Math Problem Statement

Find the missing sides or angles in the triangles as shown in the image.

Solution

Let’s analyze the two triangles in the image and solve for the missing values of xx and yy:


Left Triangle:

  1. Triangle Type: It has one angle of 6060^\circ and appears to be an isosceles triangle (two sides marked as equal, 1111).

  2. Key Equation for Angles: In any triangle, the sum of angles is 180180^\circ.

    • The given angles are 6060^\circ and 3x3x^\circ, with the third angle as 5y5y^\circ. Thus: 60+3x+5y=18060 + 3x + 5y = 180
  3. Simplify: 3x+5y=120(Equation 1).3x + 5y = 120 \quad \text{(Equation 1)}.


Right Triangle:

  1. Triangle Type: It is also isosceles, as two sides are marked equal (3x+83x + 8 and 2323). Additionally, one angle is 6060^\circ.

  2. Equal Sides: 3x+8=233x + 8 = 23 Simplify to solve for xx: 3x=15    x=5.3x = 15 \quad \implies \quad x = 5.

  3. Key Equation for Angles: Again, the sum of angles is 180180^\circ.

    • The angles are 6060^\circ, 6060^\circ, and 6y6y^\circ. Thus: 60+60+6y=18060 + 60 + 6y = 180 Simplify to solve for yy: 6y=60    y=10.6y = 60 \quad \implies \quad y = 10.

Final Values:

  • Left Triangle: Equation 3x+5y=1203x + 5y = 120 is still unsolved; substituting x=5x = 5 from the right triangle: 3(5)+5y=120    15+5y=120    5y=105    y=21.3(5) + 5y = 120 \quad \implies \quad 15 + 5y = 120 \quad \implies \quad 5y = 105 \quad \implies \quad y = 21.

  • Right Triangle: x=5x = 5, y=10y = 10.


Answers:

  1. Left Triangle: x=5x = 5, y=21y = 21.
  2. Right Triangle: x=5x = 5, y=10y = 10.

Do you want a detailed step-by-step breakdown for any part?

Here are 5 related questions to deepen your understanding:

  1. How can you identify if a triangle is isosceles or equilateral?
  2. What is the sum of angles in any polygon, and why does it hold true?
  3. How do side ratios in isosceles triangles relate to their angles?
  4. How does substituting variables into equations work systematically?
  5. Why are 6060^\circ triangles commonly seen in geometry problems?

Tip: Always confirm relationships between sides and angles using triangle properties like the Pythagorean theorem or angle-sum rules.

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Math Problem Analysis

Mathematical Concepts

Triangle Geometry
Isosceles Triangles
Angle Sum Property
Linear Equations

Formulas

Sum of angles in a triangle: A + B + C = 180°
Equality of sides in isosceles triangles
Solving for variables in linear equations

Theorems

Angle Sum Property of a Triangle
Isosceles Triangle Theorem

Suitable Grade Level

Grade 8-10