Apply Theorem 5.9 from Section 5.3, and remember that if A> B then A=B+x for some x> 0.
Statements
Reasons
1.
PR≅ PQ
1.
Given
2.
∠ PRQ ≅ ∠ PQR
2.
Isosceles Triangle Theorem
3.
m∠ PRQ = m∠ PQR
3.
Definition of Congruent Angles
4.
m∠ PRQ = m∠ 1 + m∠ 4 and m∠ PQR = m∠ 2 + m∠ 3
4.
Angle Addition Postulate
5.
m∠ 1 + m∠ 4 = m∠ 2 + m∠ 3
5.
Substitution
6.
SQ > SR
6.
Given
7.
m∠ 4 > m∠ 3
7.
Theorem 5.9
8.
m∠ 4 = m∠ 3 + x
8.
Definition of inequality
9.
m∠ 1 = m∠ 2 - x
9.
Subtracting equation in 8 from equation in 5
10.
m∠ 1 + x = m∠ 2
10.
Solving for m∠ 2
11.
m∠ 1 < m∠ 2
11.
Definition of inequality
Practice makes perfect
Let's begin by highlighting the given information in the diagram.
By applying the Isosceles Triangle Theorem we obtain that ∠ PRQ ≅ ∠ PQR, which implies that m∠ PRQ = m∠ PQR. Also, by the Angle Addition Postulate we can write the following two equations.
m∠ PRQ = m ∠ 1 + m ∠ 4
m∠ PQR = m ∠ 2 + m ∠ 3
Since m∠ PRQ = m∠ PQR we can equate both equations above.
m ∠ 1 + m ∠ 4 = m ∠ 2 + m ∠ 3
Remember that SQ > SR, which implies that m ∠ 4 > m ∠ 3 (Theorem 5.9). By the definition of inequality, we can write the equation below.
m ∠ 4 = m ∠ 3 + x , for somex > 0
Next, we will subtract the latter two equations.
m ∠ 1 + m ∠ 4 &= m ∠ 2 + m ∠ 3
^- m ∠ 4 &= m ∠ 3 + x
m ∠ 1 &= m ∠ 2 - x
From the resulting equation we obtain that m ∠ 1 + x = m ∠ 2, and by definition of inequality it means that m ∠ 1 < m ∠ 2.
Two-Column Proof Table
In the table below, we will summarize the proof we just did before.
