Angles and Relationships Honors Assignment
Answers:
1. Using what you have learned about straight angles and linear pairs, explain why
vertical angles are always congruent.
Vertical angles are two angles that share a vertex and are formed by two
intersecting lines. One line will act as the shared side of the linear pair formed by
the other line. Since a linear pair always has a sum of 180°, then those two
angles (called a and b) would have a sum of 180°. If you used angle b with its
adjacent angle on the other side (called c), then the two angles would still have a
sum of 180°. Since angle b did not change measurement, angle c would have to
equal angle a. Angle a and angle c share a vertex and are formed by
intersecting lines, making them vertical angles.
2. Explain why a linear pair is always considered a pair of supplementary angles.
A linear pair is two angles that have a common side and their non-common sides
are opposite rays. Opposite rays are two rays with the same starting point that
extend in opposite directions, creating a straight line, which would be a straight
angle. A straight angle always has a measure of 180°, so a linear pair would
always have a sum of 180°. Supplementary angles are a pair of angles that add
up to 180°.
3. Explain why there are always four right angles when two lines are perpendicular.
When two intersecting lines are perpendicular, they create four angles. It is
given that at least one of those angles is a right angle. That right angle would be
vertical to another angle of the four, giving two right angles. The first right angle
would form a linear pair with another angle. Since a linear pair always equals
180°, then the adjacent angle would also be a right angle. The last angle would
be vertical to the adjacent angle, making it a right angle as well.
4. Give a reason for why measuring tools are not used in constructions. Be sure to
back up your reason with mathematical facts.
Constructions are used to create shapes, angles, lines, etc. in geometry. By not
using measuring tools, you have to truly understand the concept and properties
behind each object in order to construct it properly. If measuring tools were
allowed, you would just be measuring out the object instead of truly constructing
it from a blank slate.