# Circles, Angles, Triangles, and Transformations

## Circles

**Circle**: a set of points in a plane that are the same distance from a point called the center**Chord**:A chord may or may not go through the center of a circle

**Diameter**:Longest chord of a circle

**Radius**: a line segment joining the center of a circle to any point on the circleTwo radii (plural of radius) end-to-end form a diameter

The length of the diameter is twice the length of the radius

**Circumference**: the distance around or “perimeter” of a circleCircumference approximately equals 3 * diameter or 6 * radius

### Example

If the radius of a circle is 5 inches, how long is the diameter?

The diameter of a circle is twice the diameter, so 2 * 5 inches = 10 inches.

## Angles

An angle is made up of two rays that share a vertex. Intersecting lines and line segments can also form angles.

Angles are measured in degrees (°). An angle that is open wider has a greater number of degrees. (For example, in the picture, the angle on the right has a greater number of degrees.)

A degree is 1/360 of a rotation of a full circle. There are 360° in a circle.

Types of angles: right, acute, obtuse, straight

A right angle measures exactly 90°.

An acute angle measures greater than 0° but less than 90°.

An obtuse angle measures greater than 90° but less than 180°.

A straight angle measures exactly 180°.

You should be able to roughly tell the different types of angles just by looking at them.

### Example

Angle C is the sum of angles A and B. Angle A is 72° and angle C is 162°. What is the measure of angle B and what type of angle is it?

We can write an equation for this problem.

A + B = C

Then we can plug in values.

72° + B = 162°

B = 90°

Since the measure of angle B is equal to 90°, it is a right angle.

## Triangles

**Congruent**: this word means that two things are the same; for example, two side lengths, two angles, two shapes

Congruent sides are marked with the same number of hatch marks on each congruent side.

**IM****PORTANT:**

The sum of the interior angles of a triangle always adds up to 180°.

**Classification of triangles:**

- By angles:

A right triangle has one right angle

An obtuse triangle has one obtuse angle

An acute triangle has three acute angles

- By side length:

A scalene triangle has no congruent sides

An isosceles triangle has at least two congruent sides

An equilateral triangle is a type of isosceles triangle that has 3 congruent sides and angles, and each angle measures 60°

### Example

Categorize the triangle on the left.

By sides: It is an isosceles triangle because there are two congruent sides (we know because of the hatch marks).

By angles: All the angles are less than 90°, so it is an acute triangle.

The triangle is an acute isosceles triangle.

### Example 2

What is the measure of angle C in the triangle on the right?

We know that the sum of the interior angles of a triangle is 180°. Since we know two of the angles, we can write an equation.

30° + 37° + C = 180°, so C = 113°.

## Transformations

A transformation of a figure changes the size, shape, or position of the figure to a new figure.

Congruent figures have the same size and shape and the orientation of shapes doesn't affect whether or not they're congruent.

**Types of transformations:**

A

**translation**(shift) is a transformation in which every point on the shape is moved the same distance in the same direction.A

**reflection**(flip) is a transformation in which the shape is reflected over the line of reflection. All corresponding points in the original and new images are the same distance from the line of reflection.A

**rotation**(spin) is a transformation in which the shape is rotated about a point called the center of rotation. The center of rotation may or may not be on the original image.

The resulting figure of a translation, reflection, or rotation is always congruent to the original figure.

## Quiz

## Sources Used and Helpful Links

https://www.khanacademy.org/math/basic-geo/basic-geo-angle/angle-intro/a/angle-basics-review?modal=1

https://www.aplustopper.com/different-types-of-angles/

https://www.onlinemathlearning.com/types-of-triangles.html

https://www.tes.com/lessons/u-U_Nfhwv3RsTA/transformations-translations-reflections-rotations

https://www.wyzant.com/resources/lessons/math/geometry/lines_and_angles/introduction_to_angles

http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/2016/cf/grade5math-cf.pdf