Pre-Algebra Homework

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NEW!Student

Learning Goal Chart

Lesson Reflection for

Chapter 5

Pre-Algebra Learning GoalStudents 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.

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 Important Notes

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.

3. Name a right angle in the figure.

4. Name a pair of complementary angles.

5. If m1 47°, then find m3.

2. Name two lines in the figure.

Possible answer: A, B, and C

Possible answer: AGF

Possible answer: 1 and 2

47°

Possible answer: AD and BE

5-2 Parallel and Perpendicular Lines

Learning Goal Assignment

Learn to identify parallel and perpendicular lines and the angles formed by a transversal.

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 PROPERTIES OF TRANSVERSALS TO PARALLEL LINES 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 Important Notes

5-2 FAST TRACK Quiz

In the figure a || b.

1. Name the angles congruent to 3.

2. Name all the angles supplementary to 6.

3. If m1 = 105° what is m3?

4. If m5 = 120° what is m2?

1, 5, 7

1, 3, 5, 7

105°

60°

5-3 TrianglesLearning Goal Assignment

Learn to find unknown angles in triangles.

Pre-Algebra

5-3 Triangles

VocabularyTriangle 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 three angles in the triangle can be arranged to form a straight line or 180°.

The sides of the triangle are transversals to the parallel lines.

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°

P = 63°

–117° –117°

Pre-Algebra

5-3 Triangles

Try This: Example 1A

Find a in the acute triangle.

88° + 38° + a = 180°

126° + a = 180°

a = 54°

–126° –126°

88°

38°

a°

Pre-Algebra

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°

c = 48°

–132° –132°

Pre-Algebra

5-3 Triangles

Find b in the right triangle.

38° + 90° + b = 180°

128° + b = 180°

b = 52°

–128° –128°

38°

b°

Try This: Example 1B

Pre-Algebra

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°

m = 95°

–85° –85°

Pre-Algebra

5-3 Triangles

Find c in the obtuse triangle.

24° + 38° + c = 180°

62° + c = 180°

c = 118°

–62° –62° c°24°

38°

Try This: Example 1C

Pre-Algebra

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°

b° = 60°

3b° 180°3 3

=

Triangle Sum Theorem

All three angles measure 60°.

Divide both sides by 3.

Pre-Algebra

5-3 Triangles

Try This: Example 2A

39° + t° + t° = 180°39° + 2t° = 180°

2t° = 141°

–39° –39°

Find angle measures in the isosceles triangle.

2t° = 141°2 2

t° = 70.5°

Triangle Sum TheoremCombine like terms.Subtract 39° from both sides.

Divide both sides by 2

t°t°

39°

The angles labeled t° measure 70.5°.

Pre-Algebra

5-3 Triangles

Additional Example 2B: Finding Angles in Equilateral, Isosceles, and Scalene Triangles

62° + t° + t° = 180°62° + 2t° = 180°

2t° = 118°

–62° –62°

Find angle measures in the isosceles triangle.

2t° = 118°2 2

t° = 59°

Triangle Sum TheoremCombine like terms.Subtract 62° from both sides.

Divide both sides by 2.

The angles labeled t° measure 59°.

Pre-Algebra

5-3 Triangles

3x° + 7x° + 10x° = 180°

20x° = 180°

x = 9°

20 20

Find angle measures in the scalene triangle.

Triangle Sum Theorem

Combine like terms.Divide both sides by 20.

3x° 7x°

10x°

Try This: Example 2B

The angle labeled 3x° measures 3(9°) = 27°, the angle labeled 7x° measures 7(9°) = 63°, and the angle labeled 10x° measures 10(9°) = 90°.

Pre-Algebra

5-3 Triangles

Additional Example 2C: Finding Angles in Equilateral, Isosceles, and Scalene Triangles

2x° + 3x° + 5x° = 180°

10x° = 180°

x = 18°

10 10

Find angle measures in the scalene triangle.

Triangle Sum Theorem

Combine like terms.Divide both sides by 10.

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

Find angle measures in the equilateral triangle.

3x° = 180°

x° = 60°

3x° 180°3 3

=

Triangle Sum Theorem

All three angles measure 60°.

Try This: Example 2C

x° x°

x°

Pre-Algebra

5-3 Triangles

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 (6x°) = 3x° =

third angle measure.

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Additional Example 3: Finding Angles in a Triangle that Meets Given Conditions

Pre-Algebra

5-3 Triangles

Additional Example 3 Continued

x° + 6x° + 3x° = 180°

10x° = 180° 10 10

x° = 18°

Triangle Sum Theorem

Combine like terms.Divide both sides by 10.

Let x° = the first angle measure. Then 6x° = second angle measure, and (6x°) = 3x° = third angle.

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Pre-Algebra

5-3 Triangles

X° = 18°

x° = 18°

6 • 18° = 108°3 • 18° = 54°

The angles measure 18°, 54°, and 108°. The triangle is an obtuse scalene triangle.

Additional Example 3 Continued

Let x° = the first angle measure. Then 6x° = second angle measure, and (6x°) = 3x° = third angle.

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Pre-Algebra

5-3 Triangles

Lesson Quiz: Part 1

1. Find the missing angle measure in the acute triangle shown.

2. Find the missing angle measure in the right triangle shown.

38°

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°.

4. Find the missing angle measure in an obtuse triangle with angle measures of 10° and 15°.

50°

155°