Right angle FCD intersects Line A B and Ray C E  at point C. AngleFCE is congruent to AngleECD. AngleECD is complementary to AngleDCB. 

A horizontal line has points A, C, B. Right angle F C D intersects the line at point C. A line extends from point C vertically to point E. Angle A C F is 45 degrees and angle F C D is 90 degrees.
Which statement is true about AngleDCB and AngleACF? 

They are congruent and complementary.
They are congruent and supplementary.
They are complementary but not necessarily congruent.
They are supplementary but not necessarily congruent.

Let's break down the problem step by step:

- **Right angle FCD** intersects Line AB at point C.
- A ray extends vertically from C to E.
- **Angle FCE** is congruent to **Angle ECD**.
- **Angle ECD** is complementary to **Angle DCB**.
- **Angle ACF** is given as 45°.
- **Angle FCD** is 90°.

We are tasked with analyzing the relationship between **Angle DCB** and **Angle ACF**.

### Step 1: Analyze Angles Around Point C

- **Angle ACF** is 45°.
- Since **Angle FCE** is congruent to **Angle ECD**, each of these angles must be **45°** (because the two angles must sum to 90°, as **Angle FCD** is a right angle).

### Step 2: Complementary Angles

- It is stated that **Angle ECD** is complementary to **Angle DCB**. Therefore, the sum of these two angles must be 90°.
  - Since **Angle ECD** is 45°, **Angle DCB** must also be **45°** to satisfy the complementary condition.

### Step 3: Compare **Angle DCB** and **Angle ACF**

- **Angle DCB** is 45° and **Angle ACF** is also 45°. This means:
  - **Angle DCB** and **Angle ACF** are congruent (both equal to 45°).
  - Since their sum is 90°, they are also complementary.

### Conclusion:
The correct statement is:
- **They are congruent and complementary.**

Would you like more details or have any further questions?

### Related Questions:
1. What is the sum of complementary angles?
2. How do congruent angles differ from complementary angles?
3. Why is **Angle DCB** complementary to **Angle ECD**?
4. Can two angles be congruent and supplementary?
5. What makes an angle a right angle?

**Tip:** Congruent angles are always equal in measure, while complementary angles sum to 90°.

Right angle FCD intersects Line A B and Ray C E at point C. Angle FCE is congruent to Angle ECD. Angle ECD is complementary to Angle DCB. A horizontal line has points A, C, B. Right angle F C D intersects the line at point C. A line extends from point C vertically to point E. Angle A C F is 45 degrees and angle F C D is 90 degrees. Which statement is true about Angle DCB and Angle ACF? They are congruent and complementary. They are congruent and supplementary. They are complementary but not necessarily congruent. They are supplementary but not necessarily congruent.

Explore the relationship between Angle DCB and Angle ACF in this geometry problem. Learn how to identify congruent and complementary angles, using the sum of complementary angles and congruence theorems. Suitable for students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation