Transcript Theorems About Perpendicular Lines

Theorems About
Perpendicular Lines
Section 3.2
Objective
• Use theorems about perpendicular lines.
Theorems
• 3.1 -3.4 Perpendicular Line Theorems
Key Vocabulary (Review)
• Complementary Angles – two angles with
measures that have a sum of 90˚.
• Perpendicular Lines – two lines that
intersect to form right angles.
Theorem 3.1
All right angles are congruent.
If m∠A = 90o and m∠B = 90o, then ∠A ≅ ∠B.
B
A
Theorem 3.2
If two lines are perpendicular, then they intersect to
form four right angles
If n ⊥ m, then m∠1 = 90o, m∠2 = 90o, m∠3 = 90o, and
m∠4 = 90o.
1
4
2
3
n
m
Marking Right Angles
• Theorem 3.2 tells us that if one right
angle is marked on a pair of
intersecting lines, then the other three
angles are also right angles.
• Do not need to mark all four right
angles when two lines intersect.
Marking only one right angle is
sufficient.
Example 1
In the diagram, r  s and r  t. Determine whether
enough information is given to conclude that the
statement is true. Explain your reasoning.
a. 3  5
a. Yes, enough information is
given. Both angles are right
angles. By Theorem 3.1, they
are congruent.
Example 1
b. 4  5
b. Yes, enough information is given. Lines r and t are
perpendicular. So, by Theorem 3.2, 4 is a right
angle. By Theorem 3.1, all right angles are
congruent.
c. 2  3
c. Not enough information is given to conclude that
2  3.
Your Turn:
In the diagram, g  e and g  f. Determine whether
enough information is given to conclude that the
statement is true. Explain.
1. 6  10
ANSWER
Yes; 6 and 10 are right
angles and all right angles are
congruent.
2. 7  10
ANSWER
Yes; since g  e, 7 is a right angle; 10 is also a
right angle; all right angles are congruent.
3. 6  8
ANSWER
no
Your Turn:
In the diagram, g  e and g  f. Determine whether
enough information is given to conclude that the
statement is true. Explain.
4. 7  11
ANSWER
Yes; since g  e, 7 is a right
angle; since g  f,
11 is a right angle; all right
angles are congruent.
5. 7  9
ANSWER
no.
6. 6  11
ANSWER
Yes; 6 is a right angle; since g  f, 11 is a right
angle; all right angles are congruent.
Theorem 3.3
If two lines intersect to form adjacent congruent
angles, then the lines are perpendicular.
If
∠1 ≅ ∠2, then AC ⊥ BD.
B
A
1
2
D
C
Theorem 3.4
If two sides of adjacent acute angles are perpendicular, then
the angles are complementary.
If EF ⊥ EH, then
m∠3 + m∠4 = 90o.
F
G
3
4
E
H
Example 2
In the helicopter at the right, are
AXB and CXB right angles?
Explain.
SOLUTION
If two lines intersect to form adjacent
congruent angles, as AC and BD do,
then the lines are perpendicular
(Theorem 3.3). So, AC  BD.
Because AC and BD are perpendicular,
they form four right angles (Theorem 3.2).
So, AXB and CXB are right angles.
Example 3
In the diagram at the right, EF  EH
and mGEH = 30°. Find the value of
y.
SOLUTION
FEG and GEH are adjacent acute angles and
EF  EH. So, FEG and GEH are
complementary (Theorem 3.4).
6y° + 30° = 90°
6y = 60
y = 10
ANSWER
mFEG + mGEH = 90°
Subtract 30 from each side.
Divide each side by 6.
The value of y is 10.
Your Turn:
Find the value of the variable. Explain your reasoning.
1. EFG  HFG
ANSWER
EH and FG intersect to form adjacent
congruent angles, so the lines are
perpendicular. Perpendicular lines
intersect to form 4 right angles, so
mEFG = 90°. 5x = 90; x = 18.
Your Turn:
Find the value of the variable. Explain your reasoning.
2. AB  AD
ANSWER
BAC and CAD are adjacent acute
angles and AB  AD, so BAC and CAD
are complementary. 36° + 9y° = 90°; 9y =
54; y = 6.
Your Turn:
Find the value of the variable. Explain your reasoning.
3. KJ  KL, JKM  MKL
ANSWER
JKM and MKL are adjacent acute
angles and KJ  KL, so JKM and
MKL are complementary. z° + z° = 90°;
2z = 90; z = 45.
PRACTICE PROBLEMS
GV  AE
F
A
5
6
G
B
V
1
2
8 3
C 4
E
AE  FV
F
A
5
6
G
B
V
1
2
8 3
C 4
E
4 1
F
A
5
6
G
B
V
1
2
8 3
C 4
E
3
4
F
A
5
6
G
B
V
1
2
8 3
C 4
E
1  2  90
F
A
5
6
G
B
V
1
2
8 3
C 4
E
GVA is a
right 
F
A
5
6
G
B
V
1
2
8 3
C 4
E
6 & 3 are
supplementary
F
A
5
6
G
B
V
1
2
8 3
C 4
E
6 & 2 are
complementary
F
A
5
6
G
B
V
1
2
8 3
C 4
E
Joke Time
• Why do mother kangaroos hate rainy
days?
• Because the kids have to play inside.
• Why did the chicken go to the middle of
the road?
• To lay it on the line.
• Why did the chicken cross the basketball
court to talk with the ref?
• Because he was calling all fowls!
Assignment
• Section 3.2, pg. 117-120: #1-27 odd, 3143 odd