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Math 181 Spring 2000
Informal Geometry
Solutions to exam I
Dr. A. Bart
1. Draw and label the following geometric gures:
ABC; PQR; a circle with center C and radius R, polygon MNOPQR.
Everyone got this right! The answers are mainly in chapter 2.
2. Suppose that P is parallel to Q and that R is parallel to S. Use your
knowledge of angles to nd the measures of angles 1; 2; 3 and 4. (Do
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NOT measure them! You will get the wrong answer.)
This is problem 4 on page 46. 1 = 31o ; 2 = 86o ; 3 = 63o ; 4 = 86o .
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Figure 1: Diagram for question 2
3. a. Dene what a diagonal is.
A diagonal is a segment connecting two non-consecutive (non-adjacent) vertices.
b. How many diagonal does a triangle have? Explain carefully.
Zero, all pairs of vertices are consecutive. In other words we can't connect nonadjacent vertices, because any two vertices must necessarily already be adjacent
to one another.
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4. Draw all the diagonals in the quadrilaterals below.
a. In which quadrilaterals are the diagonals congruent?
In the square, the rectangle and the isosceles trapezoid.
b. In which of the quadrilaterals are the diagonals perpendicular?
In the rhombus, the kite and the square.
Figure 2: Question 4
5. Decide whether the following statements are true or false. Carefully
explain your reasoning.
a. Every right triangle is an isosceles triangle.
FALSE. A right triangle has one right angle, but there is no reason to expect
two of the sides to be congruent. So it is not true that every right triangle is
isosceles. (Some are: 45-45-90 triangles are isosceles, but these are the only
ones!)
b. Every equilateral triangle is an isosceles triangle.
TRUE. In an equilateral triangle all sides are congruent, so if all three are
congruent, then so are two of them. So the triangle satises the requirements
for isosceles triangles.
c. Every square is a parallelogram.
TRUE. In a square opposite sides are congruent and parallel, hence the square
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is a parallelogram. d. Every square is a rhombus.
TRUE. In a square, all sides have equal length and opposite sides are parallel.
So it's a rhombus.
6. What is the relationship between rhombuses, parallelograms and
kites? Explain. Every rhombus is a parallelogram, but not every parallelogram is a rhombus. So the set of rhombuses is a subset of the set of parallelograms. Every rhombus is also a kite, but not all kites are rhombuses. And
kites and parallelograms have nothing else in common (only the rhombuses).
7. a. How many medians does a triangle have? Where do the medians
intersect?
Any triangle has 3 medians. They always intersect in the interior of the triangle:
in the centroid.
b. Draw an acute and an obtuse triangle and draw the medians for
each.
Figure 3: Medians
8. a. Dene what the altitude of a triangle is.
The altitude is a line passing from a vertex , perpendicular to the opposite side
(or extension of the opposite side). b. Draw an acute and an obtuse triangle
and draw the altitudes.
9. a. Dene what an angle bisector is.
An angle bisector is a ray which divides an angle into two congruent angles. b.
Draw an angle, and use your compass to construct the angle bisector.
See Appendix B page 336.
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Figure 4: Altitudes
10. Decide if the following statements are True (T) or False (F). No explanation
necessary.
T Three non-colinear points uniquely determine a plane.
T If three points are colinear, then they are also coplanar.
T The oor and the ceiling represent parallel planes.
T The wall is perpendicular to the ceiling.
F Any two walls are perpendicular.
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