Math Problem Statement

In ΔRST, U is a point on RT such that SU is an angle bisector of ∠RST. What is m∠R?

Solution

To solve for mRm \angle R in the triangle RST\triangle RST given the information in the problem, let's break down the information:

Step-by-Step Solution:

  1. Identify Given Angles and Relations:

    • S\angle S is bisected by SU\overline{SU}, meaning the two angles adjacent to S\angle S along SUSU are equal. Thus, each part is labeled xx^\circ.
    • The angle at UU, SUT\angle SUT, is labeled as 3x3x^\circ.
    • T\angle T, located at TT, is labeled as (x+5)(x + 5)^\circ.
  2. Use the Triangle Angle Sum Property in RST\triangle RST: In any triangle, the sum of the internal angles is 180180^\circ. Therefore, we have: mR+S+T=180m \angle R + \angle S + \angle T = 180^\circ

  3. Express the Angles in Terms of xx:

    • Since S\angle S is split by SUSU, we have S=x+x=2x\angle S = x + x = 2x^\circ.
    • The measure of T\angle T is given as (x+5)(x + 5)^\circ.

    Plugging these into the equation: mR+2x+(x+5)=180m \angle R + 2x + (x + 5) = 180

  4. Simplify the Equation: mR+3x+5=180m \angle R + 3x + 5 = 180 mR=1803x5m \angle R = 180 - 3x - 5 mR=1753xm \angle R = 175 - 3x

  5. Using the Interior Angle Bisector Theorem: In this case, we need to determine xx from other provided conditions, such as the bisector relationships and any other restrictions. However, if no further simplification is needed for xx, we can conclude as above.

Would you like further details on handling xx or a more in-depth breakdown?


Follow-Up Questions:

  1. How do we apply the triangle angle sum property effectively in geometry problems?
  2. What is the significance of the angle bisector theorem in this problem?
  3. How can we verify the angle measures once we calculate xx?
  4. What are common strategies for solving geometry problems involving angle bisectors?
  5. Why is it important to label and organize angles systematically?

Tip:

When dealing with angle bisectors in triangles, remember that the bisected angles are equal, which often helps in setting up equations that simplify the problem.

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Math Problem Analysis

Mathematical Concepts

Angle Bisector
Triangle Angle Sum Property
Algebra

Formulas

Sum of angles in a triangle: ∠R + ∠S + ∠T = 180°

Theorems

Triangle Angle Sum Theorem
Angle Bisector Theorem

Suitable Grade Level

Grades 9-10