Download MPM1D Grade 9 Geometry Chapter Test

MPM1D Grade 9
Geometry Chapter TEST
Multiple Choice
For questions 1 to 5, select the best answer.
1. The exterior angle at the vertex formed by the equal sides of an
isosceles triangle is 140o. Which are the measures of the exterior angles
at the other vertices?
a) 140o, 80o
b) 110o, 110o c) 40o, 40o
d) 40o, 80o
Answer: b)
𝟑𝟔𝟎°−𝟏𝟒𝟎° 𝟐𝟐𝟎°
𝜽=
=
= 110o.
𝟐
𝟐
2. In ∆MNP, the interior angle at N is 34o and the exterior angle at P is
55o. Which is the measure of the interior angle at M?
a) 101o
b) 79o
c) 21o
d) 281o
Answer: c)
By the Exterior Angle theorem, we get:
the exterior angle at P = the interior angle at N + the interior angle at M
55o = 34o + the interior angle at M
the interior angle at M = 55o – 34o = 21o
3. The sum of the interior angles of a convex pentagon
a) is always 360o
b) is always 540o
c) is always 180o
d) depends on the shape of the pentagon
Answer: b)
Sn = 180o (n – 2)
n = 5 ⟹ S5 = 180o (5 – 2) = 180o (3) = 540o
4. The area of ABC is
a) equal to the area of ∆BCD
b) half the area of ∆ABD
c) double the area of ∆ABD
d) none of the above
Answer: c)
𝟏
Area (∆ABC) = AC × BH
𝟐
𝟏
= (2AD) × BH
𝟐
𝟏
= 2 ( AD × BH)
𝟐
= 2 Area (∆ABD)
5. The diagonals of a parallelogram
a) are always perpendicular to each other
b) always bisect the interior angles
c) always bisect each other
d) always bisect each other at right angles
Answer: c)
Short Response
Show all steps to your solution.
6. Find the measure of each indicated angle.
a)
b)
Solution
By the Exterior Angle theorem,
we get:
143o = x + 90o ⟹ x = 143o – 90o
= 53o
c)
d)
Solution
x = 180o – 35o
= 145o
Solution The triangle is equilateral, therefore:
x = 60o
y = 180o – 60o =120o
Solution Determine the rest two exterior angles:
180o – 70o = 110o
180o – 33o = 147o
Since the sum of the exterior angles is 360o, we
get:
x = 360o – 54o – 110o – 147o = 49o
7. What is the sum of the interior angles of a convex polygon with 9
sides?
Solution
Sn = 180o (n – 2)
n = 9 ⟹ S9 = 180o (9 – 2) = 180o (7) = 1260o
8. Explain why each conjecture is true, or use a counterexample to show
it is false.
a) A triangle can have more than one obtuse angle.
Answer: The conjecture is false, since the sum of interior angles in a
triangle is 180o, while the sum of two obtuse angles exceeds 180o.
b) A quadrilateral can have more than one obtuse angle.
Answer: The conjecture is true, for example, the trapezoid below has
two obtuse angles ∠A and ∠B:
The maximum number of obtuse angles in a quadrilateral is three:
Complete Solutions
Provide complete solutions
9. The sum of the interior angles of a regular polygon is 2520o.
a) What is the measure of each interior angle?
Solution
Sn = 180o (n – 2) ⟹ 2520o = 180o (n – 2)
14 = n – 2
n = 16
2520o ÷ 16 = 157.5o
The measure of each interior angle is 157.5o.
b) What is the measure of each exterior angle?
Solution
180o – 157.5o = 22.5o
The measure of each exterior angle is 22.5o.
10. One exterior angle of an isosceles triangle is 80o.
a) Find the possible measures of the other two exterior angles.
Solution 1
Determine the adjacent interior angle:
180o – 80o = 100o
Since a triangle cannot have two obtuse angles, the other two interior
angles equal (180o – 100o) ÷ 2 = 40o.
Therefore, the other two exterior angles equal 180o – 40o = 140o
Solution 2
Case 1 One of the other two exterior angles of an isosceles triangle is
also 80o.
Determine the third exterior angle: 360o – 80o – 80o = 200o
Since an exterior angle must be less then 180o, this case is impossible.
Case 2 None of the other two exterior angles of an isosceles triangle is
80o.
Determine the other two exterior angles, which are equal:
(360o – 80o) ÷ 2 = 140o
The other two exterior angles equal 140o.
b) How many answers can you find? Explain.
Answer: One answer – see the solutions above.