Let’s solve each triangle step by step.

Problem 1

The triangle $\triangle STU$ is isosceles, as indicated by the congruence marks. $\angle S = 54^\circ$.

Properties of an isosceles triangle:

• The base angles are equal.
• The sum of the angles in a triangle is $180^\circ$.

1. Sum of angles: $m\angle S + m\angle T + m\angle U = 180^\circ$ Substituting $m\angle S = 54^\circ$: $54^\circ + 2m\angle T = 180^\circ$
2. Solving for $m\angle T$: $2m\angle T = 126^\circ \implies m\angle T = 63^\circ$
3. Since $m\angle T = m\angle U$ (isosceles property): $m\angle U = 63^\circ$

Solution:

• $m\angle T = 63^\circ$
• $m\angle U = 63^\circ$

Problem 2

The triangle $\triangle LMN$ is also isosceles, as indicated by the congruence marks. $\angle L = 28^\circ$.

Using the same properties:

1. Sum of angles: $m\angle L + m\angle M + m\angle N = 180^\circ$ Substituting $m\angle L = 28^\circ$: $28^\circ + 2m\angle M = 180^\circ$
2. Solving for $m\angle M$: $2m\angle M = 152^\circ \implies m\angle M = 76^\circ$
3. Since $m\angle M = m\angle N$: $m\angle N = 76^\circ$

Solution:

• $m\angle M = 76^\circ$
• $m\angle N = 76^\circ$

Problem 3

The triangle $\triangle EFG$ has one angle $\angle E = 23^\circ$. The triangle is not explicitly marked as isosceles, so we must calculate accordingly.

Use the triangle sum property:

1. Sum of angles: $m\angle E + m\angle F + m\angle G = 180^\circ$ Substituting $m\angle E = 23^\circ$: $23^\circ + m\angle F + m\angle G = 180^\circ$

We need additional information (like congruence marks or relationships) to calculate $EF$ or $m\angle F$.

Could you provide more clarity on this question, or do you have additional instructions?

Problem 4

The triangle $\triangle PQR$ is equilateral, as all sides are marked as congruent.

Properties of an equilateral triangle:

• All angles are equal.
• Each angle measures $60^\circ$.

Solution:

• $m\angle P = 60^\circ$
• $m\angle Q = 60^\circ$
• $m\angle R = 60^\circ$

Would you like further clarification on Problem 3 or any of the steps above?

Questions:

1. What other properties of isosceles triangles might help solve these problems?
2. How can we use the triangle inequality theorem in such problems?
3. What happens if one angle of a triangle is obtuse? How does that change the calculations?
4. Are there any cases where an equilateral triangle's properties change?
5. Can you think of real-world applications for calculating triangle angles?

Tip: Always check if triangles are isosceles, equilateral, or scalene. This simplifies the calculations significantly!