# Common Core High School Geometry Teaching and Learning Resources

## Domain: SRT: Similarity, Right Triangles, and Trigonometry

• SRT-A: Understand similarity in terms of similarity transformations

• SRT-B: Prove theorems involving similarity

• SRT-C: Define trigonometric ratios and solve problems involving right triangles

• SRT-D: Apply trigonometry to general triangles

### SRT-A: Understand similarity in terms of similarity transformations

Standards:

• SRT-A.1. Verify experimentally the properties of dilations given by a center and a scale factor:
• SRT-A.1.a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
• SRT-A.1.b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
• SRT-A.2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
• SRT-A.3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Technology-enhanced investigations:

Mathwords.com: Math Dictionary: Key vocabulary for this domain.  Use with SRT-A, SRT-B, SRT-C, and SRT-D:

 transformation dilation congruence tests for triangles area of a triangle similar scale factor trig functions Law of Sines similarity tests for triangles congruent SOHCAHTOA definitions Law of Cosines

LearnZillion:

• Lesson set: Verify experimentally the properties of dilations given by a center and a scale factor: Five video lessons:  Understand that a dilation takes a line not passing through the center to a parallel line, draw an image of a dilation by using the center of dilation and scale factor, calculate the scale factor of a dilation, draw an image of dilation with a negative scale factor, solve word problems by doing dilations.  Aligns with SRT-A.1.
• Lesson set: Decide if two figures are similar and explain the meaning of triangle similarity: Four video lessons: Determine if two figures are similar by using transformations and dilations, understand that a dilation of a triangle creates a similar triangle, determine triangle similarity using properties of similarity transformations, find missing measures in similar figures.  Aligns with SRT-A.2.
• Lesson set: Use the properties of similarity transformations to establish the AA criterion: Five video lessons: Show triangle similarity using translation, by using translation and dilation, by checking for corresponding congruent angles; prove AA similarity by overlapping angles; understand why the third angles of similar triangles are congruent.  Aligns with SRT-A.3.

McDougal Littell ClassZone, Geometry, 2011: Select resources in Ch. 6 Similarity for interactive practice: Animations (Virtual manipulatives), Games/Vocabulary Flashcards, and PowerPoint lesson examples, and interactive problems within the eWorkbook. Use with SRT-A.

National Library of Virtual Manipulatives: Dilations

Southeast Comprehensive Center of SEDL: Video: Understand similarity in terms of similarity transformations: This video aligns with standard SRT-A.2.  View online or download.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives.  Note: Within the Wolfram Demonstration Project are 5 manipulatives addressing SRT-A.1, 16 manipulatives for STR-A.2, and 2 manipulatives for SRT-A.3.  Among those:

Multiple Choice:

CK-12: Dilations in the Coordinate Plane: Practice questions with review content and video.  Review content includes a Types of Transformations Study Guide, which is an overview of reflection, rotation, dilation, and translation.   Aligns with CO-A.2 and SRT-A.1.

Khan Academy: Practice questions with videos.

MathBitsNotebook: Geometry: Similarity includes lessons, then practice problems on dilations, dilations and lines, similarity (numerical problems), similarity measurements, side splitter theorem, angle bisector theorem, and similarity proofs.  Use with SRT-A.1, SRT-A.2, SRT-A.3, SRT-B.4, and SRT-B.5.

McDougal Littell ClassZone, Geometry, 2011:

MathsIsFun.com: Resizing: Dilation or resizing is defined and illustrated.  Learners then use a virtual manipulative to experiment with dilations involving a triangle, rectangle, and L-shape, selecting a scale factor and a center.  Following this experiment are multiple choice questions on resizing.  Aligns with SRT-A.1.

Constructed-response:

AlgebraLab.org: Similar Triangles: Lesson and fill in the blank practice problems, including word problems.  Be sure to show the related AlgebraLab documents for additional practice problems.

Khan Academy: Practice questions with videos.

• Dilations: Learners construct; and enter their answers based on the construction. Aligns with SRT-A.1.

Illustrative Mathematics: Geometry:

• Rhombuses: Students use properties of the rhombus and Pythagorean theorem, and prove similarity of rhombuses.  Aligns with standards SRT-A.2, SRT-B.5.
• Hopewell Geometry: Students work with the Pythagorean theorem, find angles using trig ratios, and prove similarity involving triangles.  Aligns with standards SRT-A.2, SRT-B.5, and SRT-C.6, SRT-C.7, SRT-C.8.

Mathematics Assessment Project: Standards: High School: Geometry: https://www.map.mathshell.org/stds.php?standardid=1367

Mathematics Vision Project, Secondary 2 Student Edition:

### SRT-B: Prove theorems involving similarity

Standards:

• 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.
• SRT-B.5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Technology-enhanced investigations:

Cut the Knot: Pythagorean Theorem and its Many Proofs: Written proofs.

GeoGebra Book of Proofs Without Words for the Pythagorean Theorem by Steve Phelps is a series of manipulatives for exploration.

Proofs of Pythagoras' Theorem Using Similarity: Aligns with SRT-B.4.

McDougal Littell ClassZone, Geometry, 2011: Select resources in Ch. 6 Similarity for interactive practice: Animations (Virtual manipulatives, including for proportionality), Games/Vocabulary Flashcards, and PowerPoint lesson examples, and interactive problems within the eWorkbook. Use with SRT-B.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives.  Note: Within the Wolfram Demonstration Project are 10 manipulatives addressing SRT-B.4, 1 manipulative for STR-B.5.  Among those:

• Angle Bisector Theorem: Aligns with SRT-B.4.
• Euler line: This line passes through the incenter, circumcenter, and orthocenter of a triangle.  Aligns with SRT-B.4.
• The ratio theorem: Uses a parallel line to one side of a triangle that intersects the other two sides.  Aligns with SRT-B.4, SRT-B.5.
• How tall is that building?  Uses similar triangles to find the height of a building.  Aligns with SRT-B.4, SRT-C.6.
• Einstein's most excellent proof: "a new proof of the Pythagorean theorem based on the properties of similar triangles."  Aligns with SRT-A.2, SRT-B.4.

Multiple Choice:

Khan Academy: Practice questions with videos.

MathBitsNotebook: Geometry:

• Similarity includes lessons, then practice problems on dilations, dilations and lines, similarity (numerical problems), similarity measurements, side splitter theorem, angle bisector theorem, and similarity proofs.  Use with SRT-A.1, SRT-A.2, SRT-A.3, SRT-B.4, and SRT-B.5.
• Right Triangles includes lessons, then practice problems on the geometric mean in right triangles, Pythagorean theorem, and special right triangles.  Use with SRT-B.4, SRT-B.5, SRT-C.6, and SRT-C.8.

McDougal Littell ClassZone, Geometry, 2011: Ch. 6.6 Quiz: Use Proportionality Theorems

Constructed-response:

Khan Academy: Practice questions with videos.

Illustrative Mathematics: Geometry:

• Rhombuses: Students use properties of the rhombus and Pythagorean theorem, and prove similarity of rhombuses.  Aligns with standards SRT-A.2, SRT-B.5.
• Hopewell Geometry: Students work with the Pythagorean theorem, find angles using trig ratios, and prove similarity involving triangles.  Aligns with standards SRT-A.2, SRT-B.5, and SRT-C.6, SRT­-C.7, SRT-C.8.

Mathematics Assessment Project: Standards: High School: Geometry: https://www.map.mathshell.org/stds.php?standardid=1367

Mathematics Vision Project, Secondary 2 Student Edition:

NCTM's Reasoning and Sense Making Task Library: Bank Shot includes the task overview, teacher notes for its use in the classroom, and student activity sheet.  Aligns with CO-B.6, CO-C.9, SRT-B.5, and mathematical practice standards 1, 2, 3, and 7.

### SRT-C: Define trigonometric ratios and solve problems involving right triangles

Standards:

• 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.
• SRT-C.7. Explain and use the relationship between the sine and cosine of complementary angles.
• SRT-C.8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Technology-enhanced investigations:

GCSE Maths Tutor: Introducing Trigonometry: Videos 9-13 on Pythagorean theorem, sine, cosine, and tangent ratios for the right triangle, which align with SRT-C:

• #9 - Pythagoras'Theorem - Part 1:  "This video looks at the Pythagoras' Theorem. An appreciation of the theorem is given graphically and algebraically. Following is a formal proof, explained in some detail. Before problems are attempted, a little time is spent on 'rounding to decimal places'. The video ends with three worked examples on the Pythagoras Theorem."
• #10 - Pythagoras'Theorem - Part 2: "This video is the second of two about Pythagoras' Theorem. Another proof is described. This one uses similar triangles.
The concept of 'triples' (right-angled triangles with integral number sides) is explored.
• #11 - The Sine Ratio: "The Sine Ratio is defined and its limits explored. The sine values of common angles 30, 45 and 60 degrees are described The tutorial includes worked examples of three common problem types."
• #12 - The Cosine Ratio: "The cosine is defined and its limits explored. The values of common angles 30, 45 and 60 degrees are described.  The tutorial includes worked examples of three common problem types."
• #13 - The Tangent Ratio: "The tangent ratio is defined and its limits explored. The values of common angles 30, 45 and 60 degrees are described.  The tutorial includes worked examples of three common problem types."

LearnZillion:

McDougal Littell ClassZone, Geometry, 2011: Select resources in Ch. 7 Right Triangles and Trigonometry for interactive practice: Animations (Virtual manipulatives), Games/Vocabulary Flashcards, and PowerPoint lesson examples, and interactive problems within the eWorkbook. Use with SRT-C.

MIT BLOSSOMS: Video lesson with additional teacher and learner resources.  Description is from the video summary.  The Pythagorean Theorem: Geometry's Most Elegant Theorem: "This lesson teaches students about the history of the Pythagorean theorem, along with proofs and applications."  Aligns with CO-A.1 and SRT-C.6, SRT-C.8.

National Library of Virtual Manipulatives: Right Triangle Solver: Practice using Pythagorean Theorem and definitions of the trig functions to find sides and angles in right triangles.

Ohio Resource Center on YouTube: Tutorials for High School Mathematics: Video 22: Right Triangle Trig.  A review of trig definitions and applying trig in practical problems.

Wisc-Online, Learning Object: Evaluating Trigonometric Ratios: This is an interactive activity in which students evaluate the sine, cosine, and tangent ratios for a right triangle.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives.  Note: Within the Wolfram Demonstration Project are 3 manipulatives addressing SRT-C.6, 1 manipulative for STR-C.7, and 7 manipulatives for STR-C.8.  Among those:

Multiple Choice:

CK-12: Geometry: Video examples, practice problems, and real world applications.  The following align with SRT-C.8:

MathBitsNotebook: Geometry:

• Right Triangles includes lessons, then practice problems on the geometric mean in right triangles, Pythagorean theorem, and special right triangles.  Use with SRT-B.4, SRT-B.5, SRT-C.6, and SRT-C.8.
• Trigonometry: Right Triangles includes lessons, then practice problems on trigonometry-numerical (sides and angles), sine and cosine of complementary angles, word problems, and problems "needing" trigonometry.  Use with SRT-C.6, SRT-C.7, and SRT-C.8.

McDougal Littell ClassZone, Geometry, 2011: Use with SRT-C.8:

Constructed-response:

AlgebraLab.org:  Lesson and fill in the blank practice problems.

Khan Academy: Practice questions with videos.

Mathguide.com: Pythagorean theorem quizmaster.  The following are fill in and check answers, which are rounded to the nearest tenth.  The formula is presented with the following:

Illustrative Mathematics: Geometry:

Inside Mathematics: MARS Task: Hopewell Geometry: Students work with the Pythagorean theorem, find angles using trig ratios, and prove similarity involving triangles.  This task aligns with standards SRT-A.2, SRT-B.5, and SRT-C.6, SRT­C.7, SRT-C.8.

Mathematics Assessment Project: Standards: High School: Geometry: https://www.map.mathshell.org/stds.php?standardid=1367  Choose from the following:

Mathematics Vision Project, Secondary 2 Student Edition:

### SRT-D: Apply trigonometry to general triangles

Standards:

• SRT-D.9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
• SRT-D.10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.
• 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)

Technology-enhanced investigations:

Dave's Short Trig Course:

GCSE Maths Tutor: Introducing Trigonometry: Videos 14-16 align with SRT-D:

• #14 The Sine Rule: The Law of Sines is described and the tutorial includes worked examples of two common problem types.
• #15 The Cosine Rule: The Law of Cosines is described and the tutorial includes worked examples of two common problem types.
• #16- The Area of a Triangle: "Two different methods for calculating the area of a triangle are explained.
The video includes worked examples of two common problem types."

McDougal Littell ClassZone, Algebra 2, 2011: Select resources in Ch. 13 Trigonometric Ratios and Functions related to Law of Sines and Law of Cosines for interactive practice: Animations (Virtual manipulatives), Games/Vocabulary Flashcards, and PowerPoint lesson examples, and interactive problems within the eWorkbook. SRT-D.9, SRT-D.10, and SRT-D.11.

National Library of Virtual Manipulatives: Triangle Solver: Practice using Law of Sines and Law of Cosines.

NCTM Illuminations: Law of Sines and Law of Cosines: This unit contains two lessons.  Learners use right triangle trigonometry to develop the Law of Sines.  They use right triangle trigonometry and the Pythagorean theorem to develop the Law of Cosines. [Access for NCTM members.]

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives.  Note: Within the Wolfram Demonstration Project are 4 manipulatives addressing SRT-D.10, and 5 manipulatives for STR-D.11.  Among those:

Multiple Choice:

MathBitsNotebook: Geometry: Trigonometry: Oblique Triangles includes lessons, then practice problems on areas of triangles using trigonometry, law of sines, law of cosines, navigation and vectors.  Use with SRT-D.9, SRT-D.10, and SRT-D.11.

MathsIsFun.com: Tutorials on the Law of Sines and Law of Cosines.  Examples are shown.  Learners then complete 10 multiple choice problems on each law.  Aligns with SRT-D.10 and SRT-D.11.

McDougal Littell ClassZone, Algebra 2, 2011:

Constructed-response:

Khan Academy: Practice questions with videos.