Let me analyze the image and solve the questions for you.

Problem 1: Triangle $\triangle STU$

• Given:
  • $m\angle S = 54^\circ$
  • Triangle is isosceles, so $\angle T = \angle U$.

Solution:

• The sum of interior angles of a triangle is $180^\circ$. Thus: $m\angle S + m\angle T + m\angle U = 180^\circ$ Since $m\angle T = m\angle U$, let $m\angle T = m\angle U = x$. Substituting: $54^\circ + x + x = 180^\circ \implies 2x = 126^\circ \implies x = 63^\circ$
• Therefore: $m\angle T = 63^\circ, \, m\angle U = 63^\circ$

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Problem 2: Triangle $\triangle LMN$

• Given:
  • $m\angle L = 28^\circ$
  • Triangle is isosceles, so $\angle M = \angle N$.

Solution:

• The sum of the angles of the triangle is $180^\circ$. Let $m\angle M = m\angle N = y$. Then: $28^\circ + y + y = 180^\circ \implies 2y = 152^\circ \implies y = 76^\circ$
• Therefore: $m\angle M = 76^\circ, \, m\angle N = 76^\circ$

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Problem 3: Triangle $\triangle EFG$

• Given:
  • $m\angle E = 23^\circ$
  • Triangle is right, so $m\angle G = 90^\circ$.

Solution:

• The sum of the angles of the triangle is $180^\circ$. Let $m\angle F = z$. Then: $23^\circ + 90^\circ + z = 180^\circ \implies z = 67^\circ$
• Therefore: $m\angle F = 67^\circ$

For Side $EF$:

• $EF$ is not solvable without more information, such as trigonometric ratios or additional side lengths.

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Problem 4: Triangle $\triangle PQR$

• Triangle is equilateral (all sides and angles are equal).

Solution:

• In an equilateral triangle, all angles are $60^\circ$. Therefore: $m\angle P = 60^\circ, \, m\angle Q = 60^\circ, \, m\angle R = 60^\circ$

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Final Answers:

1. $m\angle T = 63^\circ, \, m\angle U = 63^\circ$
2. $m\angle M = 76^\circ, \, m\angle N = 76^\circ$
3. $m\angle F = 67^\circ$, $EF$ requires more data.
4. $m\angle P = 60^\circ, \, m\angle Q = 60^\circ, \, m\angle R = 60^\circ$

Would you like further clarification or more details? Here are some relative questions to explore:

1. What are the properties of isosceles and equilateral triangles?
2. How can we calculate side lengths using trigonometric ratios in a triangle?
3. What are the conditions for identifying right triangles?
4. How do we use the Pythagorean theorem in right triangles?
5. Why does the sum of interior angles in a triangle always equal $180^\circ$?

Tip: Always check whether the triangle is special (e.g., isosceles, equilateral, or right) before solving!