Download 3.6 Prove Theorems About Perpendicular Lines

3.6 Prove Theorems About Perpendicular Lines
Daily Openers ­ October 23
1. What is the distance between the points (2, 3) and (5, 7)?
2. State the postulate, theorem, or converse that proves m || n.
3. State the postulate, theorem, or converse that proves p || q.
34º
115º
n
34º
m
115º
q
p
4. State whether the lines are perpendicular, parallel, or neither.
Line 1: (­11, 7), (1, 3)
Line 2: (­2, 4), (2, 3)
5. RT bisects ∠PRS. If the m∠PRT = 15x ­ 12 and m∠TRS = 7x + 12 find x, m∠PRT, and m∠TRS.
3.6 Prove Theorems About Perpendicular Lines
3.6 Prove Theorems About Perpendicular Lines
Thm 3.8 If two lines intersect to form a linear pair of congruent angles, then
g
the lines are perpendicular.
If ∠1 ≅ ∠2, then g ⊥ h
1
h
2
Thm 3.9 If two lines are perpendicular, then they intersect to form a
four right angles.
If a ∞ b, then ∠1, ∠2, ∠3, ∠4 are right angles.
Thm 3.10
1
2
3
4
b
If two sides of two adjacent acute angles are perpendicular, then
the angles are complementary.
A
If BA ⊥ BC, then ∠1 and ∠2 are complementary.
1
2
B
C
3.6 Prove Theorems About Perpendicular Lines
Example 1: Prove theorem 3.10
Prove that if two sides of two adjacent acute angles are perpendicular, then the angles are complementary.
Given: ED ⊥ EF
Prove: ∠7 and ∠8 are complementary
D
7
8
Statements
1. ED ⊥ EF
2. ∠DEF is a right angle
3. m∠DEF = 90º
4. m∠7 + m∠8 = m∠DEF
5. m∠7 + m∠8 = 90º
6. ∠7 and ∠8 are compl. ∠s E
F
Reasons
1. Given
2. ⊥ lines intersect to form 4 rt. ∠s
(theorem 3.9)
3. Definition of a right angle
4. Angle Addition Postulate
5. Substitution POE
6. Definition of complementary ∠s
Homework: Day 1 Pages 194 – 197 #1, 3–7, 22–25, 31, 32
3.6 Prove Theorems About Perpendicular Lines
Daily Openers ­ October 26
Name the angles as corresponding, alternate interior, alternate exterior, or consecutive interior angles
1. ∠1 and ∠12
2. ∠4 and ∠8
3. ∠11 and ∠15
4. ∠7 and ∠16
5. ∠2 and ∠9
3
1
2
4
9 11
10 12
5 6
8 7
13 14
15 16
6. Write the equation of the line through point R(­2, 3) that is perpendicular to the line through the points Q(3, 8) and S(1, ­2)
7. Write the equation of the line through point A(5, 4) that is parallel to the line through the points B(­1, 5) and C(2, ­1).
3.6 Prove Theorems About Perpendicular Lines
Write the theorem that justifies the statement
1. ∠1 and ∠2 are right ∠s
2. r ⊥ s
r
1 2
1 2
s
Theorem 3.9
Theorem 3.10
3. ∠1 and ∠2 are complementary.
Theorem 3.8
1
2
3.6 Prove Theorems About Perpendicular Lines
Find m∠1
5.
4.
1
60º
75º
1
75o + m∠1 = 90o (Angle addition Postulate)
m∠1 = 15
6. Find the measure of ∠1 ­ ∠6
4
50º
3
5
45º
2 1
6
90o ­ 60o = 30 o
m∠1 = 30o (Vertical Angles Congruence Theorem)
Since ∠1 and ∠6 form a right angle, then lines a and b are perpendicular lines.
m∠1 = 40o (Vertical ∠4)
m∠2 = 45o (Vertical ∠5)
m∠3 = 45o
(Vertical 45o)
m∠4 = 40o (90o ­ 50o)
m∠5 = 45o (90o ­ 45o)
m∠6 = 50o (Vertical 50o)
3.6 Prove Theorems About Perpendicular Lines
Thm 3.11
Perpendicular Transversal Theorem
If a transversal is perpendicular to one of two parallel lines, then j
k
it is perpendicular to the other.
h
If h || k and j ⊥ h, then j ⊥ k
Thm 3.12
Lines Perpendicular to a Transversal Theorem
In a plane, if two lines are perpendicular to the same line, then
m
n
they are parallel to each other
If m ⊥ n then m || n.
distance from a point to a line :
p
the length of the perpendicular segment from the point to the line
Day 2 Pages 194 – 197 #9–14, 17–21, 29, 30, 33, 34, 40
3.6 Prove Theorems About Perpendicular Lines
7. Is r || s?
r
s
t
m
8. Is m || n?
9. Is r || t?
n