Let's analyze and solve the angles $x$, $y$, and $z$ in the triangles shown in the image.

Triangle 4.1

• Given: $\angle A = 44^\circ$, $\angle B = 56^\circ$
• Unknown: $\angle C = x$

Using the angle sum property of a triangle: $x = 180^\circ - 44^\circ - 56^\circ = 80^\circ$ So, $x = 80^\circ$.

Triangle 4.2

• Given: $\angle P = 132^\circ$, $\angle Q = 28^\circ$
• Unknown: $\angle R = y$

Using the angle sum property of a triangle: $y = 180^\circ - 132^\circ - 28^\circ = 20^\circ$ So, $y = 20^\circ$.

Triangle 4.3

• Given: $\angle D = 26^\circ$, $\angle E = 72^\circ$
• Unknown: $\angle F = z$

Using the angle sum property of a triangle: $z = 180^\circ - 26^\circ - 72^\circ = 82^\circ$ So, $z = 82^\circ$.

Summary

• $x = 80^\circ$
• $y = 20^\circ$
• $z = 82^\circ$

Would you like to go over any of these solutions in more detail, or do you have any other questions?

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Here are 5 related questions you might find interesting:

1. How would you calculate the angles in an isosceles triangle?
2. What is the significance of the exterior angle theorem in triangles?
3. Can the angles in any triangle be obtuse? Why or why not?
4. How would you solve for angles in a right triangle using trigonometric ratios?
5. How does the angle sum property apply to non-Euclidean geometry?

Tip: Always verify if the triangle is a special type (like isosceles or equilateral) before solving for angles; this can simplify the calculations.