Triangle Angle Problems and Solutions

This page presents a collection of geometry problems focusing on triangle angles, suitable for 9th-grade mathematics students. The problems cover various aspects of triangle geometry, including interior and exterior angles, special triangles, and angle calculations based on given information.

Highlight: The problems demonstrate practical applications of üçgen açı hesaplama (triangle angle calculation) techniques.

Problem 1: Interior Angle Calculation

A triangle ABC is given with angle measurements represented by variables. The problem requires solving for x using the principle that the sum of interior angles in a triangle is 180°.

Definition: The sum of interior angles in any triangle is always 180°.

Given:

• Angle A = 2k
• Angle B = 58°
• Angle C = 2k + x

Solution: 2k + 58° + $2k + x$ = 180° 4k + x + 58 = 180 4k + x = 122 x = 122 - 4k

Example: If k = 15, then x = 122 - 4(15) = 62°

Problem 2: Exterior Angle Calculation

This problem involves calculating an exterior angle of a triangle given the measures of two interior angles.

Given:

• Interior angle A = 24°
• Interior angle B = 78°
• Exterior angle at C to be calculated

Solution: Sum of interior angles A and B = 24° + 78° = 102° Exterior angle at C = 180° - 102° = 78°

Vocabulary: An exterior angle of a triangle is the angle formed between any side of a triangle and the extension of an adjacent side.

Problem 3: Isosceles Triangle Rotation

An isosceles triangle ABC with |AB| = |AC| is rotated 24° around vertex B in a positive direction, creating a new triangle A'BC'.

Given:

• Original isosceles triangle ABC
• Rotation of 24° around vertex B
• Intersection point D where [AB] intersects [A'C']

Task: Calculate m(ADB)

Solution: In an isosceles triangle, base angles are equal. Let's assume each base angle is x°. 180° - x° = x° (sum of base angles equals the vertex angle) 2x = 180° x = 90°

After rotation: m(ADB) = 90° - 24° = 66°

Highlight: This problem combines concepts of isosceles triangles and rotation, demonstrating the application of üçgen iç açı hesaplama (triangle interior angle calculation).

The page concludes with additional problems involving triangle angle calculations, reinforcing the concepts of kenar uzunluklarından açı hesaplama (angle calculation from side lengths) and üçgenin iç açıları soruları (triangle interior angle questions).