Transcript Section 2-5: Proving Angles Congruent

Section 2-5: Proving Angles
Congruent
Goal 2.02: Apply properties,
definitions, and theorems of angles
and lines to solve problems and write
proofs.
Essential Questions
1. How are vertical, complementary, and
supplementary angles identified?
2. What are the theorems about angles?
3. How are they applied?
What are adjacent angles?
Two angles side by side with a common vertex
and common side. ( no common interior
points and can’t overlap)
ex. 1
ex. 2
Vertical Angles
a. two angles whose sides form two pairs of
opposite rays.
b. when two lines intersect two pairs of vertical
angles are formed.
Complementary Angles
two angles whose measures have the sum 90.
Each angle is a complement of the other.
x = angle
ex. 1
90 – x = complement
ex. 2
Supplementary Angles
two angles whose measures have the sum 180.
Each angle is a supplement of the other.
x = angle
180 – x = supplement
ex. 1
ex. 2
Reminders
• Complementary and supplementary angles do
not have to be adjacent angles.
• Complementary will always be only 2 angles
whose sum is 90.
• Supplementary must always be 2 angles
whose sum is 180.
Examples p 100 (1-5)
1.
2.
3.
4.
5.
Supplementary to  AOD
Adjacent and congruent to AOE
Supplementary to EOA
Complementary to EOD
A pair of vertical angles
P 100 (10 -18)
10. J = D
11.  JAC = DAC
12. JAE and EAF are adjacent &
supplementary
13. m JCA = m DCA
14. m JCA + m ACD = 180
15. AJ = AD
16. C is the midpoint of JD
17. EAF and JAD are vertical angles
18. AC bisects JAD
Theorems
9. Vertical angles are congruent.
10. If two angles are supplements of the same angle
(or of congruent angles), then the two angles are
congruent.
(Supplements of the same/  angles are .)
11. If two angles are complements of the same
angle (or of congruent angles), then the two
angles are congruent.
(Complements of the same/ angles are .)
12.All right angles are congruent.
13. If two lines are perpendicular, then they
form congruent, adjacent angles.
14. If two lines form congruent, adjacent
angles, then the lines are perpendicular.
15.If the exterior sides of two adjacent acute
angles are perpendicular, then the angles are
complementary.
Together: P 102 (39 – 53 odds, 57, 59)
39.
41.
43.
45. Congruent adjacent complementary angles
47.  A and  B are complementary:
m A = 3x + 12 and B = 2x – 22
49. A is twice as large as its complement, B
51.  A is twice as large as its supplement, B
53. The measure of B, complement of A, if 4
times the measure of C, complement of
 A.
57.
59.
Groups of 2 to 3: Do p 102: 40 – 54 even, 58
Homework
• Worksheet: Practice 2-5
• Assessment
Standardized Test Prep: p 103 (60 – 66)
p 103 Mixed Review (67-74)