A: corresponding angles of similar triangles are congruentB: congruent complements theoremC: third angle theoremD: triangle proportionality theorem ### Geometry Proof Transcription **Statements and Reasons**Below is a proof table that demonstrates the congruence relationships of angles within a set of geometric figures using the properties of right angles, angle addition postulate, and complementary angles.| Statements | Reasons ||---------------------------------------------------------------------------|------------------------------------------|| 1. Point B is on line segment \( \overline{AC} \) | Given || \( \angle D \), \( \angle E \), and \( \angle DBE \) are right angles | Given || 2. \( m\angle D = m\angle E = m\angle DBE = 90^\circ \) | Definition of right angle || 3. \( m\angle A + m\angle D + m\angle ABD = 180^\circ \) | \( \angle C + m\angle E + \angle CBE = 180^\circ \) | Angle Addition Postulate || 4. | Angle Addition Postulate || \( m\angle ABD + 90^\circ + m\angle CBE = 180^\circ \) | Substitution || \( 90^\circ + m\angle ABD + \angle CBE = 180^\circ \) | Substitution || 5. | Substitution || \( m\angle C + m\angle CBE = 90^\circ \) | Subtraction Property of Equality || \( m\angle ABD + m\angle CBE = 90^\circ \) | Subtraction Property of Equality || 6. \( \angle A \) and \( \angle ABD \) are complementary | Definition of Complementary || \( \angle C \) and \( \angle CBE \) are complementary | Definition of Complementary || \( \angle ABD \) and \( \angle CBE \) are complementary | Definition of Complementary || 7. \( \angle A\) ≅ \( \angle CBE \) (by congruence) | || 8. \( \angle C\) ≅ \( \angle ABD\) | || | || \( \angle ABD \) ≅ \(\angle ABE\) | |Below the proof table, there **Multiple Choice****Question 8****What is the reason for the statement in step 8?****Given:** Point \( B \) is on line segment \( \overline{AC} \) and \( \angle D \), \( \angle E \), and \( \angle DBE \) are right angles.**Prove:** \( \triangle ADB \sim \triangle BEC \)**Diagram Description:**The diagram depicts two triangles, \( \triangle ADB \) and \( \triangle BEC \), oriented such that:- \( \triangle ADB \) has a right angle \( \angle D \), with point \( D \) situated above line \( \overline{AC} \) and closer to point \( A \). - \( \triangle BEC \) has a right angle \( \angle E \), with point \( E \) positioned above line \( \overline{AC} \) and closer to point \( C \).**Statements and Reasons:**| **Statements** | **Reasons** ||----------------------------------------|--------------------------|| 1. \( \text{Point } B \text{ is on } \overline{AC} \) | Given || 2. \( \angle D, \angle E, \text{ and } \angle DBE \text{ are all right angles} \) | Given || ... | ... |(Details of the complete proof are implied but not given in the partial table.)---**Explanation:**In this problem, the task is to prove that triangles \( \triangle ADB \) and \( \triangle BEC \) are similar, indicated by the notation \( \triangle ADB \sim \triangle BEC \). It is known that points \( D \) and \( E \), as well as the segment \( \overline{DBE} \), form right angles. With this given information and the structure of the triangles, one can use the properties of right triangles, the Angle-Angle (AA) similarity postulate, or other geometric theorems to establish similarity between the two triangles.

We have to give the reason for statement 8.

Answered: MULTIPLE CHOICE Question 8 What is the…

MULTIPLE CHOICE Question 8 What is the reason for the statement in step 8? Given: Bis a point on AC and ZD, ZE, and ZDBE are right angles. Prove: AADBABEC A B Reasons Statements Point is on AC Given

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterP: Preliminary Concepts

SectionP.CT: Test

Problem 1CT

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Question

A: corresponding angles of similar triangles are congruent

B: congruent complements theorem

C: third angle theorem

D: triangle proportionality theorem

### Geometry Proof Transcription

**Statements and Reasons**

Below is a proof table that demonstrates the congruence relationships of angles within a set of geometric figures using the properties of right angles, angle addition postulate, and complementary angles.

| Statements | Reasons |
|---------------------------------------------------------------------------|------------------------------------------|
| 1. Point B is on line segment \( \overline{AC} \) | Given |
| \( \angle D \), \( \angle E \), and \( \angle DBE \) are right angles | Given |
| 2. \( m\angle D = m\angle E = m\angle DBE = 90^\circ \) | Definition of right angle |
| 3. \( m\angle A + m\angle D + m\angle ABD = 180^\circ \) | \( \angle C + m\angle E + \angle CBE = 180^\circ \) | Angle Addition Postulate |
| 4. | Angle Addition Postulate |
| \( m\angle ABD + 90^\circ + m\angle CBE = 180^\circ \) | Substitution |
| \( 90^\circ + m\angle ABD + \angle CBE = 180^\circ \) | Substitution |
| 5. | Substitution |
| \( m\angle C + m\angle CBE = 90^\circ \) | Subtraction Property of Equality |
| \( m\angle ABD + m\angle CBE = 90^\circ \) | Subtraction Property of Equality |
| 6. \( \angle A \) and \( \angle ABD \) are complementary | Definition of Complementary |
| \( \angle C \) and \( \angle CBE \) are complementary | Definition of Complementary |
| \( \angle ABD \) and \( \angle CBE \) are complementary | Definition of Complementary |
| 7. \( \angle A\) ≅ \( \angle CBE \) (by congruence) | |
| 8. \( \angle C\) ≅ \( \angle ABD\) | |
| | |
| \( \angle ABD \) ≅ \(\angle ABE\) | |

Below the proof table, there

**Multiple Choice**

**Question 8**

**What is the reason for the statement in step 8?**

**Given:** Point \( B \) is on line segment \( \overline{AC} \) and \( \angle D \), \( \angle E \), and \( \angle DBE \) are right angles.

**Prove:** \( \triangle ADB \sim \triangle BEC \)

**Diagram Description:**

The diagram depicts two triangles, \( \triangle ADB \) and \( \triangle BEC \), oriented such that:
- \( \triangle ADB \) has a right angle \( \angle D \), with point \( D \) situated above line \( \overline{AC} \) and closer to point \( A \).
- \( \triangle BEC \) has a right angle \( \angle E \), with point \( E \) positioned above line \( \overline{AC} \) and closer to point \( C \).

**Statements and Reasons:**

| **Statements** | **Reasons** |
|----------------------------------------|--------------------------|
| 1. \( \text{Point } B \text{ is on } \overline{AC} \) | Given |
| 2. \( \angle D, \angle E, \text{ and } \angle DBE \text{ are all right angles} \) | Given |
| ... | ... |

(Details of the complete proof are implied but not given in the partial table.)

---

**Explanation:**

In this problem, the task is to prove that triangles \( \triangle ADB \) and \( \triangle BEC \) are similar, indicated by the notation \( \triangle ADB \sim \triangle BEC \). It is known that points \( D \) and \( E \), as well as the segment \( \overline{DBE} \), form right angles. With this given information and the structure of the triangles, one can use the properties of right triangles, the Angle-Angle (AA) similarity postulate, or other geometric theorems to establish similarity between the two triangles.

Figure in plane geometry formed by two rays or lines that share a common endpoint, called the vertex. The angle is measured in degrees using a protractor. The different types of angles are acute, obtuse, right, straight, and reflex.

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