Solved Problems: Using the Triangle Angle Sum Theorem and its Corollaries

📐 Understanding the Triangle Angle Sum Theorem

The Triangle Angle Sum Theorem is a fundamental concept in Euclidean geometry. It states that the sum of the interior angles of any triangle is always 180 degrees. Mathematically, if a triangle has angles A, B, and C, then:

$A + B + C = 180^{\circ}$

📜 History and Background

The theorem has been known since ancient times, with evidence of its understanding found in the works of Euclid and other Greek mathematicians. Its simplicity and fundamental nature have made it a cornerstone of geometric reasoning for millennia.

🔑 Key Principles

• 🧮 The Sum: The sum of the measures of the interior angles is exactly 180 degrees.
• ✍️ Applicability: This theorem applies to all types of triangles (acute, obtuse, right, equilateral, isosceles, scalene) in Euclidean geometry.
• 🧩 Corollaries: Corollaries are statements that follow directly from the theorem.

💡 Corollaries of the Triangle Angle Sum Theorem

A corollary is a theorem that follows directly from another theorem. Here are some important corollaries of the Triangle Angle Sum Theorem:

• ✨ Corollary 1: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
• ➕ Corollary 2: There can be at most one right angle or obtuse angle in a triangle.
• 🎯 Corollary 3: The acute angles in a right triangle are complementary.

✍️ Solved Problems

Let's apply the theorem and its corollaries to solve some problems:

1. Problem 1: In $\triangle ABC$, $\angle A = 60^{\circ}$ and $\angle B = 80^{\circ}$. Find $\angle C$.

Solution: Using the Triangle Angle Sum Theorem: $\angle A + \angle B + \angle C = 180^{\circ}$
$60^{\circ} + 80^{\circ} + \angle C = 180^{\circ}$
$140^{\circ} + \angle C = 180^{\circ}$
$\angle C = 180^{\circ} - 140^{\circ} = 40^{\circ}$

2. Problem 2: In a right triangle, one of the acute angles measures $35^{\circ}$. Find the measure of the other acute angle.

Solution: Since it's a right triangle, one angle is $90^{\circ}$. Let the other acute angle be $x$. Using Corollary 3: $90^{\circ} + 35^{\circ} + x = 180^{\circ}$
$125^{\circ} + x = 180^{\circ}$
$x = 180^{\circ} - 125^{\circ} = 55^{\circ}$

3. Problem 3: In $\triangle PQR$, $\angle P = x$, $\angle Q = 2x$, and $\angle R = 3x$. Find the measure of each angle.

Solution: Using the Triangle Angle Sum Theorem: $x + 2x + 3x = 180^{\circ}$
$6x = 180^{\circ}$
$x = 30^{\circ}$
So, $\angle P = 30^{\circ}$, $\angle Q = 2(30^{\circ}) = 60^{\circ}$, and $\angle R = 3(30^{\circ}) = 90^{\circ}$

4. Problem 4: $\triangle XYZ$ where $\angle X = 2y + 10$, $\angle Y = 4y - 5$, and $\angle Z = 5y$. Find each angle.

Solution: $(2y + 10) + (4y - 5) + 5y = 180$
$11y + 5 = 180$
$11y = 175$
$y = 15.91$ (approximately)
$\angle X = 2(15.91) + 10 = 41.82^{\circ}$
$\angle Y = 4(15.91) - 5 = 58.64^{\circ}$
$\angle Z = 5(15.91) = 79.55^{\circ}$

5. Problem 5: In $\triangle LMN$, $\angle L = 90^{\circ}$ and $\angle M = 45^{\circ}$. Find $\angle N$.

Solution: $90 + 45 + \angle N = 180$
$135 + \angle N = 180$
$\angle N = 45^{\circ}$

6. Problem 6: If two angles of a triangle are $50^{\circ}$ and $70^{\circ}$, what is the measure of the third angle?

Solution: $50 + 70 + x = 180$
$120 + x = 180$
$x = 60^{\circ}$

7. Problem 7: In $\triangle DEF$, $\angle D = x + 10$, $\angle E = x + 20$, $\angle F = x + 30$. Find the value of x and each angle.

Solution: $(x + 10) + (x + 20) + (x + 30) = 180$
$3x + 60 = 180$
$3x = 120$
$x = 40$
$\angle D = 40 + 10 = 50^{\circ}$
$\angle E = 40 + 20 = 60^{\circ}$
$\angle F = 40 + 30 = 70^{\circ}$

📝 Conclusion

The Triangle Angle Sum Theorem and its corollaries are powerful tools for solving a variety of geometric problems. Understanding and applying these concepts will enhance your problem-solving skills in geometry.