Math Problem Statement

DAC, FCB, and ABE are straight lines. Work out the size of the angle marked x. You must give reasons for your answer. Given angles are 70°, 135°.

Solution

We are given a geometric diagram with three straight lines DACDAC, FCBFCB, and ABEABE, along with specific angles. The goal is to determine the value of the angle marked xx.

Steps to solve for xx:

  1. Straight Line Rule: The angle on a straight line sums to 180180^\circ.

    • Angle FCB=135\angle FCB = 135^\circ is given.
    • Therefore, ACB=180135=45\angle ACB = 180^\circ - 135^\circ = 45^\circ.
  2. Triangle Angle Sum: The sum of angles in a triangle is always 180180^\circ. In triangle ACBACB: ACB+BAC+ABC=180\angle ACB + \angle BAC + \angle ABC = 180^\circ Substituting the known values: 45+70+x=18045^\circ + 70^\circ + x = 180^\circ Simplifying: 115+x=180115^\circ + x = 180^\circ x=180115=65x = 180^\circ - 115^\circ = 65^\circ

Therefore, the value of the angle marked xx is 6565^\circ.

Would you like any further clarification or details?


Here are 5 related questions to expand on this problem:

  1. How would the solution change if one of the given angles was different?
  2. Can you prove why the sum of angles in any triangle is always 180180^\circ?
  3. What is the relationship between exterior angles and interior angles of a triangle?
  4. How would you approach solving for an unknown angle in a non-right triangle?
  5. Can this method be applied to a quadrilateral? Why or why not?

Tip: Always remember to check that the diagram follows the rules for angles on straight lines and triangles. It helps in verifying your answers!

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Math Problem Analysis

Mathematical Concepts

Geometry
Angles on a straight line
Sum of angles in a triangle

Formulas

Sum of angles on a straight line = 180°
Sum of angles in a triangle = 180°

Theorems

Triangle Angle Sum Theorem
Linear Pair Axiom

Suitable Grade Level

Grades 7-9