Name ________________________________________ Date __________________ Class __________________
SKILL
43
Angle Relationships
Example 1
Adjacent angles are pairs of angles
that share a common vertex and a
common side.
Vertical angles are opposite
angles formed by the intersection
of two lines. Vertical angles are
congruent.
Vocabulary
Adjacent angles
Vertical angles
Congruent
Complementary
angles
Supplementary
angles
Adjacent angles:
∠1 and ∠2,
∠2 and ∠3,
∠3 and ∠4,
∠4 and ∠1
Vertical angles: ∠1 and ∠3,
∠2 and ∠4
Also, ∠1 ≅ ∠3 and ∠2 ≅ ∠4.
Check
Classify each pair of figures as adjacent or vertical angles.
1. ∠1 and ∠2 ______________
2. ∠1 and ∠3 ______________
3. ∠3 and ∠4 ______________
4. ∠2 and ∠4 ______________
5. ∠2 and ∠3 ______________
6. ∠1 and ∠4 ______________
7. If m∠4 = 80°, find m∠2. ______________
8. If m∠3 = 100°, find m∠1. ______________
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Example 2
The sum of the measures of complementary
angles is 90°.
m∠5
The sum of the measures of
supplementary angles is 180°.
m∠6 = 90°
m∠7
m∠8 = 180°
Check
Use the figure. Describe each pair of angles as complementary,
supplementary, or neither.
9. ∠1 and ∠5 ______________
10. ∠1 and ∠2 ______________
11. ∠4 and ∠5 ______________
Find the measure of x in each figure.
12.
13.
___________________________________
__________________________________
14. Explain the Error. Find the correct solution.
Each angle has a measure of 90°, so the
angles are complementary.
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Example 3
The measure of ∠3 is 23°. Find the measure of ∠1.
∠5 and ∠3 are vertical angles, so ∠5 ≅ ∠3.
∠5 ≅ ∠3
m∠5 = 23°
m∠3 = 23°
∠1 and ∠5 are complementary angles, so m∠1
m∠1
m∠1
m∠5 = 90°.
m∠5 = 90°
23° = 90°
m∠1 = 90°
m∠5 = 23°
23°
m∠1 = 67°
Subtract 23° from each side.
Simplify.
Check
The measure of ∠6 is 72°. Find the measure of each angle. Explain
your answer.
15. m∠1
_________________
16. m∠2
_________________
17. m∠3
_________________
18. m∠4
_________________
19. m∠5
_________________
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20. ∠P and ∠Q are adjacent angles. ∠P and ∠R are vertical angles. The
measure of ∠Q is 36° and the measure of ∠R is 121°. What is the
measure of ∠P?
_________________________________________________________________________________________
21. Write an equation to represent the sum of the measures of ∠ACB,
∠BCD, and ∠ECD. Then find m∠BCD.
_________________________________________________________________________________________
22. ∠A and ∠B are complementary angles. The measure of ∠A is (3x)°
and the measure of ∠B is 45°. What is the value of x?
_________________________________________________________________________________________
23. Two vertical angles are supplementary. What is the measure of each
angle?
_________________________________________________________________________________________
24. Angles S and T are complementary angles. If m∠S is x° and m∠T is
(x 10)°, find the measure of each angle.
_________________________________________________________________________________________
SKILL 43 ANGLE RELATIONSHIPS
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SKILL
49
Parallel Lines Cut by a Transversal
Example 1
A transversal is a line that intersects two or more lines in a
plane. In the figure, line t is a transversal that intersects line
m and line n.
Vocabulary
Transversal
Same-side
interior
angles
Same-side
exterior
angles
Alternate
interior
angles
Certain types of angle pairs are formed when a transversal
intersects two lines.
Same-side interior angles lie on the same side of the transversal
and inside the intersected lines. Examples: ∠3 and ∠6, ∠4 and ∠5
Same-side exterior angles lie on the same side of the transversal and
outside the intersected lines. Examples: ∠1 and ∠8, ∠2 and ∠7
Alternate
exterior
angles
Corresponding
angles
Parallel lines
Alternate interior angles lie on opposite sides of the transversal and
inside the intersected lines. Examples: ∠4 and ∠6, ∠3 and ∠5
Alternate exterior angles lie on opposite sides of the transversal
and outside the intersected lines. Examples: ∠1 and ∠7, ∠2
and ∠8
Corresponding angles lie on the same side of the transversal and
are in the same position in relation to the two intersected lines.
Examples: ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8
Check
Use the figure. Classify each pair of angles as same-side interior,
same-side exterior, alternate interior, alternate exterior, or
corresponding angles.
1. ∠1 and ∠7 __________________________________
2. ∠2 and ∠6 __________________________________
3. ∠4 and ∠8 __________________________________
4. ∠3 and ∠4 __________________________________
5. ∠6 and ∠8 __________________________________
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Example 2
Parallel lines lie in the same plane and never intersect. In the figure, line
p is parallel to line q.
Line p is parallel to line q is
written as p q.
The arrows on the line are
used to indicate parallel
li
When parallel lines are cut by a transversal, the following angle pairs
are congruent.
Alternate interior angles: ∠4 ≅ ∠6, ∠3 ≅ ∠5
Alternate exterior angles: ∠1 ≅ ∠7, ∠2 ≅ ∠8
Corresponding angles: ∠1 ≅ ∠5, ∠2 ≅ ∠6, ∠3 ≅ ∠7, ∠4 ≅ ∠8
In the figure above, m∠1 = 110°. What other angles have a measure of 110°?
Since ∠1 and ∠5 are corresponding angles, m∠5 = 110°.
Since ∠1 and ∠7 are alternate exterior angles, m∠7 = 110°.
Since ∠7 and ∠3 are corresponding angles, m∠3 = 110°.
In the figure, m∠2 = 70°. What other angles have a measure of 70°?
Since ∠2 and ∠6 are corresponding angles, m∠6 = 70°.
Since ∠2 and ∠8 are alternate exterior angles, m∠8 = 70°.
Since ∠8 and ∠4 are corresponding angles, m∠4 = 70°.
Check
In the figure, a b, m∠1 = 82° and m∠2 = 98°. Find each angle
measure. Explain your reasoning.
6. m∠3 _______________________________________________
7. m∠4 _______________________________________________
8. m∠5 _______________________________________________
9. m∠6 _______________________________________________
10. Explain the Error. Find the correct solution.
In the figure, line t is a transversal and m∠4 = 65°.
Since ∠4 and ∠8 are corresponding angles, ∠4 ≅ ∠8,
so m∠8 = 65°.
____________________________________________________
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Example 3
When parallel lines are cut by a transversal, the angles formed are either
congruent or supplementary. If you know the measure of one of the
angles, you can find the measures of all of the angles.
In the figure, m∠1 = 96°.
∠1 and ∠2 are supplementary. So, m∠2 = 84°
∠1 and ∠3 are vertical angles. So, m∠3 = 96°.
∠1 and ∠4 are supplementary. So, m∠4 = 84°.
You can find the measures of ∠5, ∠6, ∠7, and ∠8 by identifying
corresponding, alternate interior, alternate exterior, or supplementary
angles.
Check
In the figure, f g, m∠1 = 112° and m∠16 = 78°. Find each angle measure.
11. m∠2
________________________
13. m∠4
________________________
15. m∠6
________________________
17. m∠8
________________________
19. m∠10
________________________
21. m∠12
________________________
23. m∠14
________________________
12. m∠3
_______________________
14. m∠5
_______________________
16. m∠7
_______________________
18. m∠9
_______________________
20. m∠11
_______________________
22. m∠13
_______________________
24. m∠15
_______________________
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25. Find the values for x and y. Explain your reasoning.
_________________________________________________________________________________________
Use the following information for 26–27. The figure below shows the
rungs of a ladder. In the figure, AB CD EF .
26. Is ∠ACD ≅ ∠AEF? Explain your reasoning.
_________________________________________________________________________________________
27. Is ∠ACD ≅ ∠ABD? Explain your reasoning.
_________________________________________________________________________________________
28. In the figure m n and r t. The lines intersect to form a parallelogram.
What are the angle measures of the interior angles of the
parallelogram?
_________________________________________________________________________________________
SKILL 49 PARALLEL LINES CUT BY A TRANSVERSAL
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SKILL
50
Parallelograms
Example 1
A parallelogram is a special type of quadrilateral. One property of
parallelograms is that its opposite sides are congruent and parallel.
In parallelogram ABCD, AB CD, BC AD, AB ≅ CD and AD ≅ BC.
Vocabulary
Parallelogram
Quadrilateral
Consecutive
angles
Diagonal
Bisect
Find the values for x and y.
AB and CD are opposite sides, so AB ≅ CD .
AB = CD
6=x
Write an equation.
AB = 6, CD = x
AD and BC are opposite sides, so AB ≅ CD.
AD = BC
2y = 5
y = 2.5
Write an equation.
AD = 2y, BC = 5
Divide each side by 2.
So, x = 6 and y = 2.5.
Check
Find the value of x for each parallelogram.
1.
2.
________________________________________
________________________________________
3. Find the perimeter of parallelogram JKLM, if KL = 8 meters and
LM = 6 meters.
_________________________________________________________________________________________
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4. Explain the Error. Find the correct solution.
In parallelogram DEFG, DE FG and EF DG.
If DE = 11 feet and DG = 14 feet, then EF = 11 feet and FG = 14 feet.
_________________________________________________________________________________________
5. PQ and RS are opposite sides of a parallelogram. If PQ = 84 inches
and RS = 6x, what is the value of x?
_________________________________________________________________________________________
Example 2
Another property of parallelograms is that their opposite angles are congruent.
In parallelogram ABCD, ∠A ≅ ∠C and ∠B ≅ ∠D.
If m∠A = 105°, find m∠C.
∠A and ∠C are opposite angles, so ∠A ≅ ∠C.
m∠A = m∠C
Write an equation.
105° = m∠C
m∠A = 105°
If m∠B = (5x
35)° and m∠D = 75, find the value of x.
∠B and ∠D are opposite angles, so ∠B ≅ ∠D.
m∠B = m∠D
5x
Write an equation.
35 = 75
m∠B = 5x
5x = 40
Subtract 35 from each side.
x=8
35 and m∠D = 75
Divide each side by 5.
Use x = 8 to check that m∠B = 75°.
m∠B = 5x
m∠B = 5(8)
35
35
m∠B = 75
So, m∠A = m∠C = 105° and m∠B = m∠D = 75°.
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Check
For 6–9, use parallelogram ABCD. Find each measure.
6. m∠C ___________________________
7. m∠D ___________________________
8. AB
___________________________
9. DA
___________________________
Find the value of x in each parallelogram.
10.
11.
________________________________________
________________________________________
________________________________________
________________________________________
12. Find the values of x and y in parallelogram DEFG.
_________________________________________________________________________________________
For 13 and 14, use parallelogram PQRS.
13. If m∠P = (2x
15)°, m∠Q = 117°, and m∠R = 63°, find the value for x.
_________________________________________________________________________________________
14. If PQ = (5y
6) feet, RS = 19 feet, and SP = 24 feet, find the value for y.
_________________________________________________________________________________________
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Example 3
Consecutive angles of a parallelogram are supplementary. Two interior angles of a
parallelogram are consecutive if they have a common side.
In the parallelogram ABCD, the following pairs of angles are supplementary.
∠A and ∠B, ∠B and ∠C, ∠C and ∠D, ∠D and ∠A
So, m∠A
m∠B = 180, m∠B
m∠C = 180, m∠C
m∠D = 180, and m∠D
m∠A = 180.
If m∠A = 98°, find the measures of the remaining angles.
∠A and ∠C are opposite angles, so ∠A ≅ ∠C.
m∠A = m∠C = 98°
∠A and ∠B are consecutive angles. The sum of consecutive angles is 180°.
m∠A
m∠B = 180
Write an equation.
98
m∠B = 180
m∠A = 82
m∠B = 82
Subtract 82 from each side.
∠B and ∠D are opposite angles, so ∠B ≅ ∠D.
m∠B = m∠D = 82°
So, m∠A = 98°, m∠B = 82°, m∠C = 98° and m∠D = 82°.
Check
Use the given angle to name two pairs of consecutive angles.
15. ∠S ___________________________
16. ∠T ___________________________
17. ∠U ___________________________
18. ∠V ___________________________
19. Two angles of a parallelogram measure 57°. What is the measure of
the other two angles?
_________________________________________________________________________________________
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Find the unknown angle measures in each parallelogram.
20.
21.
___________________________
___________________________
22. Find the value of x. Then find the measure of each angle in the
parallelogram.
_________________________________________________________________________________________
Example 4
Diagonals of a parallelogram bisect each other. In the parallelogram ABCD below,
E is the midpoint of AC and BD. So, AE ≅ EC and BE ≅ ED.
Suppose AE = 5x
14 and EC = 3x
AE = EC
10. Solve for x.
Write an equation.
5x
14 = 3x
2x
14 = 10
Subtract 3x from each side.
2x = 24
Add 14 to each side.
10
x = 12
AE = 5x
14 and EC = 3x
10
Divide each side by 2.
Find the length of AE and EC.
Replace x with 12 in the equations AE = 5x
AE = 5(12)
14 or 46
EC = 3(12)
14 and EC = 3x
10.
10 or 46
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Check
For 23 and 24, use parallelogram GHIJ.
23. Find the values for x and y.
_________________________________________________________________________________________
24. What is the length of each diagonal?
_________________________________________________________________________________________
For 25–27, use the following information.
In parallelogram CDEF,
FG = 2x 4, FD = 20, CG = 3y 2, and GE = 5y
6.
25. Find the values for x and y.
_________________________________________________________________________________________
26. What is the length of GD ?
_________________________________________________________________________________________
27. What is the length of CE ?
_________________________________________________________________________________________
SKILL 50 PARALLELOGRAMS
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SKILL
58
Writing Equations of Parallel, Perpendicular,
Vertical and Horizontal Lines
Example 1
Vocabulary
Horizontal
Lines that do not intersect are parallel lines. Parallel lines always have
the same slope.
Parallel
A line is given by the equation y = 3x
going through point (1, 2).
Vertical
4. Write a line parallel to this line
Step 1:
Identify the needed slope.
m=3
Step 2:
Substitute the slope and
coordinates of the point into
y = mx b. Then solve for b.
y = mx b
2 = 3(1) b
2=3 b
3=3 3 b
5=b
2
Step 3:
y = 3x
Now write the equation of the
parallel line.
Perpendicular
5
Check
Write the equation of the line that is parallel to the given line and goes
through the given point.
1. Line given by y = 4x
5 through the point ( 3, 0)
___________________________________
2. Line given by y = 2x
4 through the point (1, 3)
___________________________________
3. Line given by y =
1
x
2
2 through the point (8, 7)
___________________________________
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Example 2
Lines that intersect at right angles are called perpendicular lines.
Perpendicular lines have slopes that are negative reciprocals.
A line is given by the equation y = 2x
this line going through point (2, 4).
5. Write a line perpendicular to
1
2
Step 1:
Identify the needed slope.
m=
Step 2:
Substitute the slope and coordinates of the
point into y = mx b. Then solve for b.
y = mx b
1
4=
2 b
2
4= 1 b
3=b
Step 3:
Now write the equation of the parallel line.
y=
1
x 3
2
Check
Write the equation of the line that is perpendicular to the given line
and goes through the given point.
4. Line given by y =
1
x
2
4 through the point ( 2, 9)
___________________________________
5. Line given by y = x
5 through the point (4, 2)
___________________________________
6. Line given by y = 3x
9 through the point (6, 4)
___________________________________
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Example 3
Different vertical lines are always parallel to each other. Different
horizontal lines are always parallel to each other.
A vertical line and a horizontal line are always perpendicular to each other.
Name a line parallel to y = 3
that passes through the
point (4, 2).
Name a line parallel to
x = 2 that passes through
the point ( 3, 1).
Name a line perpendicular to
the line y = 3 that passes
through the point ( 2, 1).
Choose the y-coordinate of
the ordered pair to create
the parallel line.
Choose the x-coordinate of
the ordered pair to create
the parallel line.
The line given by y = 3 is
horizontal. Choose the
x-coordinate to create the
perpendicular line.
y= 2
x=2
x= 2
Check
Write the equation of the line that is parallel to the given line and goes
through the given point.
7. Line given by y = 6 through the point ( 3, 0)
___________________________________
8. Line given by x = 1 through the point (4, 5)
___________________________________
Write the equation of the line that is perpendicular to the given line
and goes through the given point.
9. Line given by y = 0 through the point (7, 8)
___________________________________
10. Line given by x = 3 through the point (4, 1)
___________________________________
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11. Find the Error. Explain the solution.
Question: Write the equation of the line
1
that is perpendicular to y = x 2 and
3
goes through (6, 1).
_____________________________________________
_____________________________________________
1
so the
3
perpendicular slope is 3. Substituting
gives 1 = 3(6) b or b = 17. The
equation is y = 3x 17.
_____________________________________________
Answer: The slope is
_____________________________________________
12. Write an equation that is parallel to y = 5x
and goes through (10, 0).
4
___________________________________
13. Write an equation that is perpendicular to y = 5x
and goes through (1, 0).
4
___________________________________
14. A company rents movies that can be streamed online or downloaded
to a computer. The company charges a one-time sign-up fee and a
charge for each movie x. The equation to represent this is y = 5x 20.
What is the sign-up fee and cost per movie?
___________________________________
15. The owner is advertising a special in which the cost per movie stays
the same but the sign-up fee decreases. For example, a customer will
pay a total of $13 to rent two movies.
A Write an equation that is parallel to y = 5x
and goes through point (2, 13).
B What is the new sign-up fee?
20
___________________________________
SKILL 58 WRITING EQUATIONS OF PARALLEL, PERPENDICULAR,
VERTICAL, AND HORIZONTAL LINES
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