Geometry Chapter 5 Quiz

Segment Lengths

• Midpoint of a segment divides it into two equal parts.
• Bisectors intersect a segment at its midpoint, allowing for equal partitioning.
• Formula for segment length: ( AB = AM + MB ).

Angle Measures

• Angles in a triangle sum to 180°.
• Exterior angle theorem states that an exterior angle equals the sum of the two opposite interior angles.
• Parallel lines create alternate interior angles that are congruent.

Algebraic Work

• Use algebraic expressions to represent segment lengths and angle measures.
• Solve equations using algebraic methods such as substitution and simplification.
• The relationship can be modeled with equations such as ( AM = \frac{1}{2} AB ).

Straightedge and Compass Constructions

• Basic constructions include:
  • Copying a segment: replicate the length of one segment to another.
  • Copying an angle: use a compass to replicate the angle.
  • Bisecting an angle: finding the angle's midpoint with a compass and straightedge.
  • Constructing a perpendicular bisector: find a line perpendicular to a segment at its midpoint.
  • Constructing an equilateral triangle using equal lengths for all sides.
• Applications of constructions can be found in solving complex geometric problems.

Problems

• For the problem involving midpoint ( M ):

  • Given ( AM = 4x + 5 ) and ( AB = 4x + 100 ).
  • Set up the equation ( AB = 2AM ) to find ( x ) and ( AB ).

• For segment ( CF ):

  • The relationship ( CD = DE = EF ) signifies equal segments.
  • With ( CE = 12 ) in, calculate ( CF ) by the equation ( CF = CD + DE + EF ).

• Finding missing angles:

  • Sum of angles: 30° + y + 50° = 180°; solve for y.
  • Next set: 40° + 110° + y = 180°; solve for y.
  • Final set: 50° + x + y = 180°; solve for x and y.