To review these concepts, go to Quadrilateral Word Problems Study Guide.

**Quadrilateral Word Problems Practice Questions**

**Practice 1**

The following practice questions test your knowledge of the properties of quadrilaterals in general and parallelograms.

**Problems**

- One side of a square is 4 meters. What is the measure of each of the other sides?
- What is the value of
*x*in the following figure? - The measure of a side of a parallelogram is 24 inches. What is the measure of the side opposite from this side?
- The measure of one angle of a rhombus is 35°. What is the measure of a consecutive angle of that angle?

**Solutions**

- Every side of a square measures the same length. Each side is 4 m.
- In a parallelogram, the opposite angles are congruent. Thus, the value of
*x*is 65°. - In a parallelogram, the sides opposite each other are congruent. The measure of the opposite side is 24 inches.
- The measures of consecutive angles of a rhombus are supplementary, or equal to 180°. The measure of the consecutive angle is 180 – 35 = 145°.

**Practice 2**

This practice set will test your skills applying the properties of trapezoids.

**Problems**

- Three angles of a trapezoid measure 100°, 90°, and 75°. What is the measure of the other angle?
- The measure of a leg of an isosceles trapezoid is 14 feet. What is the measure of the other leg?
- The length of a diagonal of an isosceles trapezoid is 50 inches. What is the measure of the other diagonal?

**Solutions**

- To find the measure of the other angle, add the known angle measures together, and subtract the sum from 360°. The three known angles add to 100 + 90 + 75 = 265; 360 – 265 = 95. The missing angle is 95°.
- The legs of an isosceles trapezoid have the same length. The other leg measures 14 feet.
- The diagonals of an isosceles trapezoid are the same length. The measure of the other diagonal is 50 inches.

**Practice 3**

Use the steps to solving word problems and your knowledge of quadrilaterals to solve each of the questions in the following set.

**Problems**

- The angles of a quadrilateral are 65°, 95°, and 110°. What is the measure of the other angle?
- The measure of a diagonal of a square is represented by the expression 4
*x*– 10. If the measure of the other diagonal is 10 meters, what is the value of*x*? - The measures of two consecutive angles of a parallelogram are represented by the expressions 2
*x*and 7*x*, respectively. What is the value of*x*? - The opposite angles of a parallelogram are represented by the expressions
*x*+ 18 and 2*x*– 2, respectively. What is the measure of each angle? - The measures of the base angles of an isosceles trapezoid are each 100°. What is the measure of each of the other angles in the figure?

**Solutions**

*Read and understand the question*. This question is looking for the missing angle of a quadrilateral when three of the angle measures are given.*Read and understand the question*. This question is looking for the value of*x*when information is given about the diagonals of a square.*Read and understand the question*. This question asks for the value of*x*when expressions for two consecutive angles of a parallelogram are given.*Read and understand the question*. This question is looking for the measure of two opposite angles in a parallelogram.*Read and understand the question*. This question is looking for the measure of each of the unknown angles in an isosceles trapezoid. The measure of the base angles is given.

*Make a plan*. Add the three known angle measures, and then subtract this amount from the total of 360° in the quadrilateral.

*Carry out the plan*. To find the measure of the other angle, add the known angle measures together, and subtract the sum from 360°. The three known angles add to 65 + 95 + 110 = 270; 360 – 270 = 90. The missing angle is 90°.

*Check your answer*. To check this result, add the four angles to be sure that the sum is 360° : 65 + 95 + 110 + 90 = 360, so this answer is checking.

*Make a plan*. The lengths of the diagonals of a square are equal. Set the given expressions equal to each other, and solve for *x*.

*Carry out the plan*. Set the expression equal to 10: 4*x* – 10 = 10. Add 10 to each side of the equation to get 4*x* = 20. Divide each side of the equation by 4 to get *x* = 5.

*Check your answer*. To check this answer, substitute *x* = 5 into the expression 4*x* – 10 to be sure the value is 10.

- 4(5) – 10 = 20 – 10 = 10

This solution is checking.

*Make a plan*. Two consecutive angles of a parallelogram have a sum of 180°; they are supplementary. Add the two expressions, set the sum equal to 180, and solve for *x*.

*Carry out the plan*. The equation becomes 2*x* + 7*x* = 180. Combine like terms to get 9*x* = 180. Divide each side of the equation by 9 to get *x* = 20.

*Check your answer*. To check this solution, substitute the value of *x* into each expression. Then, add the two angle measures to be sure that the total is 180°. The angles are 2(20) = 40° and 7(20) = 140°. The sum of the angles is 40 + 140 = 180°. This result is checking.

*Make a plan*. The measures of opposite angles of a parallelogram are equal.

Set the given expressions equal to each other, and solve for *x*. Then, substitute the value of *x* into one of the expressions to find the measure of each angle.

*Carry out the plan*. Set the expressions equal to each other: *x* + 18 = 2*x* – 2. Subtract *x* from each side of the equation to get 18 = *x* – 2. Add 2 to each side of the equation to get 20 = *x*. Substitute *x* = 20 into the expression *x* + 18.

- 20 + 18 = 38

The measure of each opposite angle is 38°.

*Check your answer*. To check this answer, substitute *x* = 20 into the expression 2*x* – 2 to be sure the value is also 38.

- 2(20) – 2 = 40 – 2 = 38

This solution is checking.

*Make a plan*. Use the problem solving strategy of drawing a picture to help with this question. In an isosceles trapezoid, the two legs are congruent and the base angles are congruent. The following figure represents this trapezoid.

*Carry out the plan*. The base angles of the trapezoid are congruent, so each of their measures is 100°. To find the measure of the other angles, subtract the sum of the base angles from 360 and divide the result by 2.

- 360 – 300 = 160

- = 80

degrees in each of the other angles. Thus, the angles measure 100, 100, 80, and 80°, respectively.

*Check your answer*. To check this answer, add the measures of the four angles to be sure that the total number of degrees is 360.

- 100 + 100 + 80 + 80 = 360°

so this answer is checking.

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