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Let's break down the problem shown in the image into individual exercises. I will help you solve each one step by step.

### For the first diagram (with the intersecting lines):
You are tasked with finding the measures of all eight angles, labeled from \( \angle 1 \) to \( \angle 8 \), given the following information:

1. \( m\angle 2 = 91^\circ \)
2. \( m\angle 3 = 75^\circ \)
3. \( m\angle 8 = x \)

#### Step-by-Step Process:
1. **If \( m\angle 2 = 91^\circ \):**
   - \( \angle 2 \) and \( \angle 7 \) are **vertically opposite angles**, so \( m\angle 7 = 91^\circ \).
   - \( \angle 2 \) and \( \angle 3 \) are **supplementary angles** because they form a straight line. Therefore:
     \[
     m\angle 3 + m\angle 2 = 180^\circ
     \]
     Since \( m\angle 2 = 91^\circ \), we get:
     \[
     m\angle 3 = 180^\circ - 91^\circ = 89^\circ
     \]
   - Vertically opposite angles \( \angle 1 \) and \( \angle 2 \) are also equal, so \( m\angle 1 = 89^\circ \).

2. **If \( m\angle 3 = 75^\circ \):**
   - \( \angle 3 \) and \( \angle 6 \) are **vertically opposite angles**, so \( m\angle 6 = 75^\circ \).
   - \( \angle 3 \) and \( \angle 2 \) are supplementary, but since \( m\angle 2 \) has already been determined as 91° (not 105°), we cannot use the information to revise this measure directly.


Find the measures of all eight angles given the following information:
1. m∠2 = 91°
2. m∠3 = 75°
3. m∠8 = x
4. m∠1 = 2x and m∠2 = 3x
5. m∠1 = 5x - 10 and m∠2 = 8x + 34
6. If m∠A = 52°, find m∠B, m∠C, and m∠D in a quadrilateral.
7. If m∠D = 2m∠C, find m∠A, m∠B, m∠C, and m∠D in a quadrilateral.

Solve a set of geometric problems involving the measurement of angles formed by intersecting lines and quadrilaterals. Learn key concepts such as vertically opposite angles, supplementary angles, and the sum of angles in a quadrilateral. Suitable for Grade 8-10 students.

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