Question

In triangle PQR if angle P- angle Q=42° and angle Q - angle R=21°, find angle P ,angle Q and angle R .​

Answer 1

`\bold{\huge{\underline{ Solution }}}`

Given :-

• We have triangle PQR
• 
  `\angle{P - {\angle}Q = 42{\degree}}`
• 
  `\angle{Q - {\angle} R = 21{\degree}}`

To Find :-

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

Let's Solve :-

According to the question,

`\sf{\angle{P - {\angle}Q = 42{\degree}}}`

`\sf{\angle{P = 42{\degree} + {\angle}Q ...eq(i) }}`

`\sf{\angle{Q - {\angle} R = 21{\degree} }}`

`\sf{\angle{Q - 21{\degree} = {\angle}R}}`

`\sf{\angle{R = {\angle}Q - 21{\degree}...eq(ii) }}`

We know that,

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

Therefore ,

By using angle sum property :-

`\sf{{\angle} P + {\angle}Q + {\angle} R = 180 {\degree}}`

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

`\sf{42 {\degree} + {\angle} Q + {\angle} Q + {\angle} Q - 21 {\degree}= 180 {\degree}}`

`\sf{ {\angle} 3Q + 21 {\degree}= 180 {\degree}}`

`\sf{ {\angle} 3Q = 180 {\degree} - 21{\degree}}`

`\sf{ {\angle} 3Q = 159 {\degree}}`

`\sf{ {\angle} Q = {(159)/(3)}}`

`\sf{ {\angle} Q = 53{\degree}}`

Thus , The value of Q is 53° .

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

`\sf{\angle{P = 42{\degree} + 53{\degree} }}`

`\sf{ {\angle} P = 95 {\degree}}`

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

`\sf{\angle{R = 53 {\degree} - 21{\degree} }}`

`\sf{ {\angle} P = 32 {\degree}}`

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

Answer 2

• P = 95°
• Q = 53°
• R = 32°

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.