# MAFS.912.G-CO.3.10

Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
General Information
Subject Area: Mathematics
Domain-Subdomain: Geometry: Congruence
Cluster: Level 3: Strategic Thinking & Complex Reasoning
Cluster: Prove geometric theorems. (Geometry - Major Cluster) -

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

Date of Last Rating: 02/14
Status: State Board Approved
Assessed: Yes
Test Item Specifications

• Assessment Limits :
Items may assess theorems and their converses for interior triangle
sum, base angles of isosceles triangles, mid-segment of a triangle,
concurrency of medians, concurrency of angle bisectors, concurrency
of perpendicular bisectors, triangle inequality, and the Hinge
Theorem.

Items may include narrative proofs, flow-chart proofs, two-column
proofs, or informal proofs.

In items that require the student to justify, the student should not be
required to recall from memory the formal name of a theorem.

• Calculator :

Neutral

• Clarification :
Students will prove theorems about triangles.

Students will use theorems about triangles to solve problems

• Stimulus Attributes :
Items may be set in a real-world or mathematical context
• Response Attributes :
Items may require the student to give statements and/or
justifications to complete formal and informal proofs.

Items may require the student to justify a conclusion from a
construction.

Sample Test Items (1)
• Test Item #: Sample Item 1
• Question:

Drag statements from the statements column and reasons from the reasons column to their correct location to complete the proof.

• Difficulty: N/A
• Type: DDHT: Drag-and-Drop Hot Text

## Related Courses

This benchmark is part of these courses.
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1206310: Geometry (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1206315: Geometry for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
7912065: Access Geometry (Specifically in versions: 2015 - 2022 (current), 2022 and beyond)

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MAFS.912.G-CO.3.AP.10a: Measure the angles and sides of equilateral, isosceles, and scalene triangles to establish facts about triangles.

## Related Resources

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

## Assessments

Sample 1 - High School Geometry State Interim Assessment:

This is a State Interim Assessment for 9th-12th grade.

Type: Assessment

Sample 3 - High School Geometry State Interim Assessment:

This is a State Interim Assessment for 9th-12th grade.

Type: Assessment

## Formative Assessments

The Measure of an Angle of a Triangle:

Students are given the measure of one interior angle of an isosceles triangle and are asked to find the measure of another interior angle.

Type: Formative Assessment

Proving the Triangle Inequality Theorem:

Students are asked to prove the Triangle Inequality Theorem.

Type: Formative Assessment

An Isosceles Trapezoid Problem:

Students are asked to explain why the sum of the lengths of the diagonals of an isosceles trapezoid is less than its perimeter.

Type: Formative Assessment

Triangles and Midpoints:

Students are asked to explain why a quadrilateral formed by drawing the midsegments of a triangle is a parallelogram and to find the perimeter of the triangle formed by the midsegments.

Type: Formative Assessment

Interior Angles of a Polygon :

Students are asked to explain why the sum of the measures of the interior angles of a convex n-gon is given by the formula (n – 2)180°.

Type: Formative Assessment

The Third Side of a Triangle:

Students are given the lengths of two sides of a triangle and asked to describe all possible lengths of the remaining side.

Type: Formative Assessment

Locating the Missing Midpoint:

Students are given a triangle in which the midpoints of two sides are shown and are asked to describe a method for locating the midpoint of the remaining side using only a straight edge and pencil.

Type: Formative Assessment

Median Concurrence Proof:

Students are asked to prove that the medians of a triangle are concurrent.

Type: Formative Assessment

Triangle Sum Proof:

Students are asked prove that the measures of the interior angles of a triangle sum to 180°.

Type: Formative Assessment

Isosceles Triangle Proof:

Students are asked to prove that the base angles of an isosceles triangle are congruent.

Type: Formative Assessment

Triangle Midsegment Proof:

Students are asked to prove that the segment joining the midpoints of two sides of a triangle is parallel to the third side of the triangle and half of its length.

Type: Formative Assessment

## Lesson Plans

Engineering Design Challenge: Exploring Structures in High School Geometry :

Students explore ideas on how civil engineers use triangles when constructing bridges. Students will apply knowledge of congruent triangles to build and test their own bridges for stability.

Type: Lesson Plan

Triangle Mid-Segment Theorem:

This lesson provides a straightforward way to show the steps and the thought process of a proof involving the Triangle Mid-Segment Theorem and algebra.

Type: Lesson Plan

Keeping Triangles in Balance: Discovering Triangle Centroid is Concurrent Medians:

In this lesson, students identify, analyze, and understand the Triangle Centroid Theorem. Students discover that the centroid is the point of concurrency for the medians of a triangle and recognize its associated usage with the center of gravity or barycenter. This set of instructional materials provides the teacher with hands-on activities using technology as well as paper-and-pencil methods.

Type: Lesson Plan

Triangles: To B or not to B?:

Students will explore triangle inequalities that exist between side lengths with physical models of segments. They will determine when a triangle can/cannot be created with given side lengths and a range of lengths that can create a triangle.

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

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

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 lead 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

Geometer Sherlock: Triangle Investigations:

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

Type: Lesson Plan

Proofs of the Pythagorean Theorem:

This lesson is intended to help you assess how well students are able to produce and evaluate geometrical proofs. In particular, this unit is intended to help you identify and assist students who have difficulties in:

• Interpreting diagrams.
• Identifying mathematical knowledge relevant to an argument.
• Linking visual and algebraic representations.
• Producing and evaluating mathematical arguments.

Type: Lesson Plan

Evaluating Statements About Length and Area:

This lesson unit is intended to help you assess how well students can understand the concepts of length and area, use the concept of area in proving why two areas are or are not equal and construct their own examples and counterexamples to help justify or refute conjectures.

Type: Lesson Plan

Intersecting Medians and the Resulting Ratios:

This lesson leads students to discover empirically that the distance from each vertex to the intersection of the medians of a triangle is two-thirds of the total length of each median.

Type: Lesson Plan

Shape It Up:

Students will draw diagonals for different polygons, separating the polygons into triangles. Using the fact that the sum of the measures of the interior angles of a triangle is 180 degrees, and the fact the angles of the triangles are used to form the angles of the polygons, students will derive the formula for finding the sum of the measures of the angles of a polygon with n sides. Students will also learn to use this formula, along with the fact that all angles of a regular polygon are congruent, to find the measures of the angles of a regular polygon.

Type: Lesson Plan

Right turn, Clyde!:

Students will develop their knowledge of perpendicular bisectors & point of concurrency of a triangle, as well as construct perpendicular bisectors through real world problem solving with a map.

Type: Lesson Plan

Halfway to the Middle!:

Students will develop their knowledge of mid-segments of a triangle, construct and provide lengths of mid-segments.

Type: Lesson Plan

Detemination 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

Finding the area of an equilateral triangle:

This problem solving task asks students to find the area of an equilateral triangle.

Seven Circles I:

This task is intended to help model a concrete situation with geometry. Placing the seven pennies in a circular pattern is a concrete and fun experiment which leads to a genuine mathematical question: does the physical model with pennies give insight into what happens with seven circles in the plane?

## Tutorials

Sum of Exterior Angles of an Irregular Pentagon:

In this video, students will learn how to use what they know about the sum of angles in a triangle to determine the sum of the exterior angles of an irregular pentagon.

Type: Tutorial

Proof: Sum of Measures of Angles in a Triangle Are 180:

Lets prove that the sum of interior angles of a triangle are equal to 180 degrees.

Type: Tutorial

Triangle Angle Example 1:

Let's find the measure of an angle, using interior and exterior angle measurements.

Type: Tutorial

## MFAS Formative Assessments

An Isosceles Trapezoid Problem:

Students are asked to explain why the sum of the lengths of the diagonals of an isosceles trapezoid is less than its perimeter.

Interior Angles of a Polygon :

Students are asked to explain why the sum of the measures of the interior angles of a convex n-gon is given by the formula (n – 2)180°.

Isosceles Triangle Proof:

Students are asked to prove that the base angles of an isosceles triangle are congruent.

Locating the Missing Midpoint:

Students are given a triangle in which the midpoints of two sides are shown and are asked to describe a method for locating the midpoint of the remaining side using only a straight edge and pencil.

Median Concurrence Proof:

Students are asked to prove that the medians of a triangle are concurrent.

Proving the Triangle Inequality Theorem:

Students are asked to prove the Triangle Inequality Theorem.

The Measure of an Angle of a Triangle:

Students are given the measure of one interior angle of an isosceles triangle and are asked to find the measure of another interior angle.

The Third Side of a Triangle:

Students are given the lengths of two sides of a triangle and asked to describe all possible lengths of the remaining side.

Triangle Midsegment Proof:

Students are asked to prove that the segment joining the midpoints of two sides of a triangle is parallel to the third side of the triangle and half of its length.

Triangle Sum Proof:

Students are asked prove that the measures of the interior angles of a triangle sum to 180°.

Triangles and Midpoints:

Students are asked to explain why a quadrilateral formed by drawing the midsegments of a triangle is a parallelogram and to find the perimeter of the triangle formed by the midsegments.

## Student Resources

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

Finding the area of an equilateral triangle:

This problem solving task asks students to find the area of an equilateral triangle.

Seven Circles I:

This task is intended to help model a concrete situation with geometry. Placing the seven pennies in a circular pattern is a concrete and fun experiment which leads to a genuine mathematical question: does the physical model with pennies give insight into what happens with seven circles in the plane?

## Tutorials

Sum of Exterior Angles of an Irregular Pentagon:

In this video, students will learn how to use what they know about the sum of angles in a triangle to determine the sum of the exterior angles of an irregular pentagon.

Type: Tutorial

Proof: Sum of Measures of Angles in a Triangle Are 180:

Lets prove that the sum of interior angles of a triangle are equal to 180 degrees.

Type: Tutorial

Triangle Angle Example 1:

Let's find the measure of an angle, using interior and exterior angle measurements.

Type: Tutorial

## Parent Resources

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