#### Geometry

Rapidly teaches geometrical concepts through custom-designed problems that are understandable to students who have not yet taken a proof-based high school geometry course. Covers geometry which appears in MATHCOUNTS, AMC 8, and AMC 10.

If you haven't yet seen our revolutionary teaching method, visit our home page first, to **shatter your expectations** of what online math looks like.

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### Who Is This Course For?

Are you taking any math class from 6th Grade Math through Algebra, and already know the content in our Introduction class? If so, then you are already ready to understand this course. It will efficiently teach you how to think creatively about Geometry, and solve problems beyond the curriculum.

Diagnostic Tests:

### Course Structure

This course contains 20 LIVE 1-hour video sessions, of which 16 are lessons, and 4 are exam review sessions that discuss the course's 4 practice 1-hour exams (homework). Each class is co-taught by 2 of our Stars. In total, there are over 150 problems.

Each lesson contains several problems which are meant to be unfamiliar to you, so that they inspire you to invent your own solutions. In our LIVE video class format, you propose your ideas, and we brainstorm together with you to construct a variety of alternative solutions using the class's own ideas. This actively involves you in discovering mathematics.

Each practice exam contains 20 problems, of which half are of average difficulty, and the other half are very challenging.

### Reviews

This is really helpful. I used to suck at geometry and had little to no interest in it. After I used this course, I had drastic improvements in Geometry. definitely would recommend

Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!

Geometry Tools is a great course if you want to learn more about Geometry. Professor Po Shen Loh teaches a lot of great tricks that you can use to solve some difficult problems. I have learned a lot of tricks and tips, including how to square a number, that ends in 5, in just a few seconds. After I completed only 15% of the course, I learned a lot of material that I had not learned in school. I improved greatly on most of my geometry tests. I am very happy about my improvement in Geometry and plan to take the Module 3: Combinatorics Tools and Module 4: Algebra Tools. Thank you, Professor Po Shen Loh, for taking your time to make videos to try to help younger people, and I hope you make more courses!

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.

I really like the seminar, it is interesting and challenging; and hard of course. Dr. Po did a great job explaining the question using different approaches.

muy bien

This is really helpful. I used to suck at geometry and had little to no interest in it. After I used this course, I had drastic improvements in Geometry. definitely would recommend

Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!

Geometry Tools is a great course if you want to learn more about Geometry. Professor Po Shen Loh teaches a lot of great tricks that you can use to solve some difficult problems. I have learned a lot of tricks and tips, including how to square a number, that ends in 5, in just a few seconds. After I completed only 15% of the course, I learned a lot of material that I had not learned in school. I improved greatly on most of my geometry tests. I am very happy about my improvement in Geometry and plan to take the Module 3: Combinatorics Tools and Module 4: Algebra Tools. Thank you, Professor Po Shen Loh, for taking your time to make videos to try to help younger people, and I hope you make more courses!

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.

I really like the seminar, it is interesting and challenging; and hard of course. Dr. Po did a great job explaining the question using different approaches.

muy bien

This is really helpful. I used to suck at geometry and had little to no interest in it. After I used this course, I had drastic improvements in Geometry. definitely would recommend

Before I took this course, geometry was a part of math that I was okay at, but didn't consider one of my top aspects of mathematics. As I took this course, I began to discover the other side of geometry, and how it was in fact a very important part of math. Now, I consider geometry as one of my favorite parts of math to work with, thanks to this course!

Geometry Tools is a great course if you want to learn more about Geometry. Professor Po Shen Loh teaches a lot of great tricks that you can use to solve some difficult problems. I have learned a lot of tricks and tips, including how to square a number, that ends in 5, in just a few seconds. After I completed only 15% of the course, I learned a lot of material that I had not learned in school. I improved greatly on most of my geometry tests. I am very happy about my improvement in Geometry and plan to take the Module 3: Combinatorics Tools and Module 4: Algebra Tools. Thank you, Professor Po Shen Loh, for taking your time to make videos to try to help younger people, and I hope you make more courses!

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

This is a very fun (but hard!) course. I like how the problems are given to introduce new concepts and the solutions are very clear, and how even though there's a lot of stuff covered, it's presented in only a few problems.

I really like the seminar, it is interesting and challenging; and hard of course. Dr. Po did a great job explaining the question using different approaches.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

This comprehensive module helped me a lot with my geometric skills. Highly recommended.

muy bien

### Syllabus

**Day 1**

Exterior and interior angles; degrees in a circle; symmetry; sum of angles in triangle; altitude and area of an equilateral triangle; Pythagorean Thm; simplifying radicals; similar triangles and area scaling; 30°-60°-90° triangles; trick for squaring numbers ending in 5

**Day 2**

Regular octagons; right isosceles triangles; similar triangles; Pythagorean Thm; simplifying radical denominators; variable equations; polygon area; midpoints; corresponding and alternate interior angles; angle-chasing method

**Day 3**

Areas of circular sections; areas of intersections of circles; radius; equilateral triangles; inscribed angles and arcs; area of a triangle; 30°-60°-90° right triangles; area of a lune; ratios of areas of similar figures

**Day 4**

Area of parallelograms; altitude of a triangle; special triangles (3-5-7, right, obtuse); exterior angles; manipulating diagrams; 30°-60°-90° triangles; isosceles trapezoids; factoring; variable equations

**Day 5**

Right trapezoids; 8-15-17 right triangles; Pythagorean Thm; AAA similarity and scaling; corresponding angles; simplifying ratios; kite area and diagonals; angle chasing; symmetry; complementary angles

**Day 6**

Inscribed and central angles; inscribed angle thm; isosceles triangles; supplementary angles; exterior angle thm; variables; arc measure; subtended arcs; secant lines; intersecting secant angle thm; angles of intersecting chords

**Day 7**

Chords; midpoints; 8-15-17 and 3-4-5 right triangles; ratios and scaling; inscribed angles and arcs; congruency; Pythagorean Thm; similar triangles; intersecting chords thm; difference of squares; common chords; circumscribed circle on three points

**Day 8**

Cyclic quadrilaterals; arc measure; inscribed angles; degrees in a circle; exterior angle thm; supplementary angles; intersecting chord angle thm; counting; secant and tangent lines to a circle; secant-tangent product thm

**Day 9**

Concentric circles; right triangles; Pythagorean Thm; symmetry; angle between radius and tangent of a circle; congruent triangles; simplifying radicals; externally tangent circles; equilateral triangles; areas of circular sectors

**Day 10**

Angle between tangent and chord to a circle; angle between radius and tangent; intersecting chord angle thm; secant-tangent and tangent-tangent inscribed angles; angles of cyclic quadrilaterals; inscribed angle thm; proof techniques

**Day 11**

External and internal tangents to two circles; parallel lines and corresponding angles; Pythagorean Thm; 5-12-13 and 3-4-5 right triangles; similar triangles and scaling; prime numbers; manipulations of diagrams

**Day 12**

Tangent-tangent segments to a circle; Pythagorean Thm; kites; similar triangles; angle bisector; area and sums of pairs of sides of quadrilateral with inscribed circle; area of a triangle

**Day 13**

Angle bisector thm; isosceles and congruent triangles; area and altitude of a triangle; ratios and scaling; golden ratio; simplifying radical denominators; proof of angle bisector thm; inscribed circle in a triangle; estimation

**Day 14**

Reflected angles; minimizing path length; triangle inequality; opposite interior angles; congruency; transversals and corresponding angles; Pythagorean Thm; 3-D path length; manipulating 3-D diagrams; estimation

**Day 15**

Triangle inequality; counting; case analysis

**Day 16**

Special triangles; area of a triangle; Pythagorean triples; scaling; proof of Heron's formula; semiperimeter; Pythagorean Thm; polynomials; common chord to two circles; 30°-60°-90° triangle

### Enroll

We will offer the LIVE version of this course in Summer 2023.

For students who want to learn a subject sooner than a LIVE course matches their schedule, our self-paced version contains all of Prof. Loh's teaching on the subject (without live teaching). This is available on our self-paced course platform, where thousands of students have learned over the past 3 years.