Let's begin by reviewing the idea of a flow proof. Arrows show the logical connections between the statements and reasons are written below the statements. We can begin our proof by using the given information, ∠ 3 ≅ ∠ 5.
Statement1
∠ 3 ≅ ∠ 5
Given
By the definition of congruent angles, we can conclude that the measures of these angles are equal.
Statement2
m ∠ 3 = m ∠ 5
Definition of congruent angles
Next, we can use the Triangle Angle-Sum Theorem. This states that the sum of the measures of the angles in each triangle is 180. Therefore, we can write that the sum of the angles also equal 180.
Statement3
m ∠ 1 + m ∠ 3 + m ∠ 3 = 180
m ∠ 5+ m ∠ 6 + m ∠ 7 = 180
Triangle Angle-Sum Theorem
Continuing, we can use the Transitive Property of Equality because both equations equal the same number, which in this case is 180. Write the equation so that the left-hand sides of each equations are equal.
Statement4
m ∠ 1 + m ∠ 2 + m ∠ 3= m ∠ 5 + m ∠ 6+ m ∠ 7
Transitive Property of Equality
Notice that we can substitute m ∠ 5 for m ∠ 3 in our equation.
Statement5
m ∠ 1 + m ∠ 2 + m ∠ 5= m ∠ 5 + m ∠ 6+ m ∠ 7
Substitution Property of Equality
Now, we can subtract m ∠ 5 from both sides. This gives us m ∠ 1 + m ∠ 2 = m ∠ 6 + m ∠ 7, which is what we wanted to prove.
Statement6
m ∠ 1 + m ∠ 2 = m ∠ 6+ m ∠ 7
Subtraction Property of Equality
Final Proof
&Given:∠ 3 ≅ ∠ 5
&Prove: m ∠ 1+ m ∠ 2= m ∠ 6 + m ∠ 7
Proof:
