Download SD School Geometry

SD School Geometry
Program 3: Triangles
QuikNotes
The reflexive property states that a value is equal to
itself.
The substitution property says that we can switch two
values that are equal. For example, if the measure of
angle 1 plus the measure of angle 3 equals 180, and the
measure of angle 1 is equal to the measure of angle 2,
the measure of angle 2 plus the measure of angle 3
equals 180. Switching the values of the measure of
angle 1 and the measure of angle 2 is substitution.
The transitive property says that if A equals B, and B
equals C, then A equals C.
Theorem: If two angles are the complements of two
congruent angles, then the two angles are congruent.
For example: If angle 1 and angle 2 are the
complements of angle 3 and angle 4, and angle 3 and
angle 4 are each 60 degrees, then angle 1 and angle 2
must each be 30 degrees.
Theorem: If two angles are the supplements of two
congruent angles, then the two angles are congruent.
Theorem: every angle has exactly one bisector. As angle
CUT is bisected by ray UP, we get two congruent
angles, CUP and TUP
The parallel postulate is one of the most important
postulates in geometry. It says: given a line, L, and a
point, P, not on that line, there’s exactly one line, M,
through the point, P, that is parallel to line L.”
Student Notes
SD School Geometry
Program 3: Triangles
QuikNotes
When parallel lines are cut by the transversal, eight
different angles are formed. There are four different
ways to categorize these angles. The most general way
is as corresponding angles. Angles 1 and 5, 2 and 6, 3
and 7, and 4 and 8 are corresponding angles because
they are in the same position on either line. They
correspond to each other.
Angles 3 and 5, and 4 and 6 are alternate interior
angles. They’re all interior angles because they’re
between the two lines. The pairs we mentioned are
alternate interior angles because they are on alternate,
or opposite, sides of the transversal.
Angles 1 and 7, and 2 and 8 are alternate exterior
angles. These guys are all exterior angles because they
are on the outside of the lines. Again, the pairs are
alternate exterior angles because they are on opposite
sides of the transversal.”
Angles 3 and 6, and 4 and 5 are same-side interior
angles. Because, well, the pairs are all interior angles
on the same side of the transversal
The Corresponding Angle Postulate: says that lines
are parallel if and only if corresponding angles are
congruent.
Student Notes
SD School Geometry
Program 3: Triangles
QuikNotes
A triangle is the figure formed by three segments
joining three non-collinear points, which means it’s a
three-sided shape.
The three segments are called the triangle’s sides.
Each side meets another at a vertex, where they form
an angle.
There are two ways to classify triangles. Either by
lengths of their sides, or by their angles.
Equilateral triangles have three congruent sides. Like
triangle A here. All sides measure 4 inches.
A
Isosceles triangles have two congruent sides. Triangle
B has two sides that equal 4 inches. We call the other
side the base. The base can be on any side. For
example, if we turn triangle B on its side, or upsidedown, its base is still 2 inches.
B
C
Triangle C is scalene, which means it has no congruent
sides. It’s just a mess. It’s all over the place.
Equiangular triangles have all three angles congruent.
In triangle A, each angle measures 60 degrees. All
equiangular angles measure 60 degrees. All
equiangular triangles are equilateral, and vice-versa.
Student Notes
SD School Geometry
Program 3: Triangles
QuikNotes
Right triangles have one right angle. Other parts of
right triangle are its two legs, which form the right
angle, and the side opposite the right angle is called the
hypotenuse.
Obtuse triangles have one obtuse angle. This triangle
has angles that measure 30 degrees, 35 degrees, and the
obtuse angle, which is 115 degrees.
Acute triangles have three acute angles. This triangle
has measures of 40, 60, and 80 degrees.
Student Notes
SD School Geometry
Program 3: Triangles
QuikNotes
Complementary angles are two angles whose
measures add up to 90 degrees. The measure of angle
Y is 33 degrees, and angle O is 57 degrees. If we add
the measures of angle Y and angle O together, we get
90 degrees, the measure of a right angle. So angles Y
and O are complementary.
Supplementary angles are two angles whose measures
add up to 180 degrees. The measure of angle D is 99
degrees, and angle A is 81 degrees. We add angle D
and angle A together, for a total of 180 degrees, which
is the measure of a straight angle. So, angle D and
angle A are supplementary
Complementary and supplementary angles do NOT
need to be adjacent.
Vertical angles are what we get when two lines
intersect. Vertical angles are two angles that occur
opposite each other when two lines intersect. When
these two lines intersect, they form four angles. In this
diagram, angles 1 and 2 are vertical angles. So are
angles 3 and 4. Vertical angles are congruent, if you
know the measure of one, you also know the measure
of the other.
Student Notes
SD School Geometry
Program 3: Triangles
QuikNotes
Example: If we’re given the measure of one of these angles,
we can figure out the other three.
If angle 1 is 55 degrees, then angle 2 is also 55 degrees,
because they’re vertical angles.
Angles 1 and 3 are supplementary angles. Together, their
measures will add up to 180 degrees. We know the
measure of angle 1 is 55 degrees, so angle 3 measures 180
degrees minus 55 degrees, or 125 degrees.
Angle 4 also measures 125 degrees because angles 3 and 4
are vertical angles
An angle bisector is a ray that splits an angle in the
middle, forming two congruent adjacent angles. Here, ray
QR bisects angle SQT. Angle SQR is congruent to angle
TQR. Note the curved symbol inside each of the angles. If
there’s a slash in each one, then that means the angles are
congruent.
Perpendicular lines are intersecting lines that form
right angles. Rays and segments can also be
perpendicular. Here we have two perpendicular
lines, L and M. So all four angles, D, O, G, and S,
are right angles. When we write that two lines are
perpendicular, we use a special symbol. L is
perpendicular to M.
l⊥m
Student Notes
SD School Geometry
Program 3: Triangles
QuikCheck
True or False
1. The transitive property says that if X equals Y, and Y equals Z, then X equals Z. _______
2. When two lines are cut by a line, we call this intersecting line a bisector. __________
3. Corresponding angles are in the same position between two parallel lines and their
transversal. _____________
4. Obtuse triangles have three obtuse angles. ____________
5. Supplementary angles are two angles whose measures add up to 180 degrees. ________
6. An angle bisector is a ray that splits an angle in the middle, forming two congruent adjacent
angles. __________
Q&A
7. If angles J and K are alternate exterior angles, and angle J measures 540 , what is the measure
of angle K?
8. If angles M, N, and O are in an acute triangle, and angles M and N are both acute, is angle O
acute or obtuse?
9. If angle UVW is 940 , and line TV is UVW’s bisector, what is the measure of angle UVT?
10. Angle 4 is 610 and angles 4 and 5 are vertical angles, what is the measure of angle 5?
11. If an equilateral triangle has angles A, B, and C, and angle A is 600 , what is the measure of
angle B?
12. If an isosceles triangle has one angle of 1400 , and one angle 200 , what is the third angle?
13. If angle G is 790 , what is the measure of its supplement, angle H?
14. If a right triangle has one angle 330 and one angle 570 , what is the third angle?
SD School Geometry
Program 3: Triangles
QuikCheck Answer Key
True or False
1. The transitive property says that if X equals Y, and Y equals Z, then X equals Z. True
2. When two lines are cut by a line, we call this intersecting line a bisector. False When two lines
are cut by a line, we call this intersecting line a transversal.
3. Corresponding angles are in the same position between two parallel lines and their
transversal. True
4. Obtuse triangles have three obtuse angles. FalseObtuse triangles have one obtuse angle.
5. Supplementary angles are two angles whose measures add up to 180 degrees. True
6. An angle bisector is a ray that splits an angle in the middle, forming two congruent adjacent
angles. True
Q&A
7. If angles J and K are alternate exterior angles, and angle J measures 540 , what is the measure
of angle K?
54 0
Alternate exterior angles are the same size.
8. If angles M, N, and O are in an acute triangle, and angles M and N are both acute, is angle O
acute or obtuse?
Acute. All the angles in an acute triangle are acute.
9. If angle UVW is 940 , and line TV is UVW’s bisector, what is the measure of angle UVT?
It is one half of 940 .
47 0
10. Angle 4 is 610 and angles 4 and 5 are vertical angles, what is the measure of angle 5?
61 0
Vertical angles have the same measurement.
11. If an equilateral triangle has angles A, B, and C, and angle A is 600 , what is the measure of
angle B?
60 0
All the angles of an equilateral triangle are 600 .
12. If an isosceles triangle has one angle of 1400 , and one angle 200 , what is the third angle?
20 0
An isosceles triangle has two angles of the same measurement,
and a triangle can only ever have one obtuse angle.
0
13. If angle G is 79 , what is the measure of its supplement, angle H?
Supplementary angles add up to 1800 .
81 0
14. If a right triangle has one angle 330 and one angle 570 , what is the third angle?
90 0
All right triangles have one right angle.
Distributed by
United Learning
1560 Sherman Avenue, Suite 100
Evanston, IL 60201
800-323-9084
www.unitedstreaming.com
[email protected]