Math Problem Statement

QS bisects angle PQT, the measurement of angle SQT= 8x-25, the measurement of angle PQT= 9x + 34,and the measurement of angle SQR = 112, find each measure

Solution

Let's solve the problem step by step.

Given:

  • QS bisects PQT\angle PQT, which means PQS=SQT\angle PQS = \angle SQT.
  • mSQT=8x25m \angle SQT = 8x - 25
  • mPQT=9x+34m \angle PQT = 9x + 34
  • mSQR=112m \angle SQR = 112^\circ

Step 1: Use the bisecting property

Since QS bisects PQT\angle PQT, we have: mPQS=mSQTm \angle PQS = m \angle SQT But also: mPQT=mPQS+mSQTm \angle PQT = m \angle PQS + m \angle SQT

Given that mPQS=mSQTm \angle PQS = m \angle SQT, we can express: mPQT=2×mSQTm \angle PQT = 2 \times m \angle SQT

Substituting the given expressions: 9x+34=2×(8x25)9x + 34 = 2 \times (8x - 25)

Step 2: Solve for xx

Expand and simplify the equation: 9x+34=16x509x + 34 = 16x - 50

Bring all terms involving xx to one side and constants to the other: 34+50=16x9x34 + 50 = 16x - 9x

84=7x84 = 7x

x=847=12x = \frac{84}{7} = 12

Step 3: Calculate the angle measurements

Now that we have x=12x = 12, we can find the actual measurements of the angles.

  1. mSQTm \angle SQT: mSQT=8x25=8(12)25=9625=71m \angle SQT = 8x - 25 = 8(12) - 25 = 96 - 25 = 71^\circ

  2. mPQTm \angle PQT: mPQT=9x+34=9(12)+34=108+34=142m \angle PQT = 9x + 34 = 9(12) + 34 = 108 + 34 = 142^\circ

  3. mPQSm \angle PQS: Since QS bisects PQT\angle PQT, mPQS=mSQT=71m \angle PQS = m \angle SQT = 71^\circ.

  4. mSQRm \angle SQR: Given directly as mSQR=112m \angle SQR = 112^\circ.

Final Measurements:

  • mSQT=71m \angle SQT = 71^\circ
  • mPQT=142m \angle PQT = 142^\circ
  • mPQS=71m \angle PQS = 71^\circ (since it’s equal to mSQTm \angle SQT)
  • mSQR=112m \angle SQR = 112^\circ

Would you like further explanation on any part of this solution, or do you have additional questions?

Here are five related questions you might find interesting:

  1. How do you determine if two angles are congruent?
  2. What is the relationship between the exterior and interior angles of a triangle?
  3. How do you find the unknown angles in an isosceles triangle?
  4. What is the significance of angle bisectors in a triangle?
  5. How can the Angle Sum Property of a triangle help solve angle-related problems?

Tip: When working with bisected angles, remember that the bisector divides the angle into two equal parts, which can simplify the calculations significantly.

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Math Problem Analysis

Mathematical Concepts

Geometry
Angle Bisectors

Formulas

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Theorems

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Suitable Grade Level

Grades 9-10