Geometry (& Geometry GT)
Unit 2: Triangles, Proof, and Trigonometry
Part 1: Triangle Relationships, Congruence, and Proofs
Students will use their knowledge of similarity and congruence to build an understanding of similar and congruent triangles. They will use similarity transformations to establish the AA and other Triangle Similarity Theorems and rigid motions to establish the SSS, SAS, and ASA Triangle Congruence Theorems. Students will understand that congruence is a special case of similarity, where the ratio of side lengths is 1:1. They will use multiple formats to prove various theorems about triangles throughout the unit, including those pertaining to congruence, mid-segments, and medians.
Part 2: Right Triangle Trigonometry
Students will use their knowledge of similarity of right triangles (and other triangles, in Geometry GT) to establish an understanding of the trigonometric ratios of angles in these triangles. They will understand the interrelationships between the trigonometric functions. They will use these ratios, and the Pythagorean theorem to solve right triangles, given various initial information.
In Geometry GT, students will further their exploration or triangles using the Law of Sines and the Law of Cosines. They will both solve triangles that are not right triangles and they will develop and use a general formula for the area of triangles which uses trigonometry.
Students will use their knowledge of similarity and congruence to build an understanding of similar and congruent triangles. They will use similarity transformations to establish the AA and other Triangle Similarity Theorems and rigid motions to establish the SSS, SAS, and ASA Triangle Congruence Theorems. Students will understand that congruence is a special case of similarity, where the ratio of side lengths is 1:1. They will use multiple formats to prove various theorems about triangles throughout the unit, including those pertaining to congruence, mid-segments, and medians.
Part 2: Right Triangle Trigonometry
Students will use their knowledge of similarity of right triangles (and other triangles, in Geometry GT) to establish an understanding of the trigonometric ratios of angles in these triangles. They will understand the interrelationships between the trigonometric functions. They will use these ratios, and the Pythagorean theorem to solve right triangles, given various initial information.
In Geometry GT, students will further their exploration or triangles using the Law of Sines and the Law of Cosines. They will both solve triangles that are not right triangles and they will develop and use a general formula for the area of triangles which uses trigonometry.
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What will my child learn?
Students will:
PART I: TRIANGLE RELATIONSHIPS, CONGRUENCE, AND PROOFS
Prove geometric theorems. (Note: This standard will be embedded throughout this unit.)
G.CO.C.10
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180º; 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.
Understand similarity in terms of similarity transformations.
G.SRT.A.3
Use properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Prove theorems involving similarity.
G.SRT.B.5
Use similarity criteria for triangles to solve problems and to prove relationships in geometric figures (SAT® Content - ATM.03).
Prove geometric theorems. (Note: This standard is embedded throughout this unit.)
G.CO.C.10
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180º.
Prove theorems involving similarity.
G.SRT.B.4
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
Use coordinates to prove simple geometric theorems algebraically.
G.GPE.B.6
Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
Prove geometric theorems. (Note: This standard is embedded throughout this unit.)
G.CO.C.10
Prove theorems about triangles. Theorems include: the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length.
Make geometric constructions.
G.CO.D.12
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.); bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment.
Prove geometric theorems. (Note: This standard will be embedded throughout this unit.)
G.CO.C.10
Prove theorems about triangles. Theorems include: the medians of a triangle meet at a point.
Understand congruence in terms of rigid motions.
G.CO.B.7
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
G.CO.B.8
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
Prove theorems involving similarity.
G.SRT.B.5
Use congruence criteria for triangles to solve problems and to prove relationships in geometric figures.
Prove geometric theorems.
G.CO.C.9
Prove theorems about lines and angles. Theorems include: points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints (SAT® Content - ATM.03).
Prove geometric theorems. (Note: This standard is embedded throughout this unit.)
G.CO.C.10
Prove theorems about triangles. Theorems include: base angles of isosceles triangles are congruent.
Prove theorems involving similarity.
G.SRT.B.5
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Prove geometric theorems.
G.CO.C.11
Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. (Note: This standard will be revisited in Unit 2. In this unit, the focus is on verifying relationships in the coordinate plane.)
PART II: RIGHT TRIANGLE TRIGONOMETRY
Prove theorems involving similarity.
G.SRT.B.4
Prove theorems about triangles. Theorems include: Pythagorean Theorem proved using triangle similarity.
Define trigonometric ratios and solve problems involving right triangles.
G.SRT.C.6
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
G.SRT.C.7
Explain and use the relationship between the sine and cosine of complementary angles (SAT® Content - ATM.02 | ATM.04).
G.SRT.C.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangle in applied problems (SAT® Content - ATM.02).
Apply trigonometry to general triangles. (Geometry GT [+])
G.SRT.D.10 (+)
Prove the Laws of Sines and Cosines and use them to solve problems.
G.SRT.D.11 (+)
Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
G.SRT.D.9 (+)
Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
Students will:
PART I: TRIANGLE RELATIONSHIPS, CONGRUENCE, AND PROOFS
Prove geometric theorems. (Note: This standard will be embedded throughout this unit.)
G.CO.C.10
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180º; 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.
- Background Info.
- Enrichment Tasks: Sum of Angles in a Triangle | Midpoints of Triangle Sides
Understand similarity in terms of similarity transformations.
G.SRT.A.3
Use properties of similarity transformations to establish the AA criterion for two triangles to be similar.
- Background Info.
- Check for Understanding: Defining Similarity through Angle-Preserving Transformations | Similar Triangles 1
- Enrichment Tasks: Similar Triangles
Prove theorems involving similarity.
G.SRT.B.5
Use similarity criteria for triangles to solve problems and to prove relationships in geometric figures (SAT® Content - ATM.03).
- Background Info.
- Check for Understanding: Solving Problems with Similar and Congruent Triangles | Solving Similar Triangles 2
- Enrichment Tasks: Tangent Line to Two Circles | Folding a Square into Thirds
Prove geometric theorems. (Note: This standard is embedded throughout this unit.)
G.CO.C.10
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180º.
- Background Info.
- Enrichment Tasks: Sum of Angles in a Triangle | Midpoints of Triangle Sides
Prove theorems involving similarity.
G.SRT.B.4
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
- Enrichment Tasks: Pythagorean Theorem | Joining Two Midpoints of a Triangle
Use coordinates to prove simple geometric theorems algebraically.
G.GPE.B.6
Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
- Check for Understanding: Dividing Line Segments | Midpoint Formula
- Enrichment Tasks: Scaling a Triangle in the Coordinate Plane | Finding Triangle Coordinates
Prove geometric theorems. (Note: This standard is embedded throughout this unit.)
G.CO.C.10
Prove theorems about triangles. Theorems include: the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length.
- Background Info.
- Enrichment Tasks: Sum of Angles in a Triangle | Midpoints of Triangle Sides
Make geometric constructions.
G.CO.D.12
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.); bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment.
- Check for Understanding: Compass Constructions 1
- Enrichment Tasks: Angle Bisection and Midpoints of Line Segments | Locating Warehouse
Prove geometric theorems. (Note: This standard will be embedded throughout this unit.)
G.CO.C.10
Prove theorems about triangles. Theorems include: the medians of a triangle meet at a point.
- Background Info.
- Enrichment Tasks: Sum of Angles in a Triangle | Midpoints of Triangle Sides
Understand congruence in terms of rigid motions.
G.CO.B.7
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
- Check for Understanding: Congruency Postulates | Defining Congruence through Rigid Transformations
- Enrichment Tasks: Properties of Congruent Triangles
G.CO.B.8
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
- Check for Understanding: Congruency Postulates
- Enrichment Tasks: SSS Congruence Criterion | Why Does ASA Work?
Prove theorems involving similarity.
G.SRT.B.5
Use congruence criteria for triangles to solve problems and to prove relationships in geometric figures.
- Background Info.
- Check for Understanding: Solving Problems with Similar and Congruent Triangles | Solving Similar Triangles 2
- Enrichment Tasks: Tangent Line to Two Circles | Folding a Square into Thirds
Prove geometric theorems.
G.CO.C.9
Prove theorems about lines and angles. Theorems include: points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints (SAT® Content - ATM.03).
- Check for Understanding: Line and Angle Proofs
- Enrichment Tasks: Points Equidistant From Two Points in the Plane | Congruent Angles Made By Parallel Lines and a Transverse
Prove geometric theorems. (Note: This standard is embedded throughout this unit.)
G.CO.C.10
Prove theorems about triangles. Theorems include: base angles of isosceles triangles are congruent.
- Background Info.
- Enrichment Tasks: Sum of Angles in a Triangle | Midpoints of Triangle Sides
Prove theorems involving similarity.
G.SRT.B.5
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
- Background Info.
- Check for Understanding: Solving Problems with Similar and Congruent Triangles | Solving Similar Triangles 2
- Enrichment Tasks: Tangent Line to Two Circles | Folding a Square into Thirds
Prove geometric theorems.
G.CO.C.11
Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. (Note: This standard will be revisited in Unit 2. In this unit, the focus is on verifying relationships in the coordinate plane.)
- Background Info.
- Enrichment Tasks: Midpoints of the Sides of a Parallelogram | Parallelograms and Translations
PART II: RIGHT TRIANGLE TRIGONOMETRY
Prove theorems involving similarity.
G.SRT.B.4
Prove theorems about triangles. Theorems include: Pythagorean Theorem proved using triangle similarity.
- Enrichment Tasks: Pythagorean Theorem | Joining Two Midpoints of a Triangle
Define trigonometric ratios and solve problems involving right triangles.
G.SRT.C.6
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
- Background Info.
- Check for Understanding: Trigonometric Functions and Side Ratios in Right Triangles
- Enrichment Tasks: Defining Trigonometric Ratios | Tangent of Acute Angles
G.SRT.C.7
Explain and use the relationship between the sine and cosine of complementary angles (SAT® Content - ATM.02 | ATM.04).
- Background Info.
- Check for Understanding: Trigonometric Functions and Side Ratios in Right Triangles
- Enrichment Tasks: Sine and Cosine of Complementary Angles | Trigonometric Function Values
G.SRT.C.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangle in applied problems (SAT® Content - ATM.02).
- Background Info.
- Check for Understanding: Applying Right Triangles
- Enrichment Tasks: Constructing Special Angles | Ask The Pilot
Apply trigonometry to general triangles. (Geometry GT [+])
G.SRT.D.10 (+)
Prove the Laws of Sines and Cosines and use them to solve problems.
- Check for Understanding: Law of Cosines | Law of Sines
G.SRT.D.11 (+)
Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
- Check for Understanding: Law of Cosines | Law of Sines and Law of Cosines Word Problems
G.SRT.D.9 (+)
Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
- Background Info.
- Check for Understanding: Non-Right Triangle Proofs
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What are some signs of student mastery?
Part I: Triangle Relationships, Congruence, and Proofs
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Tools & Technology
GeoGebra is a dynamic math software tool that allows users to explore geometry, algebra, tables, graphing, statistics and other areas of math in one easy-to-use package. It can be downloaded for FREE on a computer or tablet device. Geometer's Sketchpad is a similar dynamic software tool that is available in HCPSS schools. View video tutorials on how to prove that:
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More 4 U
Looking for clarification on some of the vocabulary used in the Geometry course? A slight variation in units, click here to download the MD State Department of Education's (MSDE's) geometry glossary. |
