Question

Two angles of a quadrilateral are 72° and 56° respectively and the remaining two angles are in the ratio 3:5. Find the measure of each angle of the quadrilateral.

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Answer

Given:

  • First angle: $72^\circ$
  • Second angle: $56^\circ$
  • Remaining angles: In the ratio $3:5$

Let the two remaining angles be $3x$ and $5x$.

According to the angle sum property of a quadrilateral: $$ 72^\circ + 56^\circ + 3x + 5x = 360^\circ $$

Combining the terms, we get: $$ 128^\circ + 8x = 360^\circ $$

Isolating $8x$: $$ 8x = 360^\circ - 128^\circ $$ $$ 8x = 232^\circ $$

Solving for $x$: $$ x = \frac{232^\circ}{8} $$ $$ x = 29^\circ $$

Now, we can find the measure of the remaining angles:

  • Third angle: $3x = 3 \times 29^\circ = 87^\circ$
  • Fourth angle: $5x = 5 \times 29^\circ = 145^\circ$

So, the measures of the angles of the quadrilateral are:

  • First angle: $72^\circ$
  • Second angle: $56^\circ$
  • Third angle: $87^\circ$
  • Fourth angle: $145^\circ$

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