# Geometry

Contents

## Angles between intersecting lines

Welcome. I'd like to introduce you to Mr. Angle. Nice to meet you. So nice to meet you.
This tutorial introduces us to angles. It includes how we measure them, how angles relate to each other and properties of angles created from various types of intersecting lines. Mr. Angle is actually far more fun than you might initially presume him to be.

Angles, parallel lines, & transversals

Sal draws two parallel lines and a transversal, then introduces some new vocabulary.

Parallel & perpendicular lines

Sal identifies parallel and perpendicular lines in geometric figures.

Missing angles with a transversal

Sal draws two parallel lines and a transversal, then finds some missing angle measurements.

Parallel lines

Find missing angles given two parallel lines and a transversal.

Measures of angles formed by a transversal

Sal solves an equation to find missing angles given two parallel lines and a transversal.

Equation practice with congruent angles

## Triangle angles

Do the angles in a triangle always add up to the same thing? Would I ask it if they didn't? What do we know about the angles of a triangle if two of the sides are congruent (an isosceles triangle) or all three are congruent (an equilateral)? This tutorial is the place to find out.
Common Core Standard: 8.G.A.5

Angles in a triangle sum to 180° proof

Sal gives a formal proof that the measures of interior angles of a triangle sum to 180°.

Triangle angle example

Figuring out angles in a triangle. A little about exterior angles being the sum of the remote interior angles

Triangle angle example 2

Another example finding angles in triangles

Triangle angle example 3

Multiple ways to solve for the angles of multiple triangles

Finding angle measures 1

Finding angle measures 2

Triangle angle challenge problem

Interesting problem finding the sums of particular exterior angles of an irregular pentagon

Triangle angle challenge problem 2

Example of angle hunting!

## The Pythagorean theorem

Named after the Greek philosopher who lived nearly 2600 years ago, the Pythagorean theorem is as good as math theorems get (Pythagoras also started a religious movement). It's simple. It's beautiful. It's powerful.
Common Core Standards: 8.G.B.7, 8.G.B.8

Intro to the Pythagorean theorem 1

Sal introduces the famous and super important Pythagorean theorem!

Intro to the Pythagorean theorem 2

Sal introduces the famous and super important Pythagorean theorem!

Pythagorean theorem example

Sal uses the Pythagorean theorem to find the height of a right triangle with a base of 9 and a hypotenuse of 14.

Pythagorean theorem word problem: carpet

Sal uses the Pythagorean theorem to find the width of some carpet.

Pythagorean theorem word problem: fishing boat

Sal uses the Pythagorean theorem to solve a word problem about a fishing boat.

Pythagorean theorem

Find the leg or hypotenuse of a right triangle using the Pythagorean theorem.

Pythagorean theorem word problems

Solve real-world problems that can be modeled by right triangles, using the Pythagorean Theorem!

Pythagorean theorem in 3D

Pythagorean theorem in 3D

Distance formula

How to find the distance between lines using the Pythagorean Formula

Distance between two points

Use the Pythagorean theorem to find the distance between two points on the coordinate plane.

Thiago asks: How much time does a goalkeeper have to react to a penalty kick?

Sal uses the Pythagorean theorem to answer a question posed by a soccer superstar!

## Pythagorean theorem proofs

The Pythagorean theorem is one of the most famous ideas in all of mathematics. This tutorial proves it. Then proves it again... and again... and again. More than just satisfying any skepticism of whether the Pythagorean theorem is really true (only one proof would be sufficient for that), it will hopefully open your mind to new and beautiful ways to prove something very powerful.
Common Core Standard: 8.G.B.6

Garfield's proof of the Pythagorean theorem

James Garfield's proof of the Pythagorean Theorem.

Bhaskara's proof of the Pythagorean theorem

An elegant visual proof of the Pythagorean Theorem developed by the 12th century Indian mathematician Bhaskara.

Pythagorean theorem proof using similarity

Proof of the Pythagorean Theorem using similarity

Another Pythagorean theorem proof

Visually proving the Pythagorean Theorem

## Volume

Let's see how to find the volumes of cylinders, spheres and other three dimensional shapes.
Common Core Standard: 8.G.C.9

Cylinder volume & surface area

Finding the volume and surface area of a cylinder

Volume of a sphere

Sal introduces and uses the formula for the volume of a sphere, V=4/3πr³.

Volume of a cone

Sal introduces and explains the formula for the volume of a cone, V=1/3hπr².

Volumes of cones, cylinders, and spheres

Practice applying the volume formulas for cones, cylinders, and spheres.

## Translations

Intro to geometric transformations

Sal introduces geometric transformations! Specifically, he explains what the "image" of a transformations is, what are the "rigid" transformations, and which transformations are not rigid.

Performing translations

Sal shows how to perform a translation on a triangle using our interactive widget!

Perform translations

Use the interactive transformation tool to perform translations.

## Rotations

Intro to geometric transformations

Sal introduces geometric transformations! Specifically, he explains what the "image" of a transformations is, what are the "rigid" transformations, and which transformations are not rigid.

Performing rotations

Sal shows how to perform a rotation on a pentagon using our interactive widget!

Perform rotations

Use the interactive transformation tool to perform rotations.

## Reflections

Intro to geometric transformations

Sal introduces geometric transformations! Specifically, he explains what the "image" of a transformations is, what are the "rigid" transformations, and which transformations are not rigid.

Performing reflections

Sal shows how to perform a reflection on a quadrilateral using our interactive widget!

Perform reflections

Use the interactive transformation tool to perform reflections.

## Congruence and similarity

Congruent shapes & transformations

Sal shows that a given pair of pentagons are congruent by mapping one onto the other using rigid transformations.

Non-congruent shapes & transformations

Sal shows that a given pair of pentagons are not congruent by showing it's not possible to map one onto the other using rigid transformations.

Congruence & transformations

Given a pair of figures in the coordinate plane, try to map one onto the other and determine whether they are congruent.

Dilating shapes: shrinking

Sal is given a triangle on the coordinate plane and he draws the image of the triangle under a dilation with scale factor 1/2 about the origin.

Similar shapes & transformations

Sal is given pairs of polygons, and then he determines whether they are similar by trying to map one onto the other using angle-preserving transformations.

Similarity & transformations

Given two polygons, try to map one onto the other using angle-preserving transformations, and determine whether they are similar. Transformations are done in "intuitive mode."

Side lengths after dilation

Sal dilates a shape and then compares the side lengths of the pre-image and image.