Math Problem Statement

In the figure, BA−→BA→ and BC−→−BC→ are opposite rays. BH−→−BH→ bisects ∠EBC∠EBC and BE−→BE→ bisects ∠ABF∠ABF.

  If m∠EBC=(3r+10)°m∠EBC=3r+10° and m∠ABE=(2r−20)°m∠ABE=2r-20°, find m∠EBFm∠EBF.  

Solution

We are given the following information:

  • BA\overrightarrow{BA} and BC\overrightarrow{BC} are opposite rays, meaning ABC=180\angle ABC = 180^\circ.
  • BH\overrightarrow{BH} bisects EBC\angle EBC, meaning EBC=2×HBC\angle EBC = 2 \times \angle HBC.
  • BE\overrightarrow{BE} bisects ABF\angle ABF, meaning ABF=2×ABE\angle ABF = 2 \times \angle ABE.

Information about the angles:

  • mEBC=(3r+10)m\angle EBC = (3r + 10)^\circ
  • mABE=(2r20)m\angle ABE = (2r - 20)^\circ

Step 1: Express mABCm\angle ABC

Since ( \overrightarrow{

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Math Problem Analysis

Mathematical Concepts

Geometry
Angle Bisectors
Linear Pair of Angles

Formulas

Sum of angles on a straight line: ∠ABC = 180°
Angle bisector property: m∠EBC = 2 × m∠HBC, m∠ABF = 2 × m∠ABE

Theorems

Linear Pair Theorem
Angle Bisector Theorem

Suitable Grade Level

Grades 9-10