Unit 8: Angle and Triangle Relationships

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PYTHAGOREAN THEOREM

The Pythagorean Theorem states that, in a right triangle, the two smaller squares made from the triangles legs add up to equal the biggest square made from the hypotenuse.

Pythagoras

The lengths of the legs are commonly labeled as a and b and the hypotenuse length is labeled as c.

REMEMBER:

HYPOTENUSE: c is the longest side of the triangle

LEGS: a and b are the other two sides

Click here  to find the missing hypotenuse , here  to find the missing leg, here  to find the perimeter, here  to do real world problems, and here to see if the triangle is a right triangle or not. You will use the Pythagorean Theorem online and get automatic feedback (it grades it)! 🙂

What is the Pythagorean Theorem?

In short the Pythagorean Theorem can be explain with the formula below.

a2 + b2 = c2

AGAIN, THIS IS ONLY TRUE FOR RIGHT TRIANGLES!

Here are other examples:

Click here  to find the missing hypotenuse , here  to find the missing leg, here  to find the perimeter, here  to do real world problems, and here to see if the triangle is a right triangle or not. You will use the Pythagorean Theorem online and get automatic feedback (it grades it)! 🙂

ANGLE and TRIANGLE RELATIONSHIPS

Degrees

A degree is a unit of measurement used to measure angles.

We can measure all Angles in Degrees.

There are 360 degrees in one Full Rotation (one
complete circle around

The Degree Symbol: °

We use a little circle ° following the number to mean degrees.

For example 90° means 90 degrees

Angles

types of angle

In One Diagram

This diagram might make it easier to remember:

Also: Acute, Obtuse and Reflex are in alphabetical order.

Supplementary Angles

Two Angles are Supplementary if they add up to 180 degrees.
These two angles (140° and 40°) are Supplementary Angles, because they add up to 180°.Notice that together they make astraight angle.
But the angles don’t have to be together.These two are supplementary because 60° + 120° = 180°

Complementary Angles

Two angles are Complementary when they
add up to 90 degrees
(a Right Angle right angle).

These two angles (40° and 50°) are
Complementary Angles, because they add up to 90°.Notice that together they make a right angleright angle.
But the angles don’t have to be together.These two are complementary because 27° + 63° = 90°

Angle Relationships

 Click here, here, and here to find the missing angles of triangles and parallel lines cut by a transversal online and get automatic feedback (it grades it)! 🙂

Interior Angle

An Interior Angle is an angle inside a shape.

Exterior Angle

The Exterior Angle is the angle between any side of a shape, and a line extended from the next side.

Angles On One Side of A Straight Line

Angles on one side of a straight line will always add to 180 degrees.

If a line is split into 2 and you know one angle you can always find the other one.

30° + 150° = 180°

To view straight angles created from 2 parallel lines and a transversal click here.

Interior Angles of Polygons

An Interior Angle is an angle inside a shape.

Triangles

The Interior Angles of a Triangle add up to 180°

90° + 60° + 30° = 180°

80° + 70° + 30° = 180°

It works for this triangle!

Let’s tilt a line by 10° …

It still works, because one angle went up by 10°, but the other went down by 10°

Quadrilaterals (Squares, etc)

(A Quadrilateral has 4 straight sides)

90° + 90° + 90° + 90° = 360°

80° + 100° + 90° + 90° = 360°

A Square adds up to 360°

Let’s tilt a line by 10° … still adds up to 360°!

The Interior Angles of a Quadrilateral add up to 360°

Because there are Two Triangles in a Square

The interior angles in this triangle add up to 180°(90°+45°+45°=180°)
Shape Term Definition
perpendicular line segments Two line segments which cross to form 90 degree angles.
right angle A 90 degree angle.
equilateral triangle A triangle with all sides equal and all angles equal.
scalene triangle A triangle having three unequal sides and angles.
vertex The intersection point of two sides of a plane figure.
right triangle A triangle with one internal angle equal to 90 degrees.
pentagon A polygon with 5 sides and 5 angles.
square A rectangle having all four sides of equal length.
intersecting line segments Line segments that cross each other.
acute angle An angle less than 90 degrees but greater than 0 degrees.
chord The line segment between two points on a given curve.
radius A straight line extending from the center of a circle or sphere to the circumference or surface.
line segment One part of a line.
line A continuous extent of length.
point A position in space.
parallelogram A quadrilateral having both pairs of opposite sides parallel to each other.(Note: squares and rectangles are also quadrilaterals.)
rectangle A parallelogram having four right angles.(Note: a square is also a rectangle.)
rhombus An equilateral parallelogram having oblique angles.
parallel line segments Line segments that do not intersect.
quadrilateral A polygon with four sides.(Note: squares, rectangles and trapezoids are also quadrilaterals.)
octagon A polygon having eight angles and eight sides.
circle A closed plane curve consisting of all points at a given distance from a point within it called the center.
trapezoid A quadrilateral plane figure having two parallel and two nonparallel sides.
ray The part of a straight line considered as originating at a point on the line and as extending in one direction from that point.
closed curve A curve that is continuous and has endpoints that meet at the same point.
isosceles triangle A triangle which has two sides equal.
hexagon A polygon having six sides and six angles.
diameter A straight line passing through the center of a circle or sphere and meeting the circumference or surface at each end.
obtuse angle An angle greater than 90 degrees but less than 180 degrees.
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