Give step-by-step solution with explanation and final answer:
37,3812:28 ow TED
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Proving Statements about Segments and Angles
Learning Target: Prove statements about segments and

Question

Give step-by-step solution with explanation and final answer:
37,3812:28 ow TED
X IMG_0486.jpeg ui
Proving Statements about Segments and Angles
Learning Target: Prove statements about segments and angles.
Name the property that the statement illustrates.
32. If ZDEF ZJKL, then ZJKL = ZDEF
33. LC ZC
34. If MN = PQ and PQ = RS, then MN = RS
35. Write a two-column proof for the Reflexive Property of Angle Congruence
36. A ramp at a skate park is constructed
with ZBAD = ZCDA, as pictured B Cc
Prove that ZEAB = ZFDC.
—
E A D
Proving Geometric Relationships
Learning Target: Prove geometric relationships.
37. Complete the flowchart proof. Then write a two-column proof
Given 3 and £2 are complementary
mZl + mZ£2 = 90°
Prove /3=/1
m2 + mz2 = 90 Zl and £2 are complementary £3 and £2 are complemen
38. Write a paragraph proof
A
Given mZABC = 48°, mZABD = 24 BD
Prove BD bisects ZABC.
B G
Mathematicoteiiatticttm—

VIPSolutions · Accepted Answer

# Solutions for Questions 37 and 38

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## 37. Prove: ∠3 ≅ ∠1 (Two- Proof)

**Given:**  
∠1 and ∠2 are complementary  
$ m∠1 + m∠2 = 90^\circ $  
Prove: ∠3 ≅ ∠1

**Flowchart Proof Explanation:**
- $ m∠1 + m∠2 = 90^\circ $ (Given)
- ∠1 and ∠2 are complementary (Definition of complementary angles)
- ∠3 and ∠2 are complementary (Given)
- $ m∠3 + m∠2 = 90^\circ $ (Definition of complementary angles)
- $ m∠1 + m∠2 = m∠3 + m∠2 $ (Transitive Property)
- $ m∠1 = m∠3 $ (Subtraction Property)
- ∠1 ≅ ∠3 (Definition of congruent angles)

**Two-Column Proof:**

| Statement                                | Reason                                    |
|-------------------------------------------|-------------------------------------------|
| 1. ∠1 and ∠2 are complementary           | 1. Given                                  |
| 2. ∠3 and ∠2 are complementary           | 2. Given                                  |
| 3. $ m∠1 + m∠2 = 90^\circ $            | 3. Definition of complementary angles      |
| 4. $ m∠3 + m∠2 = 90^\circ $            | 4. Definition of complementary angles      |
| 5. $ m∠1 + m∠2 = m∠3 + m∠2 $           | 5. Transitive Property of Equality         |
| 6. $ m∠1 = m∠3 $                       | 6. Subtraction Property of Equality        |
| 7. ∠1 ≅ ∠3                               | 7. Definition of congruent angles          |

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## 38. Write a Paragraph Proof

**Given:**  
$ m∠ABC = 48^\circ $  
$ m∠ABD = 24^\circ $  
Prove: $ \overrightarrow{BD} $ bisects $ ∠ABC $

**Paragraph Proof:**

We are given that $ m∠ABC = 48^\circ $ and $ m∠ABD = 24^\circ $. $ \overrightarrow{BD} $ is a ray that divides $ ∠ABC $ into two angles: $ ∠ABD $ and $ ∠DBC $. Since $ m∠ABD = 24^\circ $, it follows that $ m∠DBC = m∠ABC - m∠ABD = 48^\circ - 24^\circ = 24^\circ $. Both $ ∠ABD $ and $ ∠DBC $ have equal measures. By the definition of an angle bisector, if a ray divides an angle into two congruent angles, it bisects the angle. Therefore, $ \overrightarrow{BD} $ bisects $ ∠ABC $.

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**Final Answers:**

**37.**  
∠3 ≅ ∠1

**38.**  
$ \overrightarrow{BD} $ bisects $ ∠ABC $ because it divides the angle into two equal parts, each measuring $ 24^\circ $.

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