Math Problem Statement

In the diagram, ΔABC ≅ ΔFDE. Given angles in ΔABC are 108° and 48°, and angles in ΔFDE are labeled y° and (2x - y)°. Find the value of x.

Solution

In this problem, we have two congruent triangles, ABC\triangle ABC and FDE\triangle FDE. Since the triangles are congruent, their corresponding angles are equal.

Given Information:

  1. In ABC\triangle ABC:

    • ACB=108\angle ACB = 108^\circ
    • ABC=48\angle ABC = 48^\circ
  2. In FDE\triangle FDE:

    • EDF=y\angle EDF = y^\circ
    • DFE=(2xy)\angle DFE = (2x - y)^\circ

Steps to Solve:

  1. Find the Measure of CAB\angle CAB: Since the sum of angles in a triangle is 180180^\circ, we can find CAB\angle CAB as follows: CAB=180ACBABC=18010848=24\angle CAB = 180^\circ - \angle ACB - \angle ABC = 180^\circ - 108^\circ - 48^\circ = 24^\circ

  2. Corresponding Angles: Since ABCFDE\triangle ABC \cong \triangle FDE, the angles in FDE\triangle FDE are:

    • FDE=CAB=24\angle FDE = \angle CAB = 24^\circ
    • EDF=ABC=48\angle EDF = \angle ABC = 48^\circ
    • DFE=ACB=108\angle DFE = \angle ACB = 108^\circ
  3. Set up an Equation for 2xy2x - y: We know from the corresponding angles that DFE=108\angle DFE = 108^\circ. Since DFE=2xy\angle DFE = 2x - y: 2xy=1082x - y = 108

  4. Use y=48y = 48: From the congruence, y=48y = 48^\circ.

  5. Substitute and Solve for xx: Substitute y=48y = 48 into the equation 2xy=1082x - y = 108: 2x48=1082x - 48 = 108 2x=1562x = 156 x=78x = 78

Solution:

The value of xx is 7878.

Would you like further details, or do you have any questions?


Related Questions:

  1. How do congruent triangles help in solving for unknown angles?
  2. What is the significance of corresponding angles in congruent triangles?
  3. Can this problem be solved without using congruence properties?
  4. How can you check if two triangles are congruent in general?
  5. What other geometric properties apply when two triangles are congruent?

Tip:

Always remember that the sum of angles in any triangle is 180180^\circ, which can help find missing angles.

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Math Problem Analysis

Mathematical Concepts

Congruent Triangles
Triangle Sum Property
Corresponding Angles

Formulas

Sum of angles in a triangle = 180°

Theorems

Congruent Triangles Theorem
Corresponding Angles in Congruent Triangles

Suitable Grade Level

Grades 8-10