Download 10.1 2) a) b) c) d) PQRS and VWTU e) PQRS and PSVU f) g) P, Q, R

10.1
2)
̅̅̅̅ 𝑎𝑛𝑑 𝑅𝑆
̅̅̅̅
a) 𝑃𝑆
̅̅̅̅ 𝑎𝑛𝑑 𝑄𝑅
̅̅̅̅
b) 𝑃𝑆
̅̅̅̅ 𝑎𝑛𝑑 𝑄𝑅
̅̅̅̅
c) 𝑃𝑆
d) PQRS and VWTU
e) PQRS and PSVU
̅̅̅̅ 𝑎𝑛𝑑 ̅̅̅̅
f) ̅̅̅̅
𝑃𝑆, 𝑅𝑆
𝑈𝑆
g) P, Q, R and S
h) Q, X and R
10) ∠DHG and ∠IHJ
12) ∠DHF
14) GHKM, DEIH and DEIG
20) True: There are infinitely many planes through a line, so when 3 points lie on the same line
(collinear), there will be infinitely many planes through these 3 points, so they are coplanar.
26) True: The sum of the 3 angles in a triangle is 180°. In a right triangle, one angle equals 90°, so the
sum of the other 2 is 180° - 90° = 90°. This means that neither of the angles can be greater than 90°
32) No, there are infinitely many planes through these 3 points, since there are infinitely many planes
through a line (see also question 20).
10.2
2) The centroid of a triangle is the point of intersection of its medians (the lines joining each vertex with
the midpoint of the opposite side).
8) The centroid is located at 1/3 of from the length of the median from the side, so in this case the
centroid will be located on the median, 12 units away from the side and 24 units from the vertex.
12) The network has 2 odd vertices: Ranger station and Rest area, so it is traversable type 2. This means
the Ranger station and Rest area will be the starting point and end point.
34)
A
B
C
D
The diagram shows that we have 4 odd vertices. Since a network with more than 2 odd vertices is not
traversable, it’s not possible to take a walk around town, starting from home and returning to home
after having crossed each bridge just once. For this to be possible we would need a type 1 traversable
network, and thus a network without odd vertices.
10.3
2)
a) 52°
b) 128° (180° - 52°)
c) 128°
d) 52°
e) 128°
f) 52°
g) 128°
4) 4x + 10 = 6x + 4  x = 3, so both angles are 22°
6) 8x – 40 = 6x + 2  x = 21, so both angles are 128°
8) ∠6 (If two parallel lines are cut by a transversal, then the alternate interior angles are congruent) and
∠11 (If two parallel lines are cut by a transversal, the interior angles on the same side of the transversal
are supplementary)
14) ∠1 = ∠2 = ∠3 = ∠4 = 90° (The angles in a square are 90°); ∠5 (The compliment of ∠3, so 180° 90°); ∠6 = 48° (The sum of the angles of a triangle is 180°, so ∠6 = 180° - ∠5 – 42°)
18) Interior angle =
(6−2)180°
6
= 120°; Exterior angle =
360°
6
= 60°; Central angle =
360°
6
= 60°
22) The measures of the exterior and central angle of a regular polygon are equal to each other. The
measure of the central angle is equal to 180° minus the measure of the interior angle.
26) 360° - 3 x 87° = 99°
28) We know that the measures of angles ∠Q and ∠S are equal to ½ * m (arc PR) = ½ * 180° = 90°
32) m∠P = ½ * (210° - 150°) = 30°
36) When he enters the danger zone, the angle between his sightings of the 2 towers will be ½
multiplied with the arc between the 2 towers, which is 180°. So as long as the angle between his
sightings is smaller than 90°, he is in the safe zone.
48)
360°
𝑛
= 40° → 𝑛 = 9
10.4
2) Side-Side-Side: Since the 2 triangles have 3 sides of equal length, they are congruent.
10) MLI: Side-Side-Side: Since the 2 triangles have 3 sides of equal length, they are congruent.
16) The 2 triangles are similar, since they have 2 congruent angles.
26) 𝑑 = √62 + 32 = √45 ≅ 6.71
32) The hypotenuse is 10 * 2 = 20, and the longer leg is 10√3 ≅ 17.3
44)
a) √2
b) 2
c) √3
d)
√3
3
e) ½
72)
a) a² + b² becomes greater than c²
b) a² + b² becomes smaller than c²
10.5
2) Rectangle ABCD has 4 right angles; Rectangle ABCD has 2 pairs of congruent sides; The 2 diagonals of
rectangle ABCD are congruent; Rectangle ABCD consists of 2 pairs of parallel sides.
6) Rhombus
8) None: for a trapezoid the other pair of sides doesn’t need to be congruent, and for any other type the
other pair of sides must be parallel.
20)
a) Yes
b) No
c) No
d) No
e) No
f) Yes
26) A quadrilateral is a parallelogram if and only if its diagonals bisect each other.
44) If adjacent sides are congruent, we have a quadrilateral with 4 congruent sides; a rhombus.
Review
4)
a) The sum of 2 complementary angles is 90°, while the sum of 2 supplementary angles is 180°
b) A perpendicular bisector bisects a line segment in 2 equal parts, while an angle bisector bisects an
angle in 2 equal angles
c) An isosceles triangle is a triangle with 2 sides of equal length, while an equilateral triangle is a triangle
with 3 sides of equal length.
d) A rhombus has 4 congruent sides, while a kite has 2 pairs of adjacent congruent sides.
e) A rhombus has 4 congruent sides, while a parallelogram has 2 pairs of opposite congruent sides.
f) A square has 4 congruent sides and 4 right angles, while a rectangle has only 4 right angles.
g) A trapezoid has 1 pair of parallel sides, while a parallelogram has 2 pairs of parallel sides.
h) A right triangle has 1 right angle and 2 acute angles, while an obtuse triangle has 1 obtuse angle and 2
acute angles.
i) A simple closed curve has no endpoint and does not cross itself, e.g. a circle. A non-simple closed
curve has no endpoint but crosses itself, e.g. a figure-8.
j) A convex polygon is such that a segment cannot be drawn from an interior point to any other interior
point and pass outside the polygon, e.g. a square. A non-convex polygon is such that a segment can be
drawn from an interior point to another interior point and pass outside the polygon, e.g. a 5-point star.
8) The sum of the angles in a triangle is 180°, so ∠C = 180° - 39° - 85° = 56°
10) A stop sign is a hexagon, so the measure of the angles is
(6−2)180°
6
= 120°
12)
a) The incenter of a triangle is always inside the triangle.
b) The circumcenter of a triangle is the point where the perpendicular bisectors of a triangle intersect.
c) The orthocenter of a triangle is located at the intersection of the altitudes of the triangle.
d) The distance from the centroid of a triangle to the vertices of the triangle is equal for all vertices.
16)
a) Opposite sides in a parallelogram are always parallel; The diagonals of a parallelogram bisect each
other.
b) All angles in a rectangle are equal to 90°; The diagonals of a rectangle are congruent.
c) All sides of a rhombus are congruent; The diagonals of a rhombus are perpendicular.