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PC3 Geometry
1.5
Sentry theorem (part 1)
1. Alex walks along the boundary of a n-sided plot of land, writing down the number of degrees
turned at each corner. What is the sum of these n numbers for n = 3, 4, 5, . . . ?
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[Sentry theorem] The sum of the exterior angles (one per vertex) of any polygon is 360 degrees.
The sum of interior angles of a n-sided convex polygon is (n − 2) · 180◦ .
2. The sides of a polygon are cyclically extended to form rays, creating one exterior angle at
each vertex. Viewed from a great distance, what theorem does this figure illustrate?
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3. A polygon is equilateral if its sides have the same length. A polygon is equiangular if its
interior angles are the same size. Clearly, a polygon is equiangular if its exterior angles are
the same size. For a triangle, equilateral is equivalent to equiangular. For polygons with
more than 3 sides, these two concepts are not equivalent anymore. For example, rhombus
is equilateral but not necessarily equiangular. On the other hand, a rectangle is equiangular
but not necessarily equilateral. A polygon that is both equilateral and equiangular is called
regular. Square is the regular quadrilateral.
Find the angle formed by two face diagonals that intersect at a vertex of the cube.
4. Mark Y inside regular pentagon P QRST , so that P QY is equilateral. Is RY T straight?
Explain.
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Sentry theorem (part 1)
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5. Equilateral triangles T HE and HIM are attached to the outside of regular pentagon T HIN K.
Is quadrilateral T IM E a parallelogram? Justify your answer.
Why there is no SSA congruence theorem? (part 1)
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Why there is no SSA congruence theorem? (part 1)
1. Draw triangles ABC and ADE with vertices having coordinates: A(0, 0), B(2, 0), C(5, 4),
D(4, 5), E = (0, 8). Triangles are clearly not congruent, but they have some congruent parts.
List all congruences you can find.
2. Construct triangle ABC such that AB = 3 and BC = 2, and ∠BAC = 30◦ . Is there only
one triangle satisfying the given information? If no, try constructing another one.
(c) BC =
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3. (Continuation) Solve the previous problem for
3
(a) BC =
(b) BC = 1
2
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4. Explain why there is no SSA congruence theorem. Is there a case, when the SSA congruence
can be used to prove that two triangles are congruent?
5. The diagonals of quadrilateral ABCD intersect at M . It is known that AM = CM , AB =
CD, ∠BAM = 10◦ and ∠DCM = 50◦ .
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(a) List all congruences between triangles AM B and CM D. Are they congruent?
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(c) Find ∠AM B.
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(b) Construct point E on DM such that triangles AM B and CM E are congruent. Describe
triangle CDE.
Trapezoids (part 2)
1.17
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Trapezoids (part 2)
1. In a trapezoid, a midline (or a midsegment) is the line joining the midpoints of the non-parallel
sides.
(a) Prove that in a trapezoid, the midline is parallel to the bases and its length is half their
sum.
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(b) Prove that the area of a trapezoid is the distance between parallel lines times the length
of the midline.
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2. Let r1 and r2 be two rays starting at point O. Let A, B, C be points on r1 such that
OA = AB = BC and let D, E, F be points on r2 such that OD = DE = EF . Given that
AD = 7, find the lengths of the segments BE and CF .
3. Consider rays r1 and r2 starting at point O. Let A, B, C be points on r1 such that AB = BC.
Three parallel lines passing through A, B, and C intersect r2 in points D, E, F , respectively.
Prove that DE = EF by drawing a line ` passing through A and parallel to r2 .
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4. Consider a trapezoid M ARS, AR k M S, and let X, Y , Z be the midpoints of AM , RS, and
M S, respectively. Assume that XZ = Y Z.
(a) Find four congruent angles in the diagram.
(b) Show that M ARS is an isosceles trapezoid.
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5. Let ABCD be a trapezoid such that BC k AD, BC = 6, and AD = 8. Denote by M and N
the midpoints of the diagonals.
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(a) Prove that M N k BC.
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(b) Find the length of M N .
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PC3 Geometry
1.18
Similarity of triangles (part 2)
1. Draw a right triangle ABC with ∠C = 90◦ and altitude CD.
(a) List all similar triangles in your diagram.
(b) Given that AD = 3 and BD = 4, find the length of CD and the perimeter of triangle
ABC.
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2. Let ABC be a right triangle, ∠C = 90◦ and let CD the altitude. Given that AD = 9 and
AC = 12, find the lengths of CD, BD, and BC.
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3. Consider a quadrilateral ABCD with ∠B = ∠C = 90◦ , AB = 6, and CD = 12. Denote by P
the intersection of the diagonals and let Q be the perpendicular from P onto BC. Find the
length of P Q.
4. Let ABC be an acute triangle and let AD, BE, CF be the altitudes.
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(a) Given that AB = 13, AC = 15, and AD = 12, find the lengths of the altitudes BE and
CF .
(b) (Continuation) Use SAS similarity to prove that triangle AEF is similar to triangle
ABC.
(c) Suppose the lengths of segments AB, AC, and AD are not given to you. Use AA
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AF
similarity to prove that AB
= AC
. What conclusion do you make?
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5. Let ABC be a triangle and let D be a point on BC. A line parallel to BC intersects AB,
AD, and AC in points P , Q, and R, respectively. Prove that
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PQ
BD
=
.
QR
CD