10.1 POINTS, LINES, PLANES, AND ANGLES
A point, represented by a small dot, has no size and is generally represented by a capital letter. A
line, connecting two points on a straight path, has no thickness and extends infinitely in both
directions; it is represented by naming two points on the line, AB . A line segment, with endpoints at
points A and B, is represented as AB . A ray is a line with one endpoint and extends infinitely in one
direction. An angle is formed by two rays with a common endpoint, called the vertex. Angles can be
named by numbers, by the vertex letter ( B), or with three letters, where the vertex letter is in the
middle and the other two represent one point on each side. Two angles that add to 90 are
complementary; if they add to 180, they are supplementary. When two lines intersect, the opposite
angles are called vertical angles and are always equal. For two parallel lines cut by a transversal,
the alternate interior, alternate exterior, and corresponding angles are equal.
1. Use the protractor to find angle measures, then classify the angle by its measure.
.
m
m
m
m
m
2.
3.
Find the complement and supplement of the following angles:
a. 58
b. 90
c. 128
d. 180
If two lines intersect forming four angles and m< 1 = 38, what are the measures of angles 2, 3, and
4 numbered consecutively clockwise.
38

4.
BAC =
HAG =
CAF =
EAH =
BAH =
Find the measure of each of the numbered angles:
2
1
50
3
4
40
5.
Which angles are supplementary, complementary, and equal in the accompanying
diagram?
6.
In the above diagram, if m<6 = 28, what are the measures of the other angles?
7.
Find the measure of the other 7 angles on the diagram below, which shows two parallel lines
intersected by a transversal.
118
7
8.
Find the measure of the remaining angles given that rs and mn:
r
70
s
105
m
n
9.
Find the measure of the smaller angle from the diagram below:
5x+14
3x+20
10. Find the measure of each of the remaining angles given that pq.
x + 20
p
3x
x
q
11. The hour hand on a clock moves from 1 to 7 o’clock. Through how manyh degrees does it move?
Try These (10.1)
Find the measure of the indicated angles and classify them as acute, right, straight, or obtuse:
1.
CAF
4.
2.
HAF
3.
HAE
Which of the following is true:
A. z = 143
B. Y = 37
C. X = 37
D. All of these
E. None of these
93
In Drawing A above,
5. Which angles are [a] supplementary, [b] complementary.
6.
If angle 2 measures 72, find the measure of angles 1, 3, 4, and 5.
7.
If angle 4 measures 3x + 18 and angle 5 measures 5x - 32, what is the measure of the smaller
angle?
8.
In drawing B above, find the measures of angles, 2, 3, 4, 5, 6, 7, and 8.
10.2 TRIANGLES
The sum of the three angles of a triangle is 180. Similar triangles have equal corresponding angle
measures and corresponding sides are proportional. Triangles with three equal sides are equilateral,
with two equal sides are isosceles, and with no equal sides are scalene. Triangles with one angle
greater than 90 are obtuse, with one angle equal to 90 is right, and with all angles less than 90 are
acute. Two triangles are similar if corresponding angles are equal, in which case, corresponding sides
are proportional. The per8imiter of a triangle can be found by adding its sides. For right triangles
with legs a and b and hypotenuse c, missing sides can be found with the Pythagorean Theorem: a2 +
b2 = c2.
1.
A triangle has two angles measuring 48 and 32. What is the measure of the missing angle.
What type of triangle is this?
2.
Find the measure of the remaining angles given that ABCD.
C
D
60
A
40
O
B
3.
In drawing A, if angles 2 and 5 each measure 70, what are the measures of angles 1, 3, 4, and 6?
4.
If side x measures 8” and y measures 12” and side d measures 6” and side e measures 8”, find the
lengths of the other sides of the two triangles.
5.
Triangles ABC and DCE are similar. Side AB = 10, BC = 12, Side CE = 4, DE = 7. Find the length
of the other sides.
6.
A person who is 5’6” tall is standing 6 feet from a light pole. The person casts a 2 foot shadow.
How tall is the pole?
7.
The length of the diagonal of a square screen is 32”. What are the dimensions?
8.
A 20’ ladder has its base 5 feet from a building. How high up the building does it reach?
9.
Find x in the accompanying diagram:
15 ft
35 ft
27 ft
x
10. A tree casts a shadow 27 feet long. At the same time, a 6 foot tall person casts a shadow 4.5 feet
long. How tall is the tree?
Try These (10.2)
1.
Which one of the following is true for the pictured triangles? Prove it! More than one answer
may be correct.
[A]
2.
B E
[B] BC = 22/3
[C] x ≠ y
[D] AC/DC = CE/CB
[E] None of these
Which one of the following is true for the pictured triangles? Prove it! More than one answer
may be correct.
65
[A] m8  m2  m7
[D] m2  115
3.
[B] m1  50
[E] None of these
[C] m2  50
A ladder 20 feet long is leaning against a vertical wall. The bottom of the ladder is 6 feet from the
wall. How far up the wall does the ladder reach?