A worksheet to help you prepare for the midterm.

For each question, circle the unique correct answer. No justification is required.
(i)
Which of the following statements is true?
(a)
If two lines are parallel, then they intersect at exactly one point.
(b)
If two lines are both perpendicular to a third line, then they are parallel.
(c)
If two lines intersect, then they are parallel.
(d)
If two lines are not parallel, then they are perpendicular.
(ii)
In the six parts of a Euclidean proposition, which part usually states what is given?
(a)
Exposition.
(b)
Specification.
(c)
Construction.
(d)
Conclusion.
(iii)
Which of the following assertions is false?
(a)
Vertical angles are equal.
(b)
The perpendicular bisector of a segment passes through its midpoint.
(c)
If two lines are parallel, then alternate interior angles are equal.
(d)
If two segments have the same length, then they lie on the same line.
This exercise contains four “Euclid-style” proof questions.
(i)
Let \(ABCD\) be a quadrilateral such that
\[ AB \parallel CD \qquad \text {and}\qquad AB=CD. \]
Prove that
\[ AD \parallel BC \qquad \text {and}\qquad AD=BC. \]
(ii)
Let \(ABCD\) be a quadrilateral such that
\[ AB=AD \qquad \text {and}\qquad CB=CD. \]
Assume that the diagonals \(AC\) and \(BD\) meet at a point \(E\). Prove that
\[ AC\perp BD. \]
(iii)
Let \(AB\) be a diameter of a circle, and let \(C\) be another point on the circle. Prove that
\[ \angle ACB \]
is a right angle.
(iv)
Let \(ABCD\) be a cyclic quadrilateral (i.e. four points on a circle) such that
\[ AB=CD. \]
Prove that
\[ AD\parallel BC. \]

Hint. Draw the diagonals \(AC\) and \(BD\), and let \(E\) be their point of intersection. First show that the triangles \(\triangle CDE\) and \(\triangle ABE\) are congruent. Then show that

\[ \angle DAE=\angle BCE, \]
and conclude.
Using only straightedge and compass, perform the following constructions and state how many moves you used. No justification is required.
(i)
Construct the midpoint of a given segment \(AB\).
(ii)
Construct a square on a given segment \(AB\).
(iii)
Construct the perpendicular to a given line \(l\) passing through a given point \(P\) not on the line.
(iv)
Construct a circle passing through three given non-collinear points \(A\), \(B\), and \(C\).
(v)
Construct a line tangent to a given circle and passing through a given point \(A\) outside the circle.
Assume the incidence axioms I1, I2, and I3. Prove the following.
(i)
Two distinct lines intersect in at most one point.
(ii)
If \(P\) is a point and \(l\) is a line not passing through \(P\), then there exist at least two distinct lines through \(P\) which meet \(l\).
(iii)
There exist at least three distinct lines.