Answer :

sqdancefan

Answer:

  • P = 95°
  • Q = 53°
  • R = 32°

Step-by-step explanation:

In ∆PQR, you have the angle relations P-Q = 42 and Q-R = 21. You want the measures of all the angles.

Angle sum

The sum of the angles in the triangle is 180°. Using the given relation between P and Q, we have ...

  P = Q +42

Using the given relation between Q and R, we have ...

  R = Q -21

Then the angle sum is ...

  P +Q +R = 180

  (Q +42) +Q +(Q -21) = 180 . . . . . . substitute for P and R

  3Q +21 = 180 . . . . . . . . . . . . . collect terms

  Q +7 = 60 . . . . . . . . . . . . divide by 3

  Q = 53 . . . . . . . . . . . subtract 7

  P = Q +42 = 95

  R = Q -21 = 32

The angles P, Q, R are 95°, 53°, and 32°, respectively.

Starrysoul100

[tex]\bold{\huge{\underline{ Solution }}}[/tex]

Given :-

  • We have triangle PQR
  • [tex]\angle{P - {\angle}Q = 42{\degree}}[/tex]
  • [tex]\angle{Q - {\angle} R = 21{\degree}}[/tex]

To Find :-

  • We have to find the value of angle P , angle Q and angle R

Let's Solve :-

According to the question,

[tex]\sf{\angle{P - {\angle}Q = 42{\degree}}}[/tex]

[tex]\sf{\angle{P = 42{\degree} + {\angle}Q ...eq(i) }}[/tex]

[tex]\sf{\angle{Q - {\angle} R = 21{\degree} }}[/tex]

[tex]\sf{\angle{Q - 21{\degree} = {\angle}R}}[/tex]

[tex]\sf{\angle{R = {\angle}Q - 21{\degree}...eq(ii) }}[/tex]

We know that,

  • The sum of the angles of triangle equal to 180°

Therefore ,

By using angle sum property :-

[tex]\sf{{\angle} P + {\angle}Q + {\angle} R = 180 {\degree}}[/tex]

From eq(i) in eq(ii) :-

[tex]\sf{42 {\degree} + {\angle} Q + {\angle} Q + {\angle} Q - 21 {\degree}= 180 {\degree}}[/tex]

[tex]\sf{ {\angle} 3Q + 21 {\degree}= 180 {\degree}}[/tex]

[tex]\sf{ {\angle} 3Q = 180 {\degree} - 21{\degree}}[/tex]

[tex]\sf{ {\angle} 3Q = 159 {\degree}}[/tex]

[tex]\sf{ {\angle} Q = {\dfrac{159}{3}}}[/tex]

[tex]\sf{ {\angle} Q = 53{\degree}}[/tex]

Thus , The value of Q is 53° .

Substitute the value of Q in eq(i) :-

[tex]\sf{\angle{P = 42{\degree} + 53{\degree} }}[/tex]

[tex]\sf{ {\angle} P = 95 {\degree}}[/tex]

Now , For value of R --- substitute the value of Q in eq(ii) :-

[tex]\sf{\angle{R = 53 {\degree} - 21{\degree} }}[/tex]

[tex]\sf{ {\angle} P = 32 {\degree}}[/tex]

Hence , The values of angle P, angle Q and angle R are 95° , 53° and 32°.

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