GEOMETRY: LOGIC
2-8: Proving Angle Relationships
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Homework
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Today
• Angle Relationships
Angle Addition Postulate
• C is in the interior of < 𝐵𝐴𝐷 if and only if
𝑚 < 𝐵𝐴𝐶 + 𝑚 < 𝐶𝐴𝐷 =< 𝐵𝐴𝐷
Example 1
• Given: 𝑚 < 𝐴𝐵𝐷 = 135
Prove: 𝑥 = 20
You Try!
• Given: 𝑚 < 𝑄𝑆𝑇 = 150
• Prove: 𝑥 = 31
Example 2:
• Given: 𝑚 < 2 = 73, < 𝐴𝐵𝐶 is right
• Prove: 𝑚 < 1 = 17
You Try!
• Given: <ABC is a straight angle
• Prove: q = 22
Congruent Supplements Theorem:
• Angles supplementary to the same angle or to congruent
angles are congruent themselves.
Let’s Actually Prove it!
• Given: < 1 𝑎𝑛𝑑 < 2 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦; < 3 𝑎𝑛𝑑 <
2 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦
• Prove: < 1 ≅ < 3
Example 3:
• Given: <5 ≅ <6
• Prove: <4 and <6 are supplementary
5
4
6
Congruent Complements Theorem
• Angles complementary to the same angle or to congruent
angles are congruent themselves.
You Try!
Theorems:
• Vertical Angles Theorem: If two angles are vertical angles,
then they are congruent.
• Ex: < 1 ≅< 3 𝑎𝑛𝑑 < 2 ≅< 4
Example 4:
• Given: Picture below
• Prove: 𝑚 < 𝐴𝐸𝐷 = 110°
Example 5:
B
• Given:𝐷𝐵 bisects <ADC
• Prove: <2 ≅ <3
A
1
2
D
3
C
Perpendicular Line Theorems
• Perpendicular lines intersect to form four what type angles
• All right angles are ________________ (I.E. Congruent)
Theorems
• Perpendicular lines intersect to form four right angles.
• All right angles are congruent
Theorems Continued
• If two angles are BOTH congruent and supplementary,
then each angle is ___________________
• If two congruent angles form a linear pair, then they are
___________________
Theorems Continued
• If two angles are congruent and supplementary, then each
angle is a right angle.
• If two congruent angles form a linear pair, then they are
right angles.
Example 6
• Given: 𝐴𝐵 ⊥ 𝐵𝐶
• Prove: <1 and <2 are complementary angles
You Try!
• Given: 𝐽𝑀 ⊥ 𝐿𝑂; 𝑚 < 𝑀𝑃𝑁 = 50°
• Prove: 𝑚 < 𝐾𝑃𝐿 = 40°
Practice Problems
• Try some on your own!
• As always call me over if you are confused!
Exit Ticket
• Given: <4 ≅ <7
• Prove: <5 ≅ <6
5
4
6
7