Circles, Angles, Triangles, and Transformations


  • Circle: a set of points in a plane that are the same distance from a point called the center

  • Chord: a line segment connecting any two points on a circle

    • A chord may or may not go through the center of a circle

  • Diameter: a chord that goes through the center of a circle

    • Longest chord of a circle

  • Radius: a line segment joining the center of a circle to any point on the circle

    • Two 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 circle

    • Circumference approximately equals 3 * diameter or 6 * radius


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.


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.


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.


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.


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°


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°.


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.


Sources Used and Helpful Links