Download Section 4.3

Section 4.3
-A Right Angle Theorem
Michael Smertz
H Geometry- 8
30 May 2008
The Theorem
• In order to prove that lines are
perpendicular, you must first
prove that they form right
angles.
• For this reason, it is necessary
to know the following theorem:
– Theorem 23: If two angles are both
supplementary and congruent, then they are
right angles.
How The Theorem Works
•
•
Most of the problems dealing with
this theorem will be proofs.
Here is how you would use it in a
proof if you were given the diagram
at right. It is given that L1 is
congruent to L2 and you must prove
both angles are right angles.
1. Since L1 and L2 form a straight line,
then they are supplementary.
2. Then, since the angles are
congruent, you know that each must
equal 90°.
3. Therefore, you now know that both
of the angles are supplementary and
congruent. You can now use the
theorem that “If two angles are both
supplementary and congruent, then
they are right angles.”
1 2
NOTE: You can now assume
that whenever two angles
form a straight line, they are
supplementary.
Sample Problems
Solution
• Problem #1
– Given:
1. Ray CD bisects LACB
1.
Given
2.
Given
3.
If a ray bisects
an angle, then it
divides the angle
into 2 congruent
angles
4.
Reflexive
6. LCDA congruent
LCDB
5.
SAS (2.3.4)
6.
CPCTC
7. LCDA and LCDB are
right angles
7.
If two angles are
both
supplementary
and congruent,
then they are
right angles.
• Ray CD bisects LACB
2. AC congruent to CB
• Segment AC is congruent
3. LACD congruent
to segment CB
LBCD
– Prove:
• LCDA and LCDB are
right angles
C
A
4. CD congruent CD
5. ∆ACD congruent
∆BCD
B
D
Sample Problems
• Problem #2
– Given:
• DA congruent to DC
• AB congruent to BC
– Prove:
• DB altitude of AC
A
E
D
C
B
1.
• Solution
DA congruent to
DC
1.
Given
2.
AB congruent to
BC
2.
Given
3.
Reflexive
3.
DB congruent to
DB
4.
SSS (1,2,3)
5.
CPCTC
4.
∆DAB congruent
to ∆DCB
6.
Reflexive
5.
LABE congruent
to LCBE
7.
SAS (2,5,6)
8.
CPCTC
6.
EB congruent to
EB
9.
7.
∆ABE congruent
to ∆CBE
If two angles are
both
supplementary and
congruent, then
they are rt. L’s
8.
LAEB congruent
to LCEB
9.
LAEB and LCEB
rt. L’s
10. DB alt. of AC
10. An altitude of a ∆
forms right angles
with the side to
which it is drawn.
Note: This is a detour problem.
Sample Problems
• Problem #3
• Solution
– If squares A and C are folded
across the dotted segments
onto B, find the area of B that
will not be covered by either
square.
12
2
A
B
C
2
– In order to solve this problem,
you first have to find that the
top part of B is eight. Then,
fold over squares A and C to
get the top part of B to be 4.
Next, you know that the side of
B will be two because A is a
square when it is folded over.
Lastly, you multiply two and
four to find the area of B that
will not covered by either
square. The final answer is
eight.
Practice Problem #1
• Given:
–
P
– AB congruent to BC
• Prove:
– LDBC and LDBA are
right angles
A
B
D
C
Practice Problem #2
M
3x+14
2x+22
O
Is M perpendicular to O?
Justify your answer.
Practice Problem #3
• Given:
– XY congruent to
XZ
– XQ bisects LYXZ
• Prove:
– XQ is
perpendicular to
YZ
X
Y
Q
Z
Practice Problem #4
• A diameter of a circle has endpoints
with coordinates (1,6) and (5,8). Find
the coordinates of the center of the
circle.
(5,8)
(1,6)
Answer Sheet
Practice Problem #1
1. P, AB congruent to BC
1. Given
2. Draw DC
2. Two points determine a segment
3. AD congruent to DC
3. All radii of a circle are congruent
4. DB congruent to DB
4. Reflexive
5. ∆ADB congruent to
∆CDB
5. SSS (1,3,4)
6. LDBA congruent to
LDBC
7. LDBC and LDBA
are right angles
6. CPCTC
7. If two angles are both
supplementary and congruent,
then they are rt. L’s
Answer Sheet
Practice Problem #2
•
YES
3x+14=2x+22
•
•
–
X=8
38=38
This means the angles are congruent.
Theorem #23 states, “If two angles are both
supplementary and congruent, then they are
rt. L’s.” The answer is yes because right
angles are formed by perpendicular lines.
Answer Sheet
Practice Problem #3
1. XY congruent to XZ and XQ
bisects LYXZ
2. LYXQ congruent to LZXQ
3. LY congruent to LZ
4. ∆YXQ congruent to ∆ ZXQ
5. LXQY congruent to LXQZ
6. LXQY and LXQZ are rt. L’s
7. XQ is
perpendicular
to YZ
1. Given
2. If a ray bisects an L, then it
divides the L into 2 congruent L’s
3. If sides, then angles
4. ASA (1,2,3)
5. CPCTC
6. If two angles are both
supplementary and congruent,
then they are rt. L’s
7. Rt. L’s are formed by
perpendicular lines
Answer Sheet
Practice Problem #4
1+5
2
And
6+8
2
Answer: (3,7)
Works Cited
"Chapter 2 Notes." 18 Oct. 2007. 29 May 2008
<home.cvc.org/math/dgeom/Chapter_2_notes/2_8_2.pdf>.
Rhoad, Richard, George Milauskas, and Robert Whipple.
Geometry for Enjoyment and Challenge. New ed. Evanston:
McDougal, Littell & Company, 1997. 180-183.