https://file.aiquickdraw.com/mathbot/file/1727452310809lsmeryb1.jpg

The image you uploaded contains several geometry problems from Chapter 2 titled "Basic Concepts and Proofs." I'll go through each question step-by-step.

### Problem Breakdown

1. **Question 3 (part a)**:  
   - You are asked to find the measure of angle ∠2. The information provided includes that line **AB** is perpendicular to line **BC**, meaning ∠2 is a right angle.
   - **Answer**: ∠2 = 68° 17' 34'' (from the image).

2. **Question 4 (part a)**:  
   - ∠1 is five times as large as ∠2. You are asked to find the measure of ∠2.
   - Let's denote the measure of ∠2 as \( x \), then ∠1 is \( 5x \). So, the total angle is \( x + 5x = 180^\circ \).  
   - Solve for \( x \), which is ∠2.

3. **Question 5**:  
   - A point \( A \) is at (0, 4) and is rotated 90° clockwise about the origin.  
   - The transformation matrix for a 90° clockwise rotation is:
     \[
     \text{Rotation matrix} = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}
     \]
   - Apply the transformation to the point \( (0, 4) \).

4. **Question 7**:  
   - Given that ∠ACB = 90° and \( AD \perp BD \), you are asked to prove \( \angle C \cong \angle D \).
   - Use triangle congruency criteria and given information to prove this.

5. **Question 8**:  
   - You are given two angles, ∠MOR and ∠ROP, and are asked to compare them.  
   - ∠MOR = (3x + 7)° and ∠ROP = (4x - 1)°.  
   - Find which angle is larger by solving for \( x \).

Would you like me to work through a specific problem in detail or provide the solution steps for each? Let me know!

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### 5 Related Questions:
1. How can you apply the concept of rotation in geometry to solve transformations?
2. What steps are involved in proving congruence between two triangles using given conditions?
3. How do you convert between degrees, minutes, and seconds for angles?
4. What is the significance of perpendicularity in geometric proofs?
5. How can you solve systems of equations involving angles in geometry?

**Tip**: When working with angle measures involving degrees, always ensure to carefully manage conversions between degrees, minutes, and seconds.

Solve the geometry problems shown in the image (Chapter 2: Basic Concepts and Proofs).

Step-by-step solutions to geometry problems involving angle measures, rotations, and congruence proofs. Learn how to solve problems from Chapter 2: Basic Concepts and Proofs. Suitable for students in Grades 9-10.

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