Download Name: ______KEY EDUC 5525 Review for Geometry and

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EDUC 5525 Review for Geometry and Measurement Midterm
1. Explain the axiomatic nature of geometry.
Axioms are statements about relationships that are accepted as true without proof. They
provide a starting point allowing participants to “play the game.” Theorems are
statements that are based on axioms and require proof. Axioms cannot be broken down
like theorems can since they are the starting point, meaning nothing comes before them
but a lot comes after.
2. What is the difference between axioms and theorems? Give an example for each.
Axioms are accepted as always true, they are the starting points within a system. Since
they are the starting points, they do not need to be proven, they are accepted as true.
Theorems are statements of relationships that are based on axioms. They require proof,
once they are proven, they are accepted as always true.
3. What are some of the problems associated with studying geometry?
The unique notation that students do not have experience with.
The axiomatic nature of geometry.
A reasoning process that is often not experienced before studying geometry.
Geometry is not as algorithmically restricted as algebra.
4. What are the undefined terms in geometry?
There are three, point, line and plane. Although we can describe these, we do not define
them. A point has no measurable attributes (no dimension) so we can only draw a
representation of a point. When we take an infinite number of points and place them
side-by-side going in two opposite directions forever, we have a line. A line is not
measureable since it has no beginning or end, but we can measure a length of a
segment of a line given two endpoints. When we place an infinite number of lines side by
side in two opposite directions forever we get a plane. A plane is not measureable since
it has no beginning or end, but we can measure a part of a plane such as a piece of
paper or a wall.
5. Define a polygon, a prism, and a plane in your own words.
A polygon is a collection of points that are coplanar. Some definitions mention a
collection of line segments that form a closed figure. In class we stated that each point in
the collection of coplanar points has the unique characteristic of being an endpoint of
exactly two line segments. A plane is two dimensional, meaning two directional. It is the
set of all points that extend in two directions forever. It is a flat surface with no thickness
extending forever in two dimensions.
6. Name and describe the classifications for triangles.
Obtuse triangle – a triangle having an angle greater than 90˚.
Acute triangle – a triangle having all three angles less than 90˚.
Right triangle – a triangle having a 90˚ angle.
7. What are the symbols for line, segment, ray, angle, parallel, perpendicular
and congruence?
8. AB and CB are tangents to circle P. Name as many facts and relationships you
can determine from this fact and the picture given.
Angles PAB and PCB = 90˚ so they are
congruent
Angles APB and CPB are congruent
Angles ABP and CBP are congruent
PA and PC are radii and are congruent
AB and CB are congruent
Triangles ABP and CBP are congruent
9. Given that the two lines that are cut by the transversal are parallel, answer the
following:
Name 3 pairs of angles that are congruent. Justify your answer. Any 3 of the
following: Ð1 @ Ð4, Ð2 @ Ð3,Ð6 @ Ð7,Ð5 @ Ð8 Vertical angles are congruent
Ð1 @ Ð5, Ð2 @ Ð6,Ð4 @ Ð8,Ð3 @ Ð7
Ð1 @ Ð8, Ð2 @ Ð7
Corresponding angles are congruent
Alternate exterior angles are congruent
Ð3 @ Ð6, Ð4 @ Ð5 Alternate interior angles are congruent
What angle corresponds to ∠7? Angle 3
What angle is vertical to ∠5? Angle 8
Are ∠8 and ∠1 congruent? Justify your answer.
Yes, they are alternate exterior angles.
If ∠2 is 64 degrees what is the measurement of ∠5?
Name an angle supplementary to ∠6.
Ð5, Ð8, Ð4, Ð1
116˚
Any of the following:
10. Explain why the sum of the measures of the three angles in any triangle is 180
degrees. Use a visual representation with appropriate mathematical symbols.
If we create a line parallel to the base of the triangle
through the opposite vertex, the two sides of the triangle
are segments of two transversals. That means angle 3 is
congruent to angle 4 (alternate interior angles) and angle
2 is congruent to angle 5 (also alternate interior angles)
and angles 5, 1 and 4 make a straight line or an angle
sum of 180˚ so the three interior angles must also equal
180˚
11. Given the triangles are similar find the height of the tree
Since the base of small triangle is 1/10
the size of the base of the big triangle,
the height of the tree is ten times bigger
than 2 or 20 m.
12. What is the formula for finding the interior angle sums of a regular polygon?
Give an example or drawing of how you can prove this.
Sum = 180(n – 2) where n = the number of sides of the polygon
13. Reflect shape A across the y-axis and list its new coordinates. Translate shape B
in a direct path from Quadrant I to Quadrant III and list its new coordinates.
A:
B:
14. Jill says the 2 shapes below have the same area. Jack says they do not. Which
student is correct and why? What do the 2 students understand or not
understand about transformations?
Jill is correct. We could cut out the two shapes and place them on top of each other to
show that the one on the right is a rotation of the one on the left. Jill understands that translations
do not change area (unless it is a dilation). Jack either does not understand that translations
keep area the same or he does not recognize that the shapes are translation of each other.
15.
Draw at least one polygon that has: The following answers are examples of infinite
number of solutions.
a) 0 lines of reflectional symmetry
b) 1 line of reflectional symmetry
c) 2 lines of reflectional symmetry
16. A student measured ∠ABC and ∠DEF and found the measure of ∠ABC to be
25º and ∠DEF to be 20º. The student however felt she measured incorrectly
and that the angle measurements must be congruent because EF and BC are
both raised equally above vertical and point to the same spot.3
a. Explain what is wrong with the student’s thinking about the
measurement of angles.
b. How would you address this misconception?
The height of points C and F may be the same, but the angles of elevation do not have to be the
same. For example, if points C and F were on the top of the same building, the distance AB is
smaller than DE which would represent two different distances from the building. A person
standing next to the building must look up at a bigger angle of elevation to see the top of the
building compared to someone several blocks away.
17. AC, CB, and BA are tangents. Find the lengths of the following: AD, AC, AB,
EB, GD, GE, CF, CB
AD = 12 cm
AB = 17cm
GD = 4 cm
CF = 8 cm
AC = 20 cm
EB = 5 cm
GE = 4 cm
CB = 13 cm