Exercise 33 Page 306

Recall that supplementary angles are two angles whose measures have a sum of 180. Therefore, we can conclude that the measures of ∠ 1 and ∠ 2 sum to 180, and that the measures of ∠ 3 and ∠ 4 sum to 180. Statement 2) & m ∠ 1+ m ∠ 2 = 180 & m ∠ 3+ m ∠ 4 = 180 Reason 2) & Definition of supplementary & angles Notice that by the Transitive Property of Equality we get m ∠ 1 + m ∠ 2 =m ∠ 3 + m ∠ 4. Statement 3) & m ∠ 1 + m ∠ 2 =m ∠ 3 + m ∠ 4 Reason 3) & Transitive Prop. of Equality Next, we are given that ∠ 2 ≅ ∠ 3. By the definition of congruent angles they have the same measures, m ∠ 2 = m ∠ 3. Statement 4) & m ∠ 2 = m ∠ 3 Reason 4) & Definition of congruent angles Having this information, we can use the Substitution Property of Equality and substitute m ∠ 3 for m ∠ 2 in our equation. Statement 5) & m ∠ 1 + m ∠ 3 =m ∠ 3 + m ∠ 4 Reason 5) & Substitution Prop. of Equality By the Subtraction Property of Equality, we can subtract m ∠ 3 from both sides of the equation. This will give us m ∠ 1 =m ∠ 4. Statement 6) & m ∠ 1 =m ∠ 4 Reason 6) & Subtraction Prop. of Equality Angles with the same measure are congruent. Therefore, by the definition of congruent angles, ∠ 1 ≅ ∠ 4. This is what we wanted to prove! Statement 7) & ∠ 1 ≅ ∠ 4 Reason 7) & Definition of congruent angles

Completed Proof

Congruent Complements Theorem

Let's begin by looking at the given information and the desired outcome of the proof. Given:& ∠ 1 and ∠ 2 are complementary. & ∠ 3 and∠ 4 are complementary. & ∠ 2 ≅ ∠ 4 Prove:& ∠1 ≅ ∠3 We are given that ∠ 1 and ∠ 2 are complementary, ∠ 3 and ∠ 4 are complementary, and that ∠ 2 ≅ ∠ 4. This is how we will begin our proof! Statement 1)& ∠ 1 and ∠ 2 are complementary, & ∠ 3 and∠ 4 are complementary. & ∠ 2 ≅ ∠ 4 Reason 1) & Given Recall that complementary angles are two angles whose measures have a sum of 90. Therefore, we can conclude that the measures of ∠ 1 and ∠ 2 sum to 90 and that the measures of ∠ 3 and ∠ 4 sum to 90. Statement 2) & m ∠ 1+ m ∠ 2 = 90 & m ∠ 3+ m ∠ 4 = 90 Reason 2) & Definition of complementary & angles Notice that by the Transitive Property of Equality, we get m ∠ 1 + m ∠ 2 =m ∠ 3 + m ∠ 4. Statement 3) & m ∠ 1 + m ∠ 2 =m ∠ 3 + m ∠ 4 Reason 3) & Transitive Prop. of Equality Next, we are given that ∠ 2 ≅ ∠ 4. By the definition of congruent angles, they have the same measures, m ∠ 2 = m ∠ 4. Statement 4) & m ∠ 2 = m ∠ 4 Reason 4) & Definition of congruent angles Having this, we can use the Substitution Property of Equality and substitute m ∠ 4 for m ∠ 2 in our equation. Statement 5) & m ∠ 1 + m ∠ 4 =m ∠ 3 + m ∠ 4 Reason 5) & Substitution Prop. of Equality By the Subtraction Property of Equality, we can subtract m ∠ 4 from both sides of the equation. This will give us m ∠ 1 =m ∠ 3. Statement 6) & m ∠ 1 =m ∠ 3 Reason 6) & Subtraction Prop. of Equality Angles with the same measure are congruent. Therefore, by the definition of congruent angles, ∠ 1 ≅ ∠ 3. This is what we wanted to prove! Statement 7) & ∠ 1 ≅ ∠ 3 Reason 7) & Definition of congruent angles

Completed Proof