10. Which of the following quadrilaterals always have diagonals that are congruent. Circle all that apply
Parallelograms
Rectangles
Rhombi
Squares
Isosceles trapezoids
19. Given square ABCD: if the measure of angle ACB = (1x - 32) degrees, find the value of x. Round to the nearest tenth
17. Given rhombus DEFG: if GE = 42 and DH = 16, find GF
20. WXZY is a quadrilateral with W locates at (-5, 2) and Xoxated at (3, 0). What must be the slope of segment ZY in order for WXYZ to be a parallelogram
10. The quadrilaterals that always have congruent diagonals are:
- Rectangles
- Rhombi
- Squares
19. Given square ABCD, we know that all angles in a square are right angles (90 degrees). Therefore, the measure of angle ACB is 90 degrees. We can set up the equation:
1x - 32 = 90
Solving for x, we get:
1x = 90 + 32
1x = 122
x = 122 (rounded to the nearest tenth)
17. In a rhombus, all sides are congruent. Therefore, GE = GD = GH = GF. Given that GE = 42 and DH = 16, we can find GF by subtracting 42 (GE) and 16 (DH) from the combined length of both sides:
GF = GE - DH
GF = 42 - 16
GF = 26
20. For a quadrilateral to be a parallelogram, opposite sides must be parallel. In terms of slopes, this means that the slopes of WX and ZY must be equal.
The slope of a line is calculated using the formula:
slope = (change in y / change in x)
Given that W is located at (-5, 2) and X is located at (3, 0), we can find the slope of WX:
slope(WX) = (change in y / change in x)
slope(WX) = (0 - 2) / (3 - (-5))
slope(WX) = -2 / 8
slope(WX) = -1/4
In order for WXYZ to be a parallelogram, the slope of ZY must also be -1/4.
10. To determine which quadrilaterals always have congruent diagonals, we need to understand the properties of each quadrilateral.
- Parallelograms: The diagonals of a parallelogram bisect each other, but they are not necessarily congruent. Therefore, parallelograms do not always have congruent diagonals.
- Rectangles: In a rectangle, the diagonals are congruent and bisect each other. So, rectangles always have congruent diagonals.
- Rhombi: The diagonals of a rhombus bisect each other at right angles, but they are not necessarily congruent. Therefore, rhombi do not always have congruent diagonals.
- Squares: A square is a special type of rectangle and rhombus. It has all the properties of both shapes. Thus, squares always have congruent diagonals.
- Isosceles trapezoids: In an isosceles trapezoid, the diagonals are not congruent.
From the given options, the quadrilaterals that always have congruent diagonals are rectangles and squares.
19. In a square, all four angles are equal to 90 degrees. Therefore, the measure of angle ACB is also 90 degrees.
90 = (1x - 32)
Solving this equation, we can find the value of x.
17. In a rhombus, the diagonals are perpendicular bisectors of each other. Therefore, GF is equal to half the length of GE. Thus, GF = 42 / 2 = 21.
20. To prove that WXYZ is a parallelogram, we need to show that opposite sides are parallel. A slope of a line can help us determine if lines are parallel.
To find the slope of segment ZY, we need the coordinates of points Z and Y. However, the question does not provide the coordinates of point Y. Without this information, we cannot calculate the slope of segment ZY and determine if it is parallel to WX. Hence, we need more information to answer this question.