Euclidean Geometry Review

 

Angles an Intersecting Lines

 

Supplementary Angles – add to 180 degrees

 

 

 

 


Complementary Angles – add to 90 degrees

 

 

 

 

 


Angles and Triangles

 

Equilateral Triangle

 

 

 

 

 

 


All sides have equal measure

 

Isosceles Triangle

 

 

 

 

 

 

Exactly two sides have equal measure

Scalene Triangle

 

 

 

 

 

 


All sides have different measures

Acute Triangle

 

 

 

 

 

 

 


All angles have a measure less than 90 degrees

 

Right Triangle

 

 

 

 

 

 

 

Triangle containing one right angle (90 degree angle)

Obtuse Triangle

 

 

 

 

 

 

 

Triangle containing one obtuse (greatger than 90 degree) angle

 

Note: The sum of the angles of any triangle is 180 degrees.

 

Angles and Parallel Lines

 

 

 

 

 

 

 

 

 

 

 


Alternate Interior Angles

Have equal measure

Examples:

C & F, D & E

Corresponding Angles

Have equal measure

A & E, C & G, B & F, D & H

Co-interior Angles (Same-side)

Are supplementary.

C & E, D & F

 

Pythagorean Theorem

 

 Where c is the side opposite the right angle (hypotenuse)

 

 

 

 


Congruent Triangles

 

Triangles, which are congruent, have three corresponding sides and three corresponding angles, which are equal in measure.

 

 

 

To prove triangles are congruent we may use one of three congruence conditions:

 

  1. SAS – Side Angle Side (where the angle is contained (between) the sides)
  2. SSS – Side Side Side
  3. ASA – Angle Side Angle (where the side is contained (between) the angles)

 

Angles in a Circle

 

Central Angle – Angle BOC is a central angle subtended by the arc BC

(Note: O is the center of the circle)

 

 

Inscribed Angle – Angle BAC is an inscribed angle subtended by the arc BC

 

 

 

 

 

A central angle is twice as large as an inscribed angle when subtended on the same arc.

 

Two inscribed angles subtended on the same arc are equal in measure.

 

Cyclic Quadrilaterals

 

A cyclic quadrilateral is a four-sided figure whose vertices are located on the circumference of a circle.

 

The angles, which are opposite one another in a cyclic quadrilateral, are supplementary (i.e. have a sum of 180 degrees)

 

Example: A + D = 180 degrees also C + B = 180 degrees in the figure below.

 

 

 

 

 

 

 

 

Similar Triangles

 

Two triangles are similar if the measures of their corresponding angles are equal.

 

 

 

 

 

 

 

 

 

 

 

 


since , and

 

The ratios of corresponding sides of similar triangles are equal.

 

i.e.