Review Supplement Unit 2c: Triangle Stuff (PAP Geom)

Review Supplement
Unit 2c: Triangle Stuff (PAP Geom)
You will be able to find the measures of
the interior and exterior angles of a
triangle
1.
Find the value of 𝑥 and 𝑦.
46
Geometry Textbook P. 282: 4, 6-8
x
On a scale of 1-5, rate your confidence in
this objective (5 being the most
confident):
2y
x
2.
The sum of the interior angles of a triangle in Euclidean geometry is 180°. What is the sum in Elliptic geometry? Hyperbolic
geometry? Why are these answers not the same?
3.
Find the value of 𝑥.
4.
Find the value of 𝑥.
5.
Find the value of each marked angle.
6.
In terms of radians, what is the sum
of the three interior angles of a
triangle?
7.
Convert the angles to radians and
find the value of 𝑥 in radians.
8.
Find the value of 𝑥 in radians.
7π
9
x
9.
Explain why ∠𝐴 and ∠𝐵 are
complementary.
10. Find the sum of the marked angles.
You will be able to develop and prove
theorems about isosceles triangles
Geometry Textbook P. 282: 2; P. 285: 2426
11. Find the values of 𝑥 and 𝑦.
On a scale of 1-5, rate your confidence in
this objective (5 being the most
confident):
12. Find the values of 𝑥 and 𝑦.
13. Find the values of 𝑥, 𝑦, and 𝑧.
14. Find the missing coordinates.
Constructions:
Use your compass and straightedge to construct the indicated angle given ∠𝑨 and ∠𝑩 below.
B
A
15. Given ∠𝐴 and ∠𝐵 in ∆𝐴𝐵𝐶, construct ∠𝐶.
16. Given ∠𝐴 is the vertex angle in isosceles ∆𝐴𝑃𝐸, construct ∠𝑃.
17. Given ∠𝐴 and ∠𝑀 are the base angles of isosceles ∆𝐴𝑀𝐷, construct ∠𝐷.
You will be able to use and define
perpendicular bisectors, angle bisectors,
medians, and altitudes
Geometry Textbook P. 344: 1
On a scale of 1-5, rate your confidence in
this objective (5 being the most
confident):
18. Compare and contrast a
perpendicular bisector and a median
of a triangle.
You will be able to discover, use, and
prove various theorems about
perpendicular bisectors and angle
bisectors
Geometry Textbook P. 345: 9-11; P. 348:
4-7
19. Find the length of̅̅̅̅
𝑅𝑆.
On a scale of 1-5, rate your confidence in
this objective (5 being the most
confident):
20. Find the value of x.
21. Find the values of x and y.
You will be able to define various points
of concurrency
Geometry Textbook P. 344: 2-5
22. Find the approximate location of the
orthocenter by drawing the altitudes
in the triangle below.
6
C
On a scale of 1-5, rate your confidence in
this objective (5 being the most
confident):
4
2
B
5
5
A
2
4
You will be able to discover, use, and
prove various theorems about points of
concurrency
Geometry Textbook P. 345: 12-13; P. 346:
14-18
23. The perpendicular bisectors of ΔABC
are concurrent at G.
On a scale of 1-5, rate your confidence in
this objective (5 being the most
confident):
a)
What type of point is G?
b) What is GB?
24. Point L is the centroid of PQR. If
ML = 10x − 4 and MR = 12x + 18, find
the value of x.
25. Explain how you could circumscribe a
circle around a triangle.
26. Explain how you could inscribe a circle
inside a triangle.
27. A circular revolving sprinkler needs to
be set up to water every part of a
triangular garden. Where should the
sprinkler be located so that it reaches
the entire garden, but doesn’t spray
farther than necessary?
28. You need to supply electric power to
three transformers, one on each of
three roads enclosing a large
triangular track of land. Each
transformer should be the same
distance from the power-generation
plant and as close to the plant as
possible. Where should you build the
power plant, and where should you
locate each transformer?
29. Complete each statement as fully as
possible.
a) M is equidistant to: __________
b) P is equidistant to: __________
c) Q is equidistant to: __________
d) R is equidistant to: __________
Identify each statement as describing the incenter, circumcenter, orthocenter, or centroid.
30.
The point equally distant from the three sides of a triangle.
31.
32.
33.
34.
35.
36.
37.
38.
39.
The center of gravity of a thin metal triangle.
The point equidistant from the three vertices.
The intersection of the perpendicular bisectors of the sides of a triangle.
The intersection of the altitudes of a triangle.
The intersection of the angle bisectors of a triangle.
The intersection of the medians of a triangle.
A point always inside the triangle and on the Euler line.
The point on the hypotenuse of a right triangle.
The point at a vertex of a right triangle.
Compass and Straightedge Constructions
40. Construct an obtuse triangle. Construct the inscribed and the circumscribed circles. (Note that there is not enough space here.)
41. Construct a triangle. Use your compass and straight edge to construct the Euler Line
42. Inscribe a triangle in the circle on the left, and use the triangle to construct the center of the circle. Circumscribe a triangle
about the circle on the right, and use the triangle to construct the center of the circle. Label the centers of both circles. Do not
erase the arcs and other marks you make as you construct the centers. (Circles on the next page)
You will be able to discover, use, and
prove the Midsegment Theorem
Geometry Textbook P. 344: 6, 7
43. The perimeter of ∆𝐴𝐵𝐶 is 12 units. If
̅̅̅̅ , 𝐸𝐹
̅̅̅̅ , 𝐹𝐷
̅̅̅̅ , 𝐺𝐼
̅̅̅ , ̅𝐼𝐻
̅̅̅ , 𝐻𝐺
̅̅̅̅ , 𝐿𝐾
̅̅̅̅ , 𝐾𝐽
̅̅̅, and
𝐷𝐸
̅
𝐽𝐿 are midsegments, what is the
perimeter of ∆𝐿𝐾𝐽?
On a scale of 1-5, rate your confidence in
this objective (5 being the most
confident):
A
G
D
L
44. If EF = 2x + 7 and GH = 5x – 1, what is
EF?
45. WY is a midsegment. Find the value
of x.
H
B
E
K
I
J
F
C
You will be able to write a coordinate
proof
Geometry Textbook P. 344: 8
On a scale of 1-5, rate your confidence in
this objective (5 being the most
confident):
46. Using the diagram from Q47, Find the
̅̅̅̅ .
coordinates of M, the midpoint of 𝐴𝐵
̅̅̅̅̅ is a
If you connect M to C, then 𝐶𝑀
̅̅̅̅̅ is also an
median. Show that 𝐶𝑀
altitude.
47. Show ΔABC is isosceles. (Use the
distance formula to show AC = CB.)
y
C (a, b)
A (0, 0)
x
B (2a, 0)
You will be able to complete and use the
Triangle Inequality Theorems
Geometry Textbook P. 347: 19-24
On a scale of 1-5, rate your confidence in
this objective (5 being the most
confident):
48. Determine whether the 3 lengths
below could form a triangle.
a) 1 in, 2 in, 2 in.
b) 1 in, 2 in, 3 in.
c)
49. The Triangle Inequality states that
the sum of any two sides of a triangle
must be greater than the third side.
The sides of ABC are 2x + 3, 3x – 1,
and 6x – 11. Write and solve three
inequalities to find the possible
values of x.
Compass and Straightedge Constructions
̅̅̅̅ and 𝐵𝐶
̅̅̅̅ , construct the smallest possible segment
51. Using 𝐴𝐵
̅̅̅̅
𝑅𝑆 and the largest possible segment ̅̅̅̅
𝑃𝑄 such that
̅̅̅̅ is the third side of Δ𝐴𝐵𝐶.
𝑅𝑆 < 𝐴𝐶 < 𝑃𝑄, where 𝐴𝐶
1 in, 2 in, 4 in.
50. List the unknown measures in order
from greatest to least.
A
B
B
C