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In /_\PQR, bar(PR) is extended through point R to point S,m/_RPQ=(2x+19)^(@), m/_QRS=(7x+13)^(@), and m/_PQR=(x+18)^(@). Find m/_PQR. Answer:

In △PQR,PR‾ \triangle \mathrm{PQR}, \overline{P R} is extended through point R \mathrm{R} to point S,m∠RPQ=(2x+19)∘ \mathrm{S}, \mathrm{m} \angle R P Q=(2 x+19)^{\circ} , m∠QRS=(7x+13)∘ \mathrm{m} \angle Q R S=(7 x+13)^{\circ} , and m∠PQR=(x+18)∘ \mathrm{m} \angle P Q R=(x+18)^{\circ} . Find m∠PQR \mathrm{m} \angle P Q R .\newlineAnswer:

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Q. In △PQR,PR‾ \triangle \mathrm{PQR}, \overline{P R} is extended through point R \mathrm{R} to point S,m∠RPQ=(2x+19)∘ \mathrm{S}, \mathrm{m} \angle R P Q=(2 x+19)^{\circ} , m∠QRS=(7x+13)∘ \mathrm{m} \angle Q R S=(7 x+13)^{\circ} , and m∠PQR=(x+18)∘ \mathrm{m} \angle P Q R=(x+18)^{\circ} . Find m∠PQR \mathrm{m} \angle P Q R .\newlineAnswer:
  1. Write Equation: To find the measure of angle PQR, we need to use the fact that the sum of the measures of the angles in a straight line is 180180 degrees. This means that the measure of angle RPQ plus the measure of angle PQR plus the measure of angle QRS must equal 180180 degrees.\newlineSo, we can write the equation:\newlinem∠RPQ+m∠PQR+m∠QRS=180∘m\angle RPQ + m\angle PQR + m\angle QRS = 180^\circ\newlineSubstitute the given expressions for the angle measures:\newline(2x+19)∘+(x+18)∘+(7x+13)∘=180∘(2x + 19)^\circ + (x + 18)^\circ + (7x + 13)^\circ = 180^\circ
  2. Combine Like Terms: Combine like terms to simplify the equation:\newline2x+x+7x+19+18+13=1802x + x + 7x + 19 + 18 + 13 = 180\newline10x+50=18010x + 50 = 180
  3. Isolate x Term: Subtract 5050 from both sides of the equation to isolate the term with xx: \newline10x+50−50=180−5010x + 50 - 50 = 180 - 50\newline10x=13010x = 130
  4. Solve for x: Divide both sides by 1010 to solve for x:\newline10x10=13010\frac{10x}{10} = \frac{130}{10}\newlinex=13x = 13
  5. Find Angle PQR: Now that we have the value of xx, we can find the measure of angle PQR by substituting xx back into the expression for m∠PQRm\angle PQR:m∠PQR=(x+18)∘m\angle PQR = (x + 18)^\circm∠PQR=(13+18)∘m\angle PQR = (13 + 18)^\circm∠PQR=31∘m\angle PQR = 31^\circ

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