geogebra that these angles are congruent.

Type: Original Student Tutorial

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.

Points equidistant from two points in the plane:

This task asks students to show how certain points on a plane are equidistant to points on a segment when placed on a perpendicular bisector.

Reflected Triangles:

This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected.

Tangent Lines and the Radius of a Circle:

This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent.

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.

Find the Missing Angle:

This task provides students the opportunity to see how the mathematical ideas embedded in the standards and clusters mature over time. The task uses facts about supplementary, complementary, vertical, adjacent, and alternate interior angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. It is a good introduction to writing paragraphs, 2-column, and/or flow chart proofs.

Is This a Rectangle?:

The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem.

## MFAS Formative Assessments

Camping Calculations:

Students are asked to find the measure of an angle formed by the support poles of a tent using the properties of geometric shapes.

Constructions for Parallel Lines:

Students are asked to construct a line parallel to a given line through a given point.

Equidistant Points:

Students are asked to prove that a point on the perpendicular bisector of a line segment is equidistant from the endpoints of the segment.

Finding Angle Measures - 1:

Students are asked to find the measures of angles formed by three concurrent lines and to justify their answers.

Finding Angle Measures - 2:

Students are asked to find the measures of angles formed by two parallel lines and a transversal.

Finding Angle Measures - 3:

Students are asked to find the measures of angles formed by two parallel lines and two transversals.

Justifying Angle Relationships:

Students are asked to describe and justify the relationship between corresponding angles and alternate interior angles.

Proving the Alternate Interior Angles Theorem:

In a diagram involving two parallel lines and a transversal, students are asked to use rigid motion to prove that alternate interior angles are congruent.

Proving the Vertical Angles Theorem:

Students are asked to identify a pair of vertical angles in a diagram and then prove that they are congruent.

Same Side Interior Angles:

Students are asked to describe and justify the relationship between same side interior angles.

## Original Student Tutorials Mathematics - Grades 9-12

Angle UP: Player 1:

Explore the construction processes for constructing an angle bisector, copying an angle and constructing a line parallel to a given line through a point not on the line using a variety of tools in this interactive, retro video game-themed tutorial.

NOTE: This tutorial uses both the angle bisector construction and the construction to copy an angle as an extension opportunity to also construct a line parallel to a given line through a point not on the line. Students also learn to identify corresponding angles created when a transversal crosses parallel lines, and discover using Geogebra that these angles are congruent.

## Student Resources

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

## Original Student Tutorial

Angle UP: Player 1:

Explore the construction processes for constructing an angle bisector, copying an angle and constructing a line parallel to a given line through a point not on the line using a variety of tools in this interactive, retro video game-themed tutorial.

NOTE: This tutorial uses both the angle bisector construction and the construction to copy an angle as an extension opportunity to also construct a line parallel to a given line through a point not on the line. Students also learn to identify corresponding angles created when a transversal crosses parallel lines, and discover using Geogebra that these angles are congruent.

Type: Original Student Tutorial

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.

Points equidistant from two points in the plane:

This task asks students to show how certain points on a plane are equidistant to points on a segment when placed on a perpendicular bisector.

Reflected Triangles:

This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected.

Tangent Lines and the Radius of a Circle:

This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent.

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.

Find the Missing Angle:

This task provides students the opportunity to see how the mathematical ideas embedded in the standards and clusters mature over time. The task uses facts about supplementary, complementary, vertical, adjacent, and alternate interior angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. It is a good introduction to writing paragraphs, 2-column, and/or flow chart proofs.

Is This a Rectangle?:

The goal of this task is to provide an opportunity for students to apply a wide range of ideas from geometry and algebra in order to show that a given quadrilateral is a rectangle. Creativity will be essential here as the only given information is the Cartesian coordinates of the quadrilateral's vertices. Using this information to show that the four angles are right angles will require some auxiliary constructions. Students will need ample time and, for some of the methods provided below, guidance. The reward of going through this task thoroughly should justify the effort because it provides students an opportunity to see multiple geometric and algebraic constructions unified to achieve a common purpose. The teacher may wish to have students first brainstorm for methods of showing that a quadrilateral is rectangle (before presenting them with the explicit coordinates of the rectangle for this problem): ideally, they can then divide into groups and get to work straightaway once presented with the coordinates of the quadrilateral for this problem.

## Parent Resources

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

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.

Points equidistant from two points in the plane:

This task asks students to show how certain points on a plane are equidistant to points on a segment when placed on a perpendicular bisector.

Reflected Triangles:

This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected.

Tangent Lines and the Radius of a Circle:

This problem solving task challenges students to find the perpendicular meeting point of a segment from the center of a circle and a tangent.

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.