Question

1. each of the two equal angles in an isosceles triangle is 24° less than the third. find the measure of the three angles.
2. each of two larger angles of a parallelogram measure 9° more than twice the measure of each of the two smaller angles. find the measure of each angle.
3. a shoe - rack is to have three shelves including the top and bottom. the width of the shoe rack is to be 12 inches more than the height. find the width and height of the shoe rack if only 96 inches of lumber is available.

Explanation:

2.

Step1: Let the third angle be $x$.

The two equal angles are each $x - 24$.

Step2: Use the angle - sum property of a triangle.

The sum of angles in a triangle is $180^{\circ}$, so $(x-24)+(x - 24)+x=180$.

Step3: Simplify the left - hand side of the equation.

$x-24+x - 24+x=3x-48$. So, $3x-48 = 180$.

Step4: Solve for $x$.

Add 48 to both sides: $3x=180 + 48=228$. Then divide both sides by 3: $x=\frac{228}{3}=76$.

Step5: Find the measure of the equal angles.

The equal angles are $x - 24=76-24 = 52$.

Step1: Let the measure of each smaller angle be $x$.

The measure of each larger angle is $2x + 9$.

Step2: Use the angle - sum property of a parallelogram.

The sum of the interior angles of a parallelogram is $360^{\circ}$, and opposite angles are equal. So, $2x+2(2x + 9)=360$.

Step3: Expand the left - hand side of the equation.

$2x+4x+18 = 360$.

Step4: Combine like terms.

$6x+18=360$.

Step5: Solve for $x$.

Subtract 18 from both sides: $6x=360 - 18=342$. Then divide both sides by 6: $x=\frac{342}{6}=57$.

Step6: Find the measure of the larger angles.

The larger angles are $2x + 9=2\times57+9=114 + 9=123$.

Step1: Let the height of the shoe - rack be $h$ inches.

The width of the shoe - rack is $h + 12$ inches.

Step2: Determine the amount of lumber used.

The shoe - rack has 3 horizontal pieces (top, middle, bottom) of length equal to the width and 2 vertical pieces of length equal to the height. So the total length of lumber used is $3(h + 12)+2h$.

Step3: Set up the equation.

We know that the total length of lumber is 96 inches. So, $3(h + 12)+2h=96$.

Step4: Expand the left - hand side of the equation.

$3h+36+2h=96$.

Step5: Combine like terms.

$5h+36=96$.

Step6: Solve for $h$.

Subtract 36 from both sides: $5h=96 - 36=60$. Then divide both sides by 5: $h=\frac{60}{5}=12$.

Step7: Find the width.

The width is $h + 12=12+12 = 24$.

Answer:

The three angles are $52^{\circ},52^{\circ},76^{\circ}$.

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