Download Exercise sheet 1

MS2013 Exercise set 1
Angles, parallel lines
(1) The angle bisector of an angle is a half-line which splits the angle in two halves.
Prove that the angle bisectors of two opposite angles form the same line.
(2) Two angles sharing the same vertex and a common side are called supplements if
their other sides are the half-lines of the same line. Prove that the angle bisectors
of two supplements are perpendicular on each other.
(3) Prove that two lines which are parallel to a third one are parallel to each other.
[Hint: cross these lines by a fourth and consider the angles.]
\=
(4) Angles ABC and A′ B ′ C ′ satisfy AB k A′ B ′ and BC k B ′ C ′ . Prove that ABC
◦
′
′
′
′
′
′
\
\
\
A B C or ABC + A B C = 180 . Also prove that the angle bisectors of the two
angles are either parallel or perpendicular or form the same line.
Congruent triangles
(1) Prove that an isosceles triangle with an angle of 60◦ is equilateral, i.e. all of its sides
are equal.
(2) Let ABCD be a quadrilateral. Show that the following statements are equivalent:
(a) ABCD is a parallelogram, meaning that AB k CD and BC k AD.
(b) AB k CD and |AB| = |CD|.
(c) The diagonals AC and BD intersect at their midpoint.
(d) |AB| = |CD| and |BC| = |AD|.
B
C
b
b
O
b
b
A
b
D
You can solve this question by taking each of the following steps separately:
• Assume (a). Prove (b).
• Assume (b). Prove (c).
• Assume (c). Prove (d).
• Assume (d). Prove (a).
(3) Let ABCD be a quadrilateral. Show that the following statements are equivalent:
(a) ABCD is a rectangle.
(b) The diagonals AC and BD intersect at their midpoint and |AC| = |BD|.
You can solve this question by taking each of the following steps separately:
• Assume (a). Prove (b).
• Assume (b). Prove (a).
(4) Let ABC be a right-angled triangle with  = 90◦ , and let M be the midpoint of
BC.
a) Prove that |AM | = 21 |BC|. b) If B̂ = 60◦ , prove |AB| = 21 |BC|.
Hint: Construct a rectangle ABDC.
1
2
(5) Let ABCD be a quadrilateral. Show that the following statements are equivalent:
(a) ABCD is a rhombus (i.e. a diamond, i.e. a parallelogram all of whose sides are
equal) .
(b) The diagonals AC and BD intersect at their midpoint and AC ⊥ BD.
(6) Let ABC be a triangle and consider the midpoint M of the segment AB and the
midpoint N of AC.
a) Show that M N kBC and |M N | = 1/2|BC|.
b) Let G be the point of intersection of BN and CM . Show that |BG| = 2|GN |
and |CG| = 2|GM |.
c) Show that all the medians in a triangle intersect at the G point. This is called
the centroid of the triangle.
Hint: a) Extend the segment M N by a segment M P of equal length (such that
M , N and P are on the same line) and use results from (2).
b) Draw the midpoints of the segments BG and CG.
(7) Let ABC be a triangle. Show that the following statements are equivalent:
(A) |AB| = |AC|.
\=\
(B) ABC
ACB.
(C) The median from A is also a height (i.e. altitude).
\
(D) The median from A is also the angle bisector of BAC.
(E) The angle bisector from A is also a height.
(F) The heights from B and C have equal lengths.
(G) The medians from B and C have equal lengths.
(8) Let A, B and C be three points on a circle of center O. Prove that
\
BOC
\
.
BAC =
2
(9) Let OAB, OA′ B ′ and OA′′ B ′′ be three equilateral triangles such that the order of
the lines through O, read clockwise, is OA, OB, OA′ , OB ′ , OA′′ , OB ′′ . Show that
the triangles AA′ A′′ and BB ′ B ′′ are congruent.
\ = 120◦ . Take a
(10) Consider triangle ABC with |AB|=6 cm, |BC|=10 cm and |ABC|
point M on the angle bisector of \
ABC such that |M B|=16 cm. Find the measures
of all the angles of triangle M AC.
Hint: Split the segment |M B|=16 cm into two segments of 6 and 10 cm, then
find some congruent triangles.
(11) Show that any triangle having two distinct angle bisectors of equal length must be
isosceles.
Hint: In the triangle ABC with equal angle bisectors BD and CE, assume that
\′ =
Ĉ < B̂. Then draw a segment BD ′ of equal length with BD such that CBD
\ (In this way AB and CD ′ will meet to form an isosceles triangle.) Now, show
ECB.
′ CD = 2D
′ BD.
\
\
that the quadrilateral DD ′ CB is convex, has |BD| = |BD ′ | and D
Since we’re working with angles, it seems convenient to look at them on a circle.
Draw the circumcircle of CDD ′ . How is B positioned with respect to it? From here,
what can you say about the sum Ĉ + B̂?
Anca Mustata, School of Mathematical Sciences, UCC
E-mail address: [email protected]