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Theorems about Congruence in Triangles

Theorems related to circumcenters and other intersection points of three or more lines.

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Lines and Angles Vocabulary and Definitions
 Feel free to "make a copy" or print out this studyguide to complete the excercises and annotate where it helps you.

Vocabulary: Lines and Angles

Line: A line is 180 degrees, equivalent to all of the degrees in a triangle summed up.  It extends indefinitely and has infinite points.  Because it is 180 degrees, it has no curves and also has no volume. There are three main forms for graphing a line, slope intercept form, point slope form, and standard form.

Click here to see an image of a line.

Line segment: A line segment has definite endpoints. It also is straight and is 180 degrees but has restrictions on its ends.  It doesn't have arrows, but rather has dots to represent these restrictions.

Click here to see an image of a line segment.

Midpoint: The middle pointon a line segment (a line is indefinite).  This means that the distances on both sides of the midpoint are equidistant.
Point: a point is an exact location - it has no size, only a position.
Collinear: when 3 or morepoints are passing through the same line
Note: a set of all the points collinear would be the equation for that line (in most cases)

Concurrent: when 3 or more lines meet at a single point (example: triangle)

Can you think of any other concurrent shapes?

Ray: a ray is a line with one "anchor" point and one end that continues endlessly.  In other words, one half of a ray looks like a line segment and the other end looks like a line.
Tip: If it helps, think of a ray like the ray of the sun.  The ray starts specifically at the sun itself, but its stream of light continues indefinitely. 

Click here to see an image of a ray.

Angle: the amount of "turn" between two lines that intersect at a common vertex, measured in degrees.

Parallel: When two lines have the same slope, lie on the same plane and never intersect

Click here to see an image of parallel lines.

Complementary Angles: Angles that add up to 90degrees, form right angle

Supplementary Angles: Angles that add up to 180 degrees, form straight line

Click here to see an image of supplementary angles.

Tangent (Lines): a line that runs through a point on the edge of a circle but never intersects the whole circle and is perpendicular to the radius

Vertex: sometimes known as a corner, the vertex is the common endpoint of two or more rays or line segments (plural = vertices)

Transversal: a line that intersects at least two other lines

Adjacent angles: Adjacent angles are angles that are next to each other. They have a common side, a common vertex, and don't overlap.

Click here to see an image of adjacent angles.

Check Your Understanding: Fill in the Blanks

Try completing the fill in the blanks without referencing back at the definitions. If you are having trouble with a specific problem, then review the guide above, but try your best to see what you remember using just your brain.

  1. A _____ extends indefinitely.
  2. A line segment has ____ on each of its ends.
  3. If point B is the midpoint on line segment AC, then segments AB and BC are _____.
  4. The set of all points collinear would be the equation for the specific line because ______.
  5. An example of a concurrent shape is a ______.
  6. An angle is measured in _____.
  7. Parallel lines have the same _____. They _____ intersect.
  8. These angles are ______.
  9. Angles that form a _______ are supplementary.
  10. ____ lines are always perpendicular to the radius.
  11. The common endpoint of two or more rays is the ______.
  12. A transversal is a line that _____ two or more lines.
  13. Adjacent angles have a common ____ and          .

Fill in the Blanks: Answers

  1. line
  2. restrictions
  3. equidistant/congruent
  4. (because) it would contain at least three points on the same line
  5. triangle
  6. degrees
  7. slope, never
  8. complementary
  9. straight line
  10. Tangent
  11. vertex
  12. intersects
  13. side, vertex

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