27. If $$ m \angle 4=15 x^{\circ} $$ and $$ m \angle 2=(11 x+20)^{\circ} $$, find the value of $$ x, m \angle 4 $$, and $$ m \angle 2 $$. 28. In the roller coaster tower at the right, the measure of $$ \angle 1 $$ is $$ 6^{\circ} $$ less than twice the measure of $$ \angle 2 $$. Find the measures of angles 1 and 2 .

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**27.**

- We are given:
  - $$ m\angle 4 = 15x^\circ $$
  - $$ m\angle 2 = (11x + 20)^\circ $$
- Based on the diagram (not shown), angles 2 and 4 are congruent. Thus, we set:
  $$
  15x = 11x + 20
  $$
- Solve for $$ x $$:
  - Subtract $$ 11x $$ from both sides:
    $$
    15x - 11x = 20 \quad \Longrightarrow \quad 4x = 20
    $$
  - Divide by 4:
    $$
    x = 5
    $$
- Now compute the measures:
  - $$ m\angle 4 = 15(5) = 75^\circ $$
  - $$ m\angle 2 = 11(5) + 20 = 55 + 20 = 75^\circ $$

**28.**

- We are told that the measure of $$\angle 1$$ is $$6^\circ$$ less than twice the measure of $$\angle 2$$. This gives:
  $$
  m\angle 1 = 2\,(m\angle 2) - 6
  $$
- In the roller coaster tower diagram, $$\angle 1$$ and $$\angle 2$$ are the two acute angles in a right triangle, so they are complementary:
  $$
  m\angle 1 + m\angle 2 = 90
  $$
- Substitute the expression for $$ m\angle 1 $$:
  $$
  (2\,(m\angle 2) - 6) + m\angle 2 = 90
  $$
- Combine like terms:
  $$
  3\,(m\angle 2) - 6 = 90
  $$
- Solve for $$ m\angle 2 $$:
  - Add 6:
    $$
    3\,(m\angle 2) = 96
    $$
  - Divide by 3:
    $$
    m\angle 2 = 32^\circ
    $$
- Now, find $$ m\angle 1 $$:
  $$
  m\angle 1 = 2(32) - 6 = 64 - 6 = 58^\circ
  $$

**Answers:**

- For 27: $$ x = 5 $$, $$ m\angle 4 = 75^\circ $$, and $$ m\angle 2 = 75^\circ $$.
- For 28: $$ m\angle 1 = 58^\circ $$ and $$ m\angle 2 = 32^\circ $$.
**27.** - $$ x = 5 $$ - $$ m\angle 4 = 75^\circ $$ - $$ m\angle 2 = 75^\circ $$ **28.** - $$ m\angle 1 = 58^\circ $$ - $$ m\angle 2 = 32^\circ $$

If and , find the value of , and .

In the roller coaster tower at the right,
the measure of is less than twice
the measure of . Find the measures
of angles 1 and 2 .

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If and , find the value of , and . In the roller coaster tower at the right, the measure of is less than twice the measure of . Find the measures of angles 1 and 2 .