MA.912.GR.1.3

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

Clarifications

Clarification 1: Postulates, relationships and theorems include measures of interior angles of a triangle sum to 180°; measures of a set of exterior angles of a triangle sum to 360°; 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.

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.)
Grade: 912
Strand: Geometric Reasoning
Standard: Prove and apply geometric theorems to solve problems.
Date Adopted or Revised: 08/20
Status: State Board Approved

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))

Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.GR.1.AP.3: Use the relationships and theorems about triangles. Solve mathematical and/or real-world problems involving postulates, relationships and theorems of triangles.

Related Resources

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

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

Name That Triangle:

Students are asked to describe a triangle whose vertices are the endpoints of a segment and a point on the perpendicular bisector of a segment.

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

Pythagorean Theorem Proof:

Students are asked to prove the Pythagorean Theorem using similar triangles.

Type: Formative Assessment

Geometric Mean Proof:

Students are asked to prove that the length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse.

Type: Formative Assessment

Converse of the Triangle Proportionality Theorem:

Students are asked to prove that if a line intersecting two sides of a triangle divides those two sides proportionally, then that line is parallel to the third side.

Type: Formative Assessment

Triangle Proportionality Theorem:

Students are asked to prove that a line parallel to one side of a triangle divides the other two sides of the triangle proportionally.

Type: Formative Assessment

Justifying the Triangle Sum Theorem:

Students are asked to provide an informal justification of the Triangle Sum Theorem.

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

Lesson Plans

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

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

Location, Location, Location, Location?:

Students will use their knowledge of graphing concurrent segments in triangles to locate and identify which points of concurrency are associated by location with cities and counties within the Texas Triangle Mega-region.

Type: Lesson Plan

Original Student Tutorial

Proving Theorems About Triangles:

Use properties, postulates, and theorems to prove a theorem about a triangle. In this interactive tutorial, you'll also learn how to prove that a line parallel to one side of a triangle divides the other two proportionally. 

Type: Original Student 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.

Converse of the Triangle Proportionality Theorem:

Students are asked to prove that if a line intersecting two sides of a triangle divides those two sides proportionally, then that line is parallel to the third side.

Geometric Mean Proof:

Students are asked to prove that the length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse.

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.

Justifying the Triangle Sum Theorem:

Students are asked to provide an informal justification of the Triangle Sum Theorem.

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.

Name That Triangle:

Students are asked to describe a triangle whose vertices are the endpoints of a segment and a point on the perpendicular bisector of a segment.

Proving the Triangle Inequality Theorem:

Students are asked to prove the Triangle Inequality Theorem.

Pythagorean Theorem Proof:

Students are asked to prove the Pythagorean Theorem using similar triangles.

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 Proportionality Theorem:

Students are asked to prove that a line parallel to one side of a triangle divides the other two sides of the triangle proportionally.

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.

Original Student Tutorials Mathematics - Grades 9-12

Proving Theorems About Triangles:

Use properties, postulates, and theorems to prove a theorem about a triangle. In this interactive tutorial, you'll also learn how to prove that a line parallel to one side of a triangle divides the other two proportionally. 

Student Resources

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

Original Student Tutorial

Proving Theorems About Triangles:

Use properties, postulates, and theorems to prove a theorem about a triangle. In this interactive tutorial, you'll also learn how to prove that a line parallel to one side of a triangle divides the other two proportionally. 

Type: Original Student Tutorial

Parent Resources

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