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2-5 Objectives: 1) To make conjectures about angles and determine the validity of the conjectures. 2) To prove and apply theorems about angles. Proving Angles Congruent
2-5 Objectives: 1) To make conjectures about angles and determine the validity of the conjectures.
2) To prove and apply theorems about angles.
Proving Angles Congruent
2-5 Objectives: 1) To make conjectures about angles and determine the validity of the conjectures.
2) To prove and apply theorems about angles.
Conjecture: Vertical angles are congruent.
Proving Angles Congruent
2-5 Proving Angles Congruent
Objectives: 1) To make conjectures about angles and determine the validity of the conjectures.
2) To prove and apply theorems about angles.
A _________________________ theorem
paragraph proof A _______________________________________________________________
_______________________________________________________________
is a statement proven true.
is a convincing argument using deductive reasoning in which statements
and reasons are connected in sentences.
Vocabulary
Here is what the start of many proofs will look like.
2-5 Objectives: 1) To make conjectures about angles and determine the validity of the conjectures.
2) To prove and apply theorems about angles.
Proving Angles Congruent
Some angle pairs that have special names:
are two coplanar angles
with a common side, a common vertex,
and no common interior points.
are two angles whose
sides are opposite rays.
2
4
2
4
2-5 Objectives: 1) To make conjectures about angles and determine the validity of the conjectures.
2) To prove and apply theorems about angles.
Proving Angles Congruent
2 4
the sum of their angle measures is 180°. the sum of their angle measures is 90°.
Each angle is called the complement of the other. Each angle is called the supplement of the other.
2-5 Objectives: 1) To make conjectures about angles and determine the validity of the conjectures.
2) To prove and apply theorems about angles.
Proving Angles Congruent
Find the value of x.
Vertical Angles Theorem
APE (Addition Property of Equality
APE
MPE (Multiplication Property of Equality)
a. Find the measures of the labeled pair of vertical angles.
Refer to the diagram for Example 1.
(4x101)° = 4(52) 101
= 107°
and (2x + 3)° = 2(52) + 3
= 107°
2-5 Objectives: 1) To make conjectures about angles and determine the validity of the conjectures.
2) To prove and apply theorems about angles.
Proving Angles Congruent
Refer to the diagram for Example 1.
b. Find the measures of the other pair of vertical angles.
y° y°
=107°
=107°
y° + 107° = 180°
y° = 73°
c. Check to see that adjacent angles are supplementary.
73° + 107° = 180°
2-5 Objectives: 1) To make conjectures about angles and determine the validity of the conjectures.
2) To prove and apply theorems about angles.
Proving Angles Congruent
2-5 Objectives: 1) To make conjectures about angles and determine the validity of the conjectures.
2) To prove and apply theorems about angles.
Proving Angles Congruent
Write a paragraph proof of Theorem 2-2
using the diagram at the right.
supplementary s
m3 m2
APE (Addition Prop. Of Equality) m2
m3 1 3
Recall the proof of Theorem 2-2. Does the size of the angles in the diagram affect the
proof? Would the proof change if 1 and 3 were acute rather than obtuse? Explain.
No, the size of the angles does not affect the proof of the truth
of the theorem.
2-5 Objectives: 1) To make conjectures about angles and determine the validity of the conjectures.
2) To prove and apply theorems about angles.
Proving Angles Congruent
(refer to Geometry textbook page 112)
Important notes to remember in problem-solving: 1) Set up an equation that uses the given variable
expressions in the problem using any of the following: postulate, theorem, definition justified in the
figure/problem; 2) Solve the equation using algebra; and 3) verify if your answer is reasonable.
2-5 Objectives: 1) To make conjectures about angles and determine the validity of the conjectures.
2) To prove and apply theorems about angles.
Proving Angles Congruent
(refer to Geometry textbook page 113)
a. 90
b. 90
c. substitution
d. m3
2-5 Objectives: 1) To make conjectures about angles and determine the validity of the conjectures.
2) To prove and apply theorems about angles.
Proving Angles Congruent
(refer to Geometry textbook page 113)
10. If 𝒎𝟏 + 𝒎𝟐 = 𝟏𝟖𝟎, 𝒂𝒏𝒅 𝒎𝟐 + 𝒎𝟑 = 𝟏𝟖𝟎, then 𝒎𝟏 + 𝒎𝟐 = 𝒎𝟐 + 𝒎𝟑 by substitution property.
Subtracting 𝒎𝟐 from each side, 𝒎𝟏 = 𝒎𝟑.
Thus 𝟏 ≅ 𝟑 by definition of congruent angles.
11. The two back legs of the chair form vertical angles. The two opposite rays
that make the vertical s also form a linear pair and therefore are
supplementary s. Since each acute angle measures 72, the two obtuse
s will have to measure 108, the supplement of a 72-angle..
72
72
108
2-5 Objectives: 1) To make conjectures about angles and determine the validity of the conjectures.
2) To prove and apply theorems about angles.
Proving Angles Congruent
(refer to Geometry textbook page 113)
16. DOB ≅ AOC and
DOA ≅ BOC since
they are vertical angles.
17. EIG ≅ FIH since all
right s are congruent;
thus EIF ≅ HIG since
they are complements
of the same angle.
18. KPJ ≅ MPJ since they
are marked congruent;
thus KPL ≅ MPL since
they are supplements of
the same congruent angles.
2-5 Objectives: 1) To make conjectures about angles and determine the validity of the conjectures.
2) To prove and apply theorems about angles.
Proving Angles Congruent
(refer to Geometry textbook page 114)
End of NOTES and EXERCISES. Now complete Practice 2-5 Worksheet.
