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Let's analyze and solve each of the problems one by one:

### **Problem 27**

#### Given the diagram, the angle pairs are:

a) **NAB and EAS** - They are **corresponding angles** and are congruent (≅).

b) **NAS and BAE** - These are **alternate interior angles** and are congruent (≅).

c) **BEA and SNA** - They are **vertical angles** and are congruent (≅).

d) **NSA and EBA** - These are **alternate interior angles** and are congruent (≅).

#### Angle Measures:

- e) **EBA**: Since EBA and BEA are vertical angles and BEA is 85°, angle EBA is also **85°**.
- f) **SNA**: Since it is a vertical angle to BEA, it measures **85°**.
- g) **NAS**: Since NAS is a vertical angle to BAE, it measures **52°**.
- h) **BAE**: It is given as **52°**.

#### Similarity Theorem
The triangles are similar by **AA (Angle-Angle) Similarity**. 

#### Similarity Statement
△**NAS** ∼ △**BAE**

---

### **Problem 28**

#### a) Name for the angle pairs:
The two angles are **linear pairs**, which means they are supplementary.

#### b) Solve for x:
Given angles: 
\[2x - 15\] and \[2x - 31\]

Since they are supplementary:
\[(2x - 15) + (2x - 31) = 180\]
\[4x - 46 = 180\]
\[4x = 226\]
\[x = 56.5\]

#### c) Measure of the two angles:
Substituting \(x = 56.5\):
- First angle: \(2(56.5) - 15 = 113 - 15 = 98°\)
- Second angle: \(2(56.5) - 31 = 113 - 31 = 82°\)

The two angles are **98°** and **82°**.

---

### **Problem 29**

Angle A and B are complementary:
Let Angle A be \(x\), then Angle B is \(4x\).

Since they are complementary:
\[x + 4x = 90\]
\[5x = 90\]
\[x = 18\]

Thus, Angle A is **18°** and Angle B is **72°**.

---

### **Problem 30**

#### a) Name the angle pair:
The pair shown are **vertical angles** and are congruent.

#### b) Solve for \(x\):
\[3x + 6\] and \(8x - 18\) are equal:
\[
3x + 6 = 8x - 18
\]
\[
24 = 5x
\]
\[
x = 4.8
\]

#### Measures of all four angles:
Substituting \(x = 4.8\):
- \(3(4.8) + 6 = 14.4 + 6 = 20.4°\)
- \(8(4.8) - 18 = 38.4 - 18 = 20.4°\)

The four angles are **20.4°**, **159.6°** (since they're supplementary to each other).

---

### **Problem 31**

Given \(AD\) is parallel to \(BE\), the corresponding angle \(CGE\) is equal to \(138°\).

---

Do you want more detailed explanations for any part?

### **Related Questions:**

1. How can you verify the similarity of triangles using corresponding sides?
2. Why are vertical angles always congruent?
3. What is the significance of the AA similarity criterion in geometry?
4. How do you determine complementary and supplementary angles in a diagram?
5. How can parallel lines help determine angle measures in transversals?

### **Tip:** 
When dealing with complementary angles, always remember that their sum is 90°, and for supplementary angles, their sum is 180°.

Based on the provided geometry problems, solve for angle measures, identify angle pairs, and determine triangle similarity.

Explore detailed solutions for geometry problems including identifying angle pairs, solving for x, and determining triangle similarity.

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