Pre-Algebra Homework
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NEW!
Student Learning
Goal Chart
Lesson Reflection for
Chapter 5
Pre-Algebra Learning Goal
Students will
understand plane
geometry through
plane figures and
patterns in geometry.
Students will understand plane geometry through
plane figures and patterns in geometry by
completing the following:
•
•
•
Learn to classify and name figures (5-1)
Learn to identify parallel and perpendicular lines and the angles formed
by a transversal (5-2)
Learn to find unknown angles in triangles (5-3)
Hop On Board the
Fast Track Train!
Chapter 5 Sections 1 & 2
5-1 Points, Lines, Planes, and Angles
Learning Goal Assignment
Learn to
classify and
name figures.
5-1 Important Notes
A right angle measures 90°.
An acute angle measures less than 90°.
An obtuse angle measures greater than 90°
and less than 180°.
Complementary angles have measures that
add to 90°.
Supplementary angles have measures that
add to 180°.
5-1 Important Notes
Congruent figures have the same size and shape.
• Segments that have the same length are
congruent.
• Angles that have the same measure are
congruent.
• The symbol for congruence is , which is read “is
congruent to.”
Intersecting lines form two pairs of vertical
angles. Vertical angles are always congruent, as
shown in the next example.
5-1 FAST TRACK Quiz
In the figure, 1 and 3 are vertical angles,
and 2 and 4 are vertical angles.
1. Name three points in the figure.
Possible answer: A, B, and C
2. Name two lines in the figure.
Possible answer: AD and BE
3. Name a right angle in the figure.
Possible answer: AGF
4. Name a pair of complementary angles.
Possible answer: 1 and 2
5. If m1 47°, then find m 3.
47°
5-2 Parallel and Perpendicular Lines
Learning Goal Assignment
Learn to identify parallel
and perpendicular lines
and the angles formed
by a transversal.
5-2 Important Notes
Parallel lines are two lines in a plane
that never meet, like a set of perfectly
straight, infinite train tracks.
Perpendicular lines are lines that
intersect to form 90° angles.
5-2 Important Notes
PROPERTIES OF TRANSVERSALS
TO PARALLEL LINES
If two parallel lines are intersected by a transversal,
• the acute angles that are formed are all
congruent,
• the obtuse angles are all congruent,
• and any acute angle is supplementary to any
obtuse angle.
If the transversal is perpendicular to the parallel
lines, all of the angles formed are congruent 90°
angles.
5-2 Important Notes
If two lines are intersected by a
transversal and any of the angle pairs
shown below are congruent, then the
lines are parallel. This fact is used in the
construction of parallel lines.
5-2 FAST TRACK Quiz
In the figure a || b.
1. Name the angles congruent to 3.
1, 5, 7
2. Name all the angles supplementary to 6.
1, 3, 5, 7
3. If m1 = 105° what is m3?
105°
4. If m5 = 120° what is m2?
60°
5-3 Triangles
Learning Goal Assignment
Learn to find
unknown angles in
triangles.
5-3 Triangles
Vocabulary
Triangle Sum Theorem
acute triangle
right triangle
obtuse triangle
equilateral triangle
isosceles triangle
scalene triangle
Pre-Algebra
5-3 Triangles
If you tear off two corners of a triangle
and place them next to the third
corner, the three angles seem to form
a straight line.
Pre-Algebra
5-3 Triangles
Draw a triangle and extend one side.
Then draw a line parallel to the
extended side, as shown.
The sides of
the triangle
are
transversals to
the parallel
lines.
The three angles in the triangle can be
arranged to form a straight line or 180°.
Pre-Algebra
5-3 Triangles
An acute triangle has 3 acute angles. A
right triangle has 1 right angle. An obtuse
triangle has 1 obtuse angle.
Pre-Algebra
5-3 Triangles
Additional Example 1A: Finding Angles in Acute,
Right and Obtuse Triangles
Find p in the acute triangle.
73° + 44° + p = 180°
117° + p = 180°
–117°
–117°
P = 63°
Pre-Algebra
5-3 Triangles
Try This: Example 1A
Find a in the acute triangle.
88° + 38° + a = 180°
38°
126° + a = 180°
–126°
–126°
a = 54°
Pre-Algebra
a°
88°
5-3 Triangles
Additional Example 1B: Finding Angles in Acute,
Right, and Obtuse Triangles
Find c in the right triangle.
42° + 90° + c = 180°
132° + c = 180°
–132°
–132°
c = 48°
Pre-Algebra
5-3 Triangles
Try This: Example 1B
Find b in the right triangle.
38°
38° + 90° + b = 180°
128° + b = 180°
–128°
–128°
b = 52°
Pre-Algebra
b°
5-3 Triangles
Additional Example 1C: Finding Angles in Acute,
Right, and Obtuse Triangles
Find m in the obtuse triangle.
23° + 62° + m = 180°
85° + m = 180°
–85°
–85°
m = 95°
Pre-Algebra
5-3 Triangles
Try This: Example 1C
Find c in the obtuse triangle.
24° + 38° + c = 180°
62° + c = 180°
–62°
–62°
c = 118°
Pre-Algebra
38°
24°
c°
5-3 Triangles
An equilateral triangle has 3
congruent sides and 3 congruent
angles. An isosceles triangle has at
least 2 congruent sides and 2 congruent
angles. A scalene triangle has no
congruent sides and no congruent
angles.
Pre-Algebra
5-3 Triangles
Additional Example 2A: Finding Angles in Equilateral,
Isosceles, and Scalene Triangles
Find angle measures in the equilateral triangle.
3b° = 180°
Triangle Sum Theorem
3b° 180°
=
3
3
Divide both
sides by 3.
b° = 60°
All three angles measure 60°.
Pre-Algebra
5-3 Triangles
Try This: Example 2A
Find angle measures in the isosceles triangle.
39° + t° + t° = 180°
39° + 2t° = 180°
–39°
–39°
Triangle Sum Theorem
Combine like terms.
Subtract 39° from both sides.
2t° = 141°
2t° = 141°
Divide both sides by 2
2
2
39°
t° = 70.5°
The angles labeled t° measure 70.5°.
Pre-Algebra
t°
t°
5-3 Triangles
Additional Example 2B: Finding Angles in Equilateral,
Isosceles, and Scalene Triangles
Find angle measures in the isosceles triangle.
62° + t° + t° = 180°
62° + 2t° = 180°
–62°
–62°
Triangle Sum Theorem
Combine like terms.
Subtract 62° from both sides.
2t° = 118°
2t° = 118°
Divide both sides by 2.
2
2
t° = 59°
The angles labeled t° measure 59°.
Pre-Algebra
5-3 Triangles
Try This: Example 2B
Find angle measures in the scalene triangle.
3x° + 7x° + 10x° = 180° Triangle Sum Theorem
20x° = 180° Combine like terms.
20
20 Divide both sides by 20.
x = 9°
The angle labeled 3x° measures
3(9°) = 27°, the angle labeled 7x°
measures 7(9°) = 63°, and the
angle labeled 10x° measures
3x°
10(9°) = 90°.
Pre-Algebra
10x°
7x°
5-3 Triangles
Additional Example 2C: Finding Angles in Equilateral,
Isosceles, and Scalene Triangles
Find angle measures in the scalene triangle.
2x° + 3x° + 5x° = 180°
10x° = 180°
10
10
Triangle Sum Theorem
Combine like terms.
Divide both sides by 10.
x = 18°
The angle labeled 2x° measures
2(18°) = 36°, the angle labeled
3x° measures 3(18°) = 54°, and
the angle labeled 5x° measures
5(18°) = 90°.
Pre-Algebra
5-3 Triangles
Try This: Example 2C
Find angle measures in the equilateral triangle.
3x° = 180°
Triangle Sum Theorem
3x° 180°
=
3
3
x°
x° = 60°
All three angles measure 60°.
Pre-Algebra
x°
x°
5-3 Triangles
Additional Example 3: Finding Angles in a Triangle
that Meets Given Conditions
The second angle in a triangle is six
times as large as the first. The third
angle is half as large as the second. Find
the angle measures and draw a possible
picture.
Let x° = the first angle measure. Then 6x° =
second angle measure, and 1 (6x°) = 3x° =
2
third angle measure.
Pre-Algebra
5-3 Triangles
Additional Example 3 Continued
Let x° = the first angle measure. Then 6x° =
second angle measure, and 1 (6x°) = 3x° =
2
third angle.
x° + 6x° + 3x° = 180°
10x° = 180°
10
10
x° = 18°
Pre-Algebra
Triangle Sum Theorem
Combine like terms.
Divide both sides by 10.
5-3 Triangles
Additional Example 3 Continued
Let x° = the first angle measure. Then 6x° =
second angle measure, and 1 (6x°) = 3x° =
2
third angle.
x° = 18°
3 • 18° = 54°
6 • 18° = 108°
X° = 18°
Pre-Algebra
The angles measure 18°,
54°, and 108°. The triangle
is an obtuse scalene
triangle.
5-3 Triangles
Lesson Quiz: Part 1
1. Find the missing angle measure in the
acute triangle shown. 38°
2. Find the missing angle measure in the
right triangle shown. 55°
Pre-Algebra
5-3 Triangles
Lesson Quiz: Part 2
3. Find the missing angle measure in an acute
triangle with angle measures of 67° and 63°.
50°
4. Find the missing angle measure in an obtuse
triangle with angle measures of 10° and 15°.
155°
Pre-Algebra